non directional hypothesis test example

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Hypotheses; directional and non-directional, what is the difference between an experimental and an alternative hypothesis.

Nothing much! If the study is a laboratory experiment then we can call the hypothesis “an experimental hypothesis”, where we make a prediction about how the IV causes an effect on the DV. If we have a non-experimental design, i.e. we are not able to manipulate the IV as in a natural or quasi-experiment , or if some other research method has been used, then we call it an “alternativehypothesis”, alternative to the null.

Directional hypothesis: A directional (or one tailed hypothesis) states which way you think the results are going to go, for example in an experimental study we might say…”Participants who have been deprived of sleep for 24 hours will have more cold symptoms in the following week after exposure to a virus than participants who have not been sleep deprived”; the hypothesis compares the two groups/conditions and states which one will ….have more/less, be quicker/slower, etc.

If we had a correlational study, the directional hypothesis would state whether we expect a positive or a negative correlation, we are stating how the two variables will be related to each other, e.g. there will be a positive correlation between the number of stressful life events experienced in the last year and the number of coughs and colds suffered, whereby the more life events you have suffered the more coughs and cold you will have had”. The directional hypothesis can also state a negative correlation, e.g. the higher the number of face-book friends, the lower the life satisfaction score “

Non-directional hypothesis: A non-directional (or two tailed hypothesis) simply states that there will be a difference between the two groups/conditions but does not say which will be greater/smaller, quicker/slower etc. Using our example above we would say “There will be a difference between the number of cold symptoms experienced in the following week after exposure to a virus for those participants who have been sleep deprived for 24 hours compared with those who have not been sleep deprived for 24 hours.”

When the study is correlational, we simply state that variables will be correlated but do not state whether the relationship will be positive or negative, e.g. there will be a significant correlation between variable A and variable B.

Null hypothesis The null hypothesis states that the alternative or experimental hypothesis is NOT the case, if your experimental hypothesis was directional you would say…

Participants who have been deprived of sleep for 24 hours will NOT have more cold symptoms in the following week after exposure to a virus than participants who have not been sleep deprived and any difference that does arise will be due to chance alone.

or with a directional correlational hypothesis….

There will NOT be a positive correlation between the number of stress life events experienced in the last year and the number of coughs and colds suffered, whereby the more life events you have suffered the more coughs and cold you will have had”

With a non-directional or two tailed hypothesis…

There will be NO difference between the number of cold symptoms experienced in the following week after exposure to a virus for those participants who have been sleep deprived for 24 hours compared with those who have not been sleep deprived for 24 hours.

or for a correlational …

there will be NO correlation between variable A and variable B.

When it comes to conducting an inferential stats test, if you have a directional hypothesis , you must do a one tailed test to find out whether your observed value is significant. If you have a non-directional hypothesis , you must do a two tailed test .

Exam Techniques/Advice

Remember, a decent hypothesis will contain two variables, in the case of an experimental hypothesis there will be an IV and a DV; in a correlational hypothesis there will be two co-variables
both variables need to be fully operationalised to score the marks, that is you need to be very clear and specific about what you mean by your IV and your DV; if someone wanted to repeat your study, they should be able to look at your hypothesis and know exactly what to change between the two groups/conditions and exactly what to measure (including any units/explanation of rating scales etc, e.g. “where 1 is low and 7 is high”)
double check the question, did it ask for a directional or non-directional hypothesis?
if you were asked for a null hypothesis, make sure you always include the phrase “and any difference/correlation (is your study experimental or correlational?) that does arise will be due to chance alone”

Practice Questions:

Mr Faraz wants to compare the levels of attendance between his psychology group and those of Mr Simon, who teaches a different psychology group. Which of the following is a suitable directional (one tailed) hypothesis for Mr Faraz’s investigation?

A There will be a difference in the levels of attendance between the two psychology groups.

B Students’ level of attendance will be higher in Mr Faraz’s group than Mr Simon’s group.

C Any difference in the levels of attendance between the two psychology groups is due to chance.

D The level of attendance of the students will depend upon who is teaching the groups.

2. Tracy works for the local council. The council is thinking about reducing the number of people it employs to pick up litter from the street. Tracy has been asked to carry out a study to see if having the streets cleaned at less regular intervals will affect the amount of litter the public will drop. She studies a street to compare how much litter is dropped at two different times, once when it has just been cleaned and once after it has not been cleaned for a month.

Write a fully operationalised non-directional (two-tailed) hypothesis for Tracy’s study. (2)

3. Jamila is conducting a practical investigation to look at gender differences in carrying out visuo-spatial tasks. She decides to give males and females a jigsaw puzzle and will time them to see who completes it the fastest. She uses a random sample of pupils from a local school to get her participants.

(a) Write a fully operationalised directional (one tailed) hypothesis for Jamila’s study. (2) (b) Outline one strength and one weakness of the random sampling method. You may refer to Jamila’s use of this type of sampling in your answer. (4)

4. Which of the following is a non-directional (two tailed) hypothesis?

A There is a difference in driving ability with men being better drivers than women

B Women are better at concentrating on more than one thing at a time than men

C Women spend more time doing the cooking and cleaning than men

D There is a difference in the number of men and women who participate in sports

Revision Activity

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Directional and non-directional hypothesis: A Comprehensive Guide

Karolina Konopka

Customer support manager

In the world of research and statistical analysis, hypotheses play a crucial role in formulating and testing scientific claims. Understanding the differences between directional and non-directional hypothesis is essential for designing sound experiments and drawing accurate conclusions. Whether you’re a student, researcher, or simply curious about the foundations of hypothesis testing, this guide will equip you with the knowledge and tools to navigate this fundamental aspect of scientific inquiry.

Understanding Directional Hypothesis

Understanding directional hypotheses is crucial for conducting hypothesis-driven research, as they guide the selection of appropriate statistical tests and aid in the interpretation of results. By incorporating directional hypotheses, researchers can make more precise predictions, contribute to scientific knowledge, and advance their fields of study.

Definition of directional hypothesis

Directional hypotheses, also known as one-tailed hypotheses, are statements in research that make specific predictions about the direction of a relationship or difference between variables. Unlike non-directional hypotheses, which simply state that there is a relationship or difference without specifying its direction, directional hypotheses provide a focused and precise expectation.

A directional hypothesis predicts either a positive or negative relationship between variables or predicts that one group will perform better than another. It asserts a specific direction of effect or outcome. For example, a directional hypothesis could state that “increased exposure to sunlight will lead to an improvement in mood” or “participants who receive the experimental treatment will exhibit higher levels of cognitive performance compared to the control group.”

Directional hypotheses are formulated based on existing theory, prior research, or logical reasoning, and they guide the researcher’s expectations and analysis. They allow for more targeted predictions and enable researchers to test specific hypotheses using appropriate statistical tests.

The role of directional hypothesis in research

Directional hypotheses also play a significant role in research surveys. Let’s explore their role specifically in the context of survey research:

Objective-driven surveys : Directional hypotheses help align survey research with specific objectives. By formulating directional hypotheses, researchers can focus on gathering data that directly addresses the predicted relationship or difference between variables of interest.
Question design and measurement : Directional hypotheses guide the design of survey question types and the selection of appropriate measurement scales. They ensure that the questions are tailored to capture the specific aspects related to the predicted direction, enabling researchers to obtain more targeted and relevant data from survey respondents.
Data analysis and interpretation : Directional hypotheses assist in data analysis by directing researchers towards appropriate statistical tests and methods. Researchers can analyze the survey data to specifically test the predicted relationship or difference, enhancing the accuracy and reliability of their findings. The results can then be interpreted within the context of the directional hypothesis, providing more meaningful insights.
Practical implications and decision-making : Directional hypotheses in surveys often have practical implications. When the predicted relationship or difference is confirmed, it informs decision-making processes, program development, or interventions. The survey findings based on directional hypotheses can guide organizations, policymakers, or practitioners in making informed choices to achieve desired outcomes.
Replication and further research : Directional hypotheses in survey research contribute to the replication and extension of studies. Researchers can replicate the survey with different populations or contexts to assess the generalizability of the predicted relationships. Furthermore, if the directional hypothesis is supported, it encourages further research to explore underlying mechanisms or boundary conditions.

By incorporating directional hypotheses in survey research, researchers can align their objectives, design effective surveys, conduct focused data analysis, and derive practical insights. They provide a framework for organizing survey research and contribute to the accumulation of knowledge in the field.

Examples of research questions for directional hypothesis

Here are some examples of research questions that lend themselves to directional hypotheses:

Does increased daily exercise lead to a decrease in body weight among sedentary adults?
Is there a positive relationship between study hours and academic performance among college students?
Does exposure to violent video games result in an increase in aggressive behavior among adolescents?
Does the implementation of a mindfulness-based intervention lead to a reduction in stress levels among working professionals?
Is there a difference in customer satisfaction between Product A and Product B, with Product A expected to have higher satisfaction ratings?
Does the use of social media influence self-esteem levels, with higher social media usage associated with lower self-esteem?
Is there a negative relationship between job satisfaction and employee turnover, indicating that lower job satisfaction leads to higher turnover rates?
Does the administration of a specific medication result in a decrease in symptoms among individuals with a particular medical condition?
Does increased access to early childhood education lead to improved cognitive development in preschool-aged children?
Is there a difference in purchase intention between advertisements with celebrity endorsements and advertisements without, with celebrity endorsements expected to have a higher impact?

These research questions generate specific predictions about the direction of the relationship or difference between variables and can be tested using appropriate research methods and statistical analyses.

Definition of non-directional hypothesis

Non-directional hypotheses, also known as two-tailed hypotheses, are statements in research that indicate the presence of a relationship or difference between variables without specifying the direction of the effect. Instead of making predictions about the specific direction of the relationship or difference, non-directional hypotheses simply state that there is an association or distinction between the variables of interest.

Non-directional hypotheses are often used when there is no prior theoretical basis or clear expectation about the direction of the relationship. They leave the possibility open for either a positive or negative relationship, or for both groups to differ in some way without specifying which group will perform better or worse.

Advantages and utility of non-directional hypothesis

Non-directional hypotheses in survey s offer several advantages and utilities, providing flexibility and comprehensive analysis of survey data. Here are some of the key advantages and utilities of using non-directional hypotheses in surveys:

Exploration of Relationships : Non-directional hypotheses allow researchers to explore and examine relationships between variables without assuming a specific direction. This is particularly useful in surveys where the relationship between variables may not be well-known or there may be conflicting evidence regarding the direction of the effect.
Flexibility in Question Design : With non-directional hypotheses, survey questions can be designed to measure the relationship between variables without being biased towards a particular outcome. This flexibility allows researchers to collect data and analyze the results more objectively.
Open to Unexpected Findings : Non-directional hypotheses enable researchers to be open to unexpected or surprising findings in survey data. By not committing to a specific direction of the effect, researchers can identify and explore relationships that may not have been initially anticipated, leading to new insights and discoveries.
Comprehensive Analysis : Non-directional hypotheses promote comprehensive analysis of survey data by considering the possibility of an effect in either direction. Researchers can assess the magnitude and significance of relationships without limiting their analysis to only one possible outcome.
S tatistical Validity : Non-directional hypotheses in surveys allow for the use of two-tailed statistical tests, which provide a more conservative and robust assessment of significance. Two-tailed tests consider both positive and negative deviations from the null hypothesis, ensuring accurate and reliable statistical analysis of survey data.
Exploratory Research : Non-directional hypotheses are particularly useful in exploratory research, where the goal is to gather initial insights and generate hypotheses. Surveys with non-directional hypotheses can help researchers explore various relationships and identify patterns that can guide further research or hypothesis development.

It is worth noting that the choice between directional and non-directional hypotheses in surveys depends on the research objectives, existing knowledge, and the specific variables being investigated. Researchers should carefully consider the advantages and limitations of each approach and select the one that aligns best with their research goals and survey design.

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Research Hypothesis In Psychology: Types, & Examples

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

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8.4: The Alternative Hypothesis

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Page ID 14493

Foster et al.
University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus via University of Missouri’s Affordable and Open Access Educational Resources Initiative

If the null hypothesis is rejected, then we will need some other explanation, which we call the alternative hypothesis, $H_A$ or $H_1$. The alternative hypothesis is simply the reverse of the null hypothesis, and there are three options, depending on where we expect the difference to lie. Thus, our alternative hypothesis is the mathematical way of stating our research question. If we expect our obtained sample mean to be above or below the null hypothesis value, which we call a directional hypothesis, then our alternative hypothesis takes the form:

\[\mathrm{H}_{\mathrm{A}}: \mu>7.47 \quad \text { or } \quad \mathrm{H}_{\mathrm{A}}: \mu<7.47 \nonumber \]

based on the research question itself. We should only use a directional hypothesis if we have good reason, based on prior observations or research, to suspect a particular direction. When we do not know the direction, such as when we are entering a new area of research, we use a non-directional alternative:

\[\mathrm{H}_{\mathrm{A}}: \mu \neq 7.47 \nonumber \]

We will set different criteria for rejecting the null hypothesis based on the directionality (greater than, less than, or not equal to) of the alternative. To understand why, we need to see where our criteria come from and how they relate to $z$-scores and distributions.

Writing hypotheses in words

As we alluded to in the null hypothesis section, we can write our hypotheses in word statements (in addition to the statements with symbols). These statements should be specific enough to the particular experiment or situation being referred to. That is, don't make them generic enough so that they would apply to any hypothesis test that you would conduct.

Examples for how to write null and alternate hypotheses in words for directional and non-directional situations are given throughout the chapters.

Contributors and Attributions

Foster et al. (University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus)

8.1.2 - Hypothesis Testing

A hypothesis test for a proportion is used when you are comparing one group to a known or hypothesized population proportion value. In other words, you have one sample with one categorical variable. The hypothesized value of the population proportion is symbolized by $p_0$ because this is the value in the null hypothesis ($H_0$).

If $np_0 \ge 10$ and $n(1-p_0) \ge 10$ then the distribution of sample proportions is approximately normal and can be estimated using the normal distribution. That sampling distribution will have a mean of $p_0$ and a standard deviation (i.e., standard error) of $\sqrt{\frac{p_0 (1-p_0)}{n}}$

Recall that the standard normal distribution is also known as the z distribution. Thus, this is known as a "single sample proportion z test" or "one sample proportion z test."

If $np_0 < 10$ or $n(1-p_0) < 10$ then the distribution of sample proportions follows a binomial distribution. We will not be conducting this test by hand in this course, however you will learn how this can be conducted using Minitab using the exact method.

8.1.2.1 - Normal Approximation Method Formulas

Here we will be using the five step hypothesis testing procedure to compare the proportion in one random sample to a specified population proportion using the normal approximation method.

In order to use the normal approximation method, the assumption is that both $n p_0 \geq 10$ and $n (1-p_0) \geq 10$. Recall that $p_0$ is the population proportion in the null hypothesis.

Where $p_0$ is the hypothesized population proportion that you are comparing your sample to.

When using the normal approximation method we will be using a z test statistic. The z test statistic tells us how far our sample proportion is from the hypothesized population proportion in standard error units. Note that this formula follows the basic structure of a test statistic that you learned in the last lesson:

$test\;statistic=\dfrac{sample\;statistic-null\;parameter}{standard\;error}$

$\widehat{p}$ = sample proportion $p_{0}$ = hypothesize population proportion $n$ = sample size

Given that the null hypothesis is true, the p value is the probability that a randomly selected sample of n would have a sample proportion as different, or more different, than the one in our sample, in the direction of the alternative hypothesis. We can find the p value by mapping the test statistic from step 2 onto the z distribution.

Note that p-values are also symbolized by $p$. Do not confuse this with the population proportion which shares the same symbol.

We can look up the $p$-value using Minitab by constructing the sampling distribution. Because we are using the normal approximation here, we have a $z$ test statistic that we can map onto the $z$ distribution. Recall, the z distribution is a normal distribution with a mean of 0 and standard deviation of 1. If we are conducting a one-tailed (i.e., right- or left-tailed) test, we look up the area of the sampling distribution that is beyond our test statistic. If we are conducting a two-tailed (i.e., non-directional) test there is one additional step: we need to multiple the area by two to take into account the possibility of being in the right or left tail.

We can decide between the null and alternative hypotheses by examining our p-value. If $p \leq \alpha$ reject the null hypothesis. If $p>\alpha$ fail to reject the null hypothesis. Unless stated otherwise, assume that $\alpha=.05$.

When we reject the null hypothesis our results are said to be statistically significant.

Based on our decision in step 4, we will write a sentence or two concerning our decision in relation to the original research question.

8.1.2.1.1 - Video Example: Male Babies

8.1.2.1.2 - Example: Handedness

Research Question : Are more than 80% of American's right handed?

In a sample of 100 Americans, 87 were right handed.

$np_0 = 100(0.80)=80$

$n(1-p_0) = 100 (1-0.80) = 20$

Both $np_0$ and $n(1-p_0)$ are at least 10 so we can use the normal approximation method.

This is a right-tailed test because we want to know if the proportion is greater than 0.80.

$H_{0}\colon p=0.80$ $H_{a}\colon p>0.80$

$z=\dfrac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}$

$\widehat{p}=\dfrac{87}{100}=0.87$, $p_{0}=0.80$, $n=100$

$z= \dfrac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}= \dfrac{0.87-0.80}{\sqrt{\frac{0.80 (1-0.80)}{100}}}=1.75$

Our $z$ test statistic is 1.75.

This is a right-tailed test so we need to find the area to the right of the test statistic, $z=1.75$, on the z distribution.

Using Minitab , we find the probability $P(z\geq1.75)=0.0400592$ which may be rounded to $p\; value=0.0401$.

Distribution plot of Density vs X - Normal, Mean=0, StDev=1

$p\leq .05$, therefore our decision is to reject the null hypothesis

Yes, there is statistical evidence to state that more than 80% of all Americans are right handed.

8.1.2.1.3 - Example: Ice Cream

Research Question : Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%?

In a sample of 50 customers 60% preferred chocolate over vanilla.

$np_0 = 50(0.80) = 40$

$n(1-p_0)=50(1-0.80) = 10$

Both $np_0$ and $n(1-p_0)$ are at least 10. We can use the normal approximation method.

This is a left-tailed test because we want to know if the proportion is less than 0.80.

$H_{0}\colon p=0.80$ $H_{a}\colon p<0.80$

$\widehat{p}=0.60$, $p_{0}=0.80$, $n=50$

$z= \dfrac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}= \dfrac{0.60-0.80}{\sqrt{\frac{0.80 (1-0.80)}{50}}}=-3.536$

Our $z$ test statistic is -3.536.

This is a left-tailed test so we need to find the area to the right of our test statistic, $z=-3.536$.

From the Minitab output above, the p-value is 0.0002031

$p \leq.05$, therefore our decision is to reject the null hypothesis.

Yes, there is evidence that the percentage of all Creamery customers who prefer chocolate ice cream over vanilla is less than 80%.

8.1.2.1.4 - Example: Overweight Citizens

According to the Center for Disease Control (CDC), the percent of adults 20 years of age and over in the United States who are overweight is 69.0% (see http://www.cdc.gov/nchs/fastats/obesity-overweight.htm ). One city’s council wants to know if the proportion of overweight citizens in their city is different from this known national proportion. They take a random sample of 150 adults 20 years of age or older in their city and find that 98 are classified as overweight. Let’s use the five step hypothesis testing procedure to determine if there is evidence that the proportion in this city is different from the known national proportion.

$np_0 =150 (0.690)=103.5 $

$n (1-p_0) =150 (1-0.690)=46.5$

Both $n p_0$ and $n (1-p_0)$ are at least 10, this assumption has been met.

Research question: Is this city’s proportion of overweight individuals different from 0.690?

This is a non-directional test because our question states that we are looking for a differences as opposed to a specific direction. This will be a two-tailed test.

$H_{0}\colon p=0.690$ $H_{a}\colon p\neq 0.690$

$\widehat{p}=\dfrac{98}{150}=.653$

$ z =\dfrac{0.653- 0.690 }{\sqrt{\frac{0.690 (1- 0.690)}{150}}} = -0.980 $

Our test statistic is $z=-0.980$

This is a non-directional (i.e., two-tailed) test, so we need to find the area under the z distribution that is more extreme than $z=-0.980$.

In Minitab, we find the proportion of a normal curve beyond $\pm0.980$:

$p-value=0.163543+0.163543=0.327086$

$p>\alpha$, therefore we fail to reject the null hypothesis

There is not sufficient evidence to state that the proportion of citizens of this city who are overweight is different from the national proportion of 0.690.

8.1.2.2 - Minitab: Hypothesis Tests for One Proportion

A hypothesis test for one proportion can be conducted in Minitab. This can be done using raw data or summarized data.

If you have a data file with every individual's observation, then you have raw data .
If you do not have each individual observation, but rather have the sample size and number of successes in the sample, then you have summarized data.

The next two pages will show you how to use Minitab to conduct this analysis using either raw data or summarized data .

Note that the default method for constructing the sampling distribution in Minitab is to use the exact method. If $np_0 \geq 10$ and $n(1-p_0) \geq 10$ then you will need to change this to the normal approximation method. This must be done manually. Minitab will use the method that you select, it will not check assumptions for you!

8.1.2.2.1 - Minitab: 1 Proportion z Test, Raw Data

If you have data in a Minitab worksheet, then you have what we call "raw data." This is in contrast to "summarized data" which you'll see on the next page.

In order to use the normal approximation method both $np_0 \geq 10$ and $n(1-p_0) \geq 10$. Before we can conduct our hypothesis test we must check this assumption to determine if the normal approximation method or exact method should be used. This must be checked manually. Minitab will not check assumptions for you.

In the example below, we want to know if there is evidence that the proportion of students who are male is different from 0.50.

$n=226$ and $p_0=0.50$

$np_0 = 226(0.50)=113$ and $n(1-p_0) = 226(1-0.50)=113$

Both $np_0 \geq 10$ and $n(1-p_0) \geq 10$ so we can use the normal approximation method.

Minitab ® – Conducting a One Sample Proportion z Test: Raw Data

Research question: Is the proportion of students who are male different from 0.50?

class_survey.mpx
In Minitab, select Stat > Basic Statistics > 1 Proportion
Select One or more samples, each in a column from the dropdown
Double-click the variable Biological Sex to insert it into the box
Check the box next to Perform hypothesis test and enter 0.50 in the Hypothesized proportion box
Select Options
Use the default Alternative hypothesis setting of Proportion ≠ hypothesized proportion value
Use the default Confidence level of 95
Select Normal approximation method
Click OK and OK

The result should be the following output:

Event: Biological Sex = Male p: proportion where Biological Sex = Male Normal approximation is used for this analysis.

Summary of Results

We could summarize these results using the five-step hypothesis testing procedure:

$np_0 = 226(0.50)=113$ and $n(1-p_0) = 226(1-0.50)=113$ therefore the normal approximation method will be used.

$H_0\colon p = 0.50$

$H_a\colon p \ne 0.50$

From the Minitab output, $z$ = -1.86

From the Minitab output, $p$ = 0.0625

$p > \alpha$, fail to reject the null hypothesis

There is NOT enough evidence that the proportion of all students in the population who are male is different from 0.50.

8.1.2.2.2 - Minitab: 1 Sample Proportion z test, Summary Data

Example: overweight.

The following example uses a scenario in which we want to know if the proportion of college women who think they are overweight is less than 40%. We collect data from a random sample of 129 college women and 37 said that they think they are overweight.

First, we should check assumptions to determine if the normal approximation method or exact method should be used:

$np_0=129(0.40)=51.6$ and $n(1-p_0)=129(1-0.40)=77.4$ both values are at least 10 so we can use the normal approximation method.

Minitab ® – Performing a One Proportion z Test with Summarized Data

To perform a one sample proportion z test with summarized data in Minitab:

Select Summarized data from the dropdown
For number of events, add 37 and for number of trials add 129.
Check the box next to Perform hypothesis test and enter 0.40 in the Hypothesized proportion box
Use the default Alternative hypothesis setting of Proportion < hypothesized proportion value

Event: Event proportion Normal approximation is used for this analysis.

$H_0\colon p = 0.40$

$H_a\colon p < 0.40$

From output, $z$ = -2.62

From output, $p$ = 0.004

$p \leq \alpha$, reject the null hypothesis

There is evidence that the proportion of women in the population who think they are overweight is less than 40%.

8.1.2.2.2.1 - Minitab Example: Normal Approx. Method

Example: gym membership.

Research question: Are less than 50% of all individuals with a membership at one gym female?

A simple random sample of 60 individuals with a membership at one gym was collected. Each individual's biological sex was recorded. There were 24 females.

First we have to check the assumptions:

np = 60 (0.50) = 30

n(1-p) = 60(1-0.50) = 30

The assumptions are met to use the normal approximation method.

For number of events, add 24 and for number of trials add 60.

$np_0=60(0.50)=30$ and $n(1-p_0)=60(1-0.50)=30$ both values are at least 10 so we can use the normal approximation method.

$H_0\colon p = 0.50$

$H_a\colon p < 0.50$

From output, $z$ = -1.55

From output, $p$ = 0.061

$p \geq \alpha$, fail to reject the null hypothesis

There is not enough evidence to support the alternative that the proportion of women memberships at this gym is less than 50%.

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Non-Directional Hypothesis

A non-directional hypothesis is a two-tailed hypothesis that does not predict the direction of the difference or relationship (e.g. girls and boys are different in terms of helpfulness).

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Statistics Made Easy

What is a Directional Hypothesis? (Definition & Examples)

A statistical hypothesis is an assumption about a population parameter . For example, we may assume that the mean height of a male in the U.S. is 70 inches.

The assumption about the height is the statistical hypothesis and the true mean height of a male in the U.S. is the population parameter .

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

Whenever we perform a hypothesis test, we always write down a null and alternative hypothesis:

Null Hypothesis (H 0 ): The sample data occurs purely from chance.
Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

A hypothesis test can either contain a directional hypothesis or a non-directional hypothesis:

Directional hypothesis: The alternative hypothesis contains the less than (“<“) or greater than (“>”) sign. This indicates that we’re testing whether or not there is a positive or negative effect.
Non-directional hypothesis: The alternative hypothesis contains the not equal (“≠”) sign. This indicates that we’re testing whether or not there is some effect, without specifying the direction of the effect.

Note that directional hypothesis tests are also called “one-tailed” tests and non-directional hypothesis tests are also called “two-tailed” tests.

Check out the following examples to gain a better understanding of directional vs. non-directional hypothesis tests.

Example 1: Baseball Programs

A baseball coach believes a certain 4-week program will increase the mean hitting percentage of his players, which is currently 0.285.

To test this, he measures the hitting percentage of each of his players before and after participating in the program.

He then performs a hypothesis test using the following hypotheses:

H 0 : μ = .285 (the program will have no effect on the mean hitting percentage)
H A : μ > .285 (the program will cause mean hitting percentage to increase)

This is an example of a directional hypothesis because the alternative hypothesis contains the greater than “>” sign. The coach believes that the program will influence the mean hitting percentage of his players in a positive direction.

Example 2: Plant Growth

A biologist believes that a certain pesticide will cause plants to grow less during a one-month period than they normally do, which is currently 10 inches.

To test this, she applies the pesticide to each of the plants in her laboratory for one month.

She then performs a hypothesis test using the following hypotheses:

H 0 : μ = 10 inches (the pesticide will have no effect on the mean plant growth)
H A : μ < 10 inches (the pesticide will cause mean plant growth to decrease)

This is also an example of a directional hypothesis because the alternative hypothesis contains the less than “<” sign. The biologist believes that the pesticide will influence the mean plant growth in a negative direction.

Example 3: Studying Technique

A professor believes that a certain studying technique will influence the mean score that her students receive on a certain exam, but she’s unsure if it will increase or decrease the mean score, which is currently 82.

To test this, she lets each student use the studying technique for one month leading up to the exam and then administers the same exam to each of the students.

H 0 : μ = 82 (the studying technique will have no effect on the mean exam score)
H A : μ ≠ 82 (the studying technique will cause the mean exam score to be different than 82)

This is an example of a non-directional hypothesis because the alternative hypothesis contains the not equal “≠” sign. The professor believes that the studying technique will influence the mean exam score, but doesn’t specify whether it will cause the mean score to increase or decrease.

Additional Resources

Introduction to Hypothesis Testing Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test

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Hey there. My name is Zach Bobbitt. I have a Master of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

Directional vs. Non-Directional Hypothesis in Research

In the world of research and statistical analysis, formulating hypotheses is a crucial step in the scientific process. Hypotheses guide researchers in making predictions and testing relationships between variables. When it comes to hypotheses, there are two main types: directional and non-directional.

In this blog post, we will explore the differences between Directional vs. Non-Directional Hypothesis in Research and their implications in research.

Table of Contents

Directional Hypothesis

A directional hypothesis, also known as a one-tailed hypothesis, is formulated with a specific predicted direction of the relationship between variables. It indicates an expectation of the relationship being either positive or negative.

The directional hypothesis is often used when there is prior knowledge or theoretical reasoning supporting the predicted direction of the relationship. It allows researchers to make more specific predictions and draw conclusions based on the expected direction of the effect.

Example of Directional Hypothesis

For example, a directional hypothesis might state that “increased physical activity will lead to a decrease in body weight.” Here, the researcher expects a negative relationship between physical activity and body weight.

Advantages of Directional Hypothesis

Specific predictions: Directional hypotheses provide a clear prediction of the expected relationship between variables, allowing for a focused investigation.
Increased statistical power: By focusing on one direction of the relationship, researchers can allocate more statistical power to that specific direction, increasing the chances of detecting a significant effect if it exists.

Non-Directional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, does not make a specific prediction about the direction of the relationship between variables. Instead, it states that there is a relationship, but without indicating whether it will be positive or negative.

Non-directional hypotheses are often used when there is insufficient prior knowledge or theoretical basis to predict the direction of the relationship. It allows for a more exploratory approach, where the researcher is open to discovering the nature of the relationship through data analysis .

Read More: Internal Validity vs External Validity | Examples

Example of Non-Directional Hypothesis

For example, a non-directional hypothesis might state that “there is a relationship between caffeine consumption and reaction time.” Here, the researcher expects a relationship between the variables but does not specify the direction.

Read More: Population vs Sample | Examples

Advantages of Non-Directional Hypothesis:

Flexibility: Non-directional hypotheses provide flexibility in exploring relationships between variables without preconceived notions about the direction of the effect.
Open to unexpected findings : By not specifying the direction, researchers remain open to unexpected results or alternative explanations that may emerge during the analysis.

Difference Between Directional and Non-Directional Hypotheses

Choosing Between Directional and Non-Directional Hypotheses: The choice between a directional and non-directional hypothesis depends on the research question, existing knowledge, and theoretical background. Here are a few considerations for selecting the appropriate type of hypothesis:

Directional vs. Non-Directional Hypothesis

Prior research: If previous studies have established a clear direction of the relationship, a directional hypothesis may be more appropriate.
Theoretical reasoning: If there is a strong theoretical foundation supporting a specific direction, a directional hypothesis can provide a focused investigation.
Exploratory nature: If the research question is exploratory or lacks prior knowledge, a non-directional hypothesis allows for a more open-ended investigation.

Read More: Reliability vs Validity | Examples

Directional vs. Non-Directional Hypothesis

Formulating hypotheses is an essential step in the research process, guiding researchers in testing relationships between variables.

Directional hypotheses offer specific predictions about the expected direction of the relationship, whereas non-directional hypotheses allow for more exploratory investigations without preconceived notions of the direction.

The choice between these types of hypotheses depends on the research question, prior knowledge, and theoretical background.

By understanding the distinctions between directional and non-directional hypotheses, researchers can effectively formulate hypotheses that align with their research goals and contribute to the advancement of scientific knowledge.

Remember, hypotheses serve as a roadmap for research, and regardless of their type, they play a crucial role in scientific inquiry and the pursuit of knowledge.

10 Chapter 10: Hypothesis Testing with Z

Setting up the hypotheses.

When setting up the hypotheses with z, the parameter is associated with a sample mean (in the previous chapter examples the parameters for the null used 0). Using z is an occasion in which the null hypothesis is a value other than 0. For example, if we are working with mothers in the U.S. whose children are at risk of low birth weight, we can use 7.47 pounds, the average birth weight in the US, as our null value and test for differences against that. For now, we will focus on testing a value of a single mean against what we expect from the population.

Using birthweight as an example, our null hypothesis takes the form: H 0 : μ = 7.47 Notice that we are testing the value for μ, the population parameter, NOT the sample statistic ̅X (or M). We are referring to the data right now in raw form (we have not standardized it using z yet). Again, using inferential statistics, we are interested in understanding the population, drawing from our sample observations. For the research question, we have a mean value from the sample to use, we have specific data is – it is observed and used as a comparison for a set point.

As mentioned earlier, the alternative hypothesis is simply the reverse of the null hypothesis, and there are three options, depending on where we expect the difference to lie. We will set the criteria for rejecting the null hypothesis based on the directionality (greater than, less than, or not equal to) of the alternative.

If we expect our obtained sample mean to be above or below the null hypothesis value (knowing which direction), we set a directional hypothesis. O ur alternative hypothesis takes the form based on the research question itself. In our example with birthweight, this could be presented as H A : μ > 7.47 or H A : μ < 7.47.

Note that we should only use a directional hypothesis if we have a good reason, based on prior observations or research, to suspect a particular direction. When we do not know the direction, such as when we are entering a new area of research, we use a non-directional alternative hypothesis. In our birthweight example, this could be set as H A : μ ≠ 7.47

In working with data for this course we will need to set a critical value of the test statistic for alpha (α) for use of test statistic tables in the back of the book. This is determining the critical rejection region that has a set critical value based on α.

Determining Critical Value from α

We set alpha (α) before collecting data in order to determine whether or not we should reject the null hypothesis. We set this value beforehand to avoid biasing ourselves by viewing our results and then determining what criteria we should use.

When a research hypothesis predicts an effect but does not predict a direction for the effect, it is called a non-directional hypothesis . To test the significance of a non-directional hypothesis, we have to consider the possibility that the sample could be extreme at either tail of the comparison distribution. We call this a two-tailed test .

Figure 1. showing a 2-tail test for non-directional hypothesis for z for area C is the critical rejection region.

When a research hypothesis predicts a direction for the effect, it is called a directional hypothesis . To test the significance of a directional hypothesis, we have to consider the possibility that the sample could be extreme at one-tail of the comparison distribution. We call this a one-tailed test .

Figure 2. showing a 1-tail test for a directional hypothesis (predicting an increase) for z for area C is the critical rejection region.

Determining Cutoff Scores with Two-Tailed Tests

Typically we specify an α level before analyzing the data. If the data analysis results in a probability value below the α level, then the null hypothesis is rejected; if it is not, then the null hypothesis is not rejected. In other words, if our data produce values that meet or exceed this threshold, then we have sufficient evidence to reject the null hypothesis ; if not, we fail to reject the null (we never “accept” the null). According to this perspective, if a result is significant, then it does not matter how significant it is. Moreover, if it is not significant, then it does not matter how close to being significant it is. Therefore, if the 0.05 level is being used, then probability values of 0.049 and 0.001 are treated identically. Similarly, probability values of 0.06 and 0.34 are treated identically. Note we will discuss ways to address effect size (which is related to this challenge of NHST).

When setting the probability value, there is a special complication in a two-tailed test. We have to divide the significance percentage between the two tails. For example, with a 5% significance level, we reject the null hypothesis only if the sample is so extreme that it is in either the top 2.5% or the bottom 2.5% of the comparison distribution. This keeps the overall level of significance at a total of 5%. A one-tailed test does have such an extreme value but with a one-tailed test only one side of the distribution is considered.

Figure 3. Critical value differences in one and two-tail tests. Photo Credit

Let’s re view th e set critical values for Z.

We discussed z-scores and probability in chapter 8. If we revisit the z-score for 5% and 1%, we can identify the critical regions for the critical rejection areas from the unit standard normal table.

A two-tailed test at the 5% level has a critical boundary Z score of +1.96 and -1.96
A one-tailed test at the 5% level has a critical boundary Z score of +1.64 or -1.64
A two-tailed test at the 1% level has a critical boundary Z score of +2.58 and -2.58
A one-tailed test at the 1% level has a critical boundary Z score of +2.33 or -2.33.

Review: Critical values, p-values, and significance level

There are two criteria we use to assess whether our data meet the thresholds established by our chosen significance level, and they both have to do with our discussions of probability and distributions. Recall that probability refers to the likelihood of an event, given some situation or set of conditions. In hypothesis testing, that situation is the assumption that the null hypothesis value is the correct value, or that there is no effec t. The value laid out in H 0 is our condition under which we interpret our results. To reject this assumption, and thereby reject the null hypothesis, we need results that would be very unlikely if the null was true.

Now recall that values of z which fall in the tails of the standard normal distribution represent unlikely values. That is, the proportion of the area under the curve as or more extreme than z is very small as we get into the tails of the distribution. Our significance level corresponds to the area under the tail that is exactly equal to α: if we use our normal criterion of α = .05, then 5% of the area under the curve becomes what we call the rejection region (also called the critical region) of the distribution. This is illustrated in Figure 4.

Figure 4: The rejection region for a one-tailed test

The shaded rejection region takes us 5% of the area under the curve. Any result which falls in that region is sufficient evidence to reject the null hypothesis.

The rejection region is bounded by a specific z-value, as is any area under the curve. In hypothesis testing, the value corresponding to a specific rejection region is called the critical value, z crit (“z-crit”) or z* (hence the other name “critical region”). Finding the critical value works exactly the same as finding the z-score corresponding to any area under the curve like we did in Unit 1. If we go to the normal table, we will find that the z-score corresponding to 5% of the area under the curve is equal to 1.645 (z = 1.64 corresponds to 0.0405 and z = 1.65 corresponds to 0.0495, so .05 is exactly in between them) if we go to the right and -1.645 if we go to the left. The direction must be determined by your alternative hypothesis, and drawing then shading the distribution is helpful for keeping directionality straight.

Suppose, however, that we want to do a non-directional test. We need to put the critical region in both tails, but we don’t want to increase the overall size of the rejection region (for reasons we will see later). To do this, we simply split it in half so that an equal proportion of the area under the curve falls in each tail’s rejection region. For α = .05, this means 2.5% of the area is in each tail, which, based on the z-table, corresponds to critical values of z* = ±1.96. This is shown in Figure 5.

Figure 5: Two-tailed rejection region

Thus, any z-score falling outside ±1.96 (greater than 1.96 in absolute value) falls in the rejection region. When we use z-scores in this way, the obtained value of z (sometimes called z-obtained) is something known as a test statistic, which is simply an inferential statistic used to test a null hypothesis.

Calculate the test statistic: Z

Now that we understand setting up the hypothesis and determining the outcome, let’s examine hypothesis testing with z! The next step is to carry out the study and get the actual results for our sample. Central to hypothesis test is comparison of the population and sample means. To make our calculation and determine where the sample is in the hypothesized distribution we calculate the Z for the sample data.

Make a decision

To decide whether to reject the null hypothesis, we compare our sample’s Z score to the Z score that marks our critical boundary. If our sample Z score falls inside the rejection region of the comparison distribution (is greater than the z-score critical boundary) we reject the null hypothesis.

The formula for our z- statistic has not changed:

To formally test our hypothesis, we compare our obtained z-statistic to our critical z-value. If z obt > z crit , that means it falls in the rejection region (to see why, draw a line for z = 2.5 on Figure 1 or Figure 2) and so we reject H 0 . If z obt < z crit , we fail to reject. Remember that as z gets larger, the corresponding area under the curve beyond z gets smaller. Thus, the proportion, or p-value, will be smaller than the area for α, and if the area is smaller, the probability gets smaller. Specifically, the probability of obtaining that result, or a more extreme result, under the condition that the null hypothesis is true gets smaller.

Conversely, if we fail to reject, we know that the proportion will be larger than α because the z-statistic will not be as far into the tail. This is illustrated for a one- tailed test in Figure 6.

Figure 6. Relation between α, z obt , and p

When the null hypothesis is rejected, the effect is said to be statistically significant . Do not confuse statistical significance with practical significance. A small effect can be highly significant if the sample size is large enough.

Why does the word “significant” in the phrase “statistically significant” mean something so different from other uses of the word? Interestingly, this is because the meaning of “significant” in everyday language has changed. It turns out that when the procedures for hypothesis testing were developed, something was “significant” if it signified something. Thus, finding that an effect is statistically significant signifies that the effect is real and not due to chance. Over the years, the meaning of “significant” changed, leading to the potential misinterpretation.

Review: Steps of the Hypothesis Testing Process

The process of testing hypotheses follows a simple four-step procedure. This process will be what we use for the remained of the textbook and course, and though the hypothesis and statistics we use will change, this process will not.

Step 1: State the Hypotheses

Your hypotheses are the first thing you need to lay out. Otherwise, there is nothing to test! You have to state the null hypothesis (which is what we test) and the alternative hypothesis (which is what we expect). These should be stated mathematically as they were presented above AND in words, explaining in normal English what each one means in terms of the research question.

Step 2: Find the Critical Values

Next, we formally lay out the criteria we will use to test our hypotheses. There are two pieces of information that inform our critical values: α, which determines how much of the area under the curve composes our rejection region, and the directionality of the test, which determines where the region will be.

Step 3: Compute the Test Statistic

Once we have our hypotheses and the standards we use to test them, we can collect data and calculate our test statistic, in this case z . This step is where the vast majority of differences in future chapters will arise: different tests used for different data are calculated in different ways, but the way we use and interpret them remains the same.

Step 4: Make the Decision

Finally, once we have our obtained test statistic, we can compare it to our critical value and decide whether we should reject or fail to reject the null hypothesis. When we do this, we must interpret the decision in relation to our research question, stating what we concluded, what we based our conclusion on, and the specific statistics we obtained.

Example: Movie Popcorn

Let’s see how hypothesis testing works in action by working through an example. Say that a movie theater owner likes to keep a very close eye on how much popcorn goes into each bag sold, so he knows that the average bag has 8 cups of popcorn and that this varies a little bit, about half a cup. That is, the known population mean is μ = 8.00 and the known population standard deviation is σ =0.50. The owner wants to make sure that the newest employee is filling bags correctly, so over the course of a week he randomly assesses 25 bags filled by the employee to test for a difference (n = 25). He doesn’t want bags overfilled or under filled, so he looks for differences in both directions. This scenario has all of the information we need to begin our hypothesis testing procedure.

Our manager is looking for a difference in the mean cups of popcorn bags compared to the population mean of 8. We will need both a null and an alternative hypothesis written both mathematically and in words. We’ll always start with the null hypothesis:

H 0 : There is no difference in the cups of popcorn bags from this employee H 0 : μ = 8.00

Notice that we phrase the hypothesis in terms of the population parameter μ, which in this case would be the true average cups of bags filled by the new employee.

Our assumption of no difference, the null hypothesis, is that this mean is exactly

the same as the known population mean value we want it to match, 8.00. Now let’s do the alternative:

H A : There is a difference in the cups of popcorn bags from this employee H A : μ ≠ 8.00

In this case, we don’t know if the bags will be too full or not full enough, so we do a two-tailed alternative hypothesis that there is a difference.

Our critical values are based on two things: the directionality of the test and the level of significance. We decided in step 1 that a two-tailed test is the appropriate directionality. We were given no information about the level of significance, so we assume that α = 0.05 is what we will use. As stated earlier in the chapter, the critical values for a two-tailed z-test at α = 0.05 are z* = ±1.96. This will be the criteria we use to test our hypothesis. We can now draw out our distribution so we can visualize the rejection region and make sure it makes sense

Figure 7: Rejection region for z* = ±1.96

Step 3: Calculate the Test Statistic

Now we come to our formal calculations. Let’s say that the manager collects data and finds that the average cups of this employee’s popcorn bags is ̅X = 7.75 cups. We can now plug this value, along with the values presented in the original problem, into our equation for z:

So our test statistic is z = -2.50, which we can draw onto our rejection region distribution:

Figure 8: Test statistic location

Looking at Figure 5, we can see that our obtained z-statistic falls in the rejection region. We can also directly compare it to our critical value: in terms of absolute value, -2.50 > -1.96, so we reject the null hypothesis. We can now write our conclusion:

When we write our conclusion, we write out the words to communicate what it actually means, but we also include the average sample size we calculated (the exact location doesn’t matter, just somewhere that flows naturally and makes sense) and the z-statistic and p-value. We don’t know the exact p-value, but we do know that because we rejected the null, it must be less than α.

Effect Size

When we reject the null hypothesis, we are stating that the difference we found was statistically significant, but we have mentioned several times that this tells us nothing about practical significance. To get an idea of the actual size of what we found, we can compute a new statistic called an effect size. Effect sizes give us an idea of how large, important, or meaningful a statistically significant effect is.

For mean differences like we calculated here, our effect size is Cohen’s d :

Effect sizes are incredibly useful and provide important information and clarification that overcomes some of the weakness of hypothesis testing. Whenever you find a significant result, you should always calculate an effect size

Table 1. Interpretation of Cohen’s d

Example: Office Temperature

Let’s do another example to solidify our understanding. Let’s say that the office building you work in is supposed to be kept at 74 degree Fahrenheit but is allowed

to vary by 1 degree in either direction. You suspect that, as a cost saving measure, the temperature was secretly set higher. You set up a formal way to test your hypothesis.

You start by laying out the null hypothesis:

H 0 : There is no difference in the average building temperature H 0 : μ = 74

Next you state the alternative hypothesis. You have reason to suspect a specific direction of change, so you make a one-tailed test:

H A : The average building temperature is higher than claimed H A : μ > 74

Now that you have everything set up, you spend one week collecting temperature data:

You calculate the average of these scores to be 𝑋̅ = 76.6 degrees. You use this to calculate the test statistic, using μ = 74 (the supposed average temperature), σ = 1.00 (how much the temperature should vary), and n = 5 (how many data points you collected):

z = 76.60 − 74.00 = 2.60 = 5.78

1.00/√5 0.45

This value falls so far into the tail that it cannot even be plotted on the distribution!

Figure 7: Obtained z-statistic

You compare your obtained z-statistic, z = 5.77, to the critical value, z* = 1.645, and find that z > z*. Therefore you reject the null hypothesis, concluding: Based on 5 observations, the average temperature (𝑋̅ = 76.6 degrees) is statistically significantly higher than it is supposed to be, z = 5.77, p < .05.

d = (76.60-74.00)/ 1= 2.60

The effect size you calculate is definitely large, meaning someone has some explaining to do!

Example: Different Significance Level

First, let’s take a look at an example phrased in generic terms, rather than in the context of a specific research question, to see the individual pieces one more time. This time, however, we will use a stricter significance level, α = 0.01, to test the hypothesis.

We will use 60 as an arbitrary null hypothesis value: H 0 : The average score does not differ from the population H 0 : μ = 50

We will assume a two-tailed test: H A : The average score does differ H A : μ ≠ 50

We have seen the critical values for z-tests at α = 0.05 levels of significance several times. To find the values for α = 0.01, we will go to the standard normal table and find the z-score cutting of 0.005 (0.01 divided by 2 for a two-tailed test) of the area in the tail, which is z crit * = ±2.575. Notice that this cutoff is much higher than it was for α = 0.05. This is because we need much less of the area in the tail, so we need to go very far out to find the cutoff. As a result, this will require a much larger effect or much larger sample size in order to reject the null hypothesis.

We can now calculate our test statistic. The average of 10 scores is M = 60.40 with a µ = 60. We will use σ = 10 as our known population standard deviation. From this information, we calculate our z-statistic as:

Our obtained z-statistic, z = 0.13, is very small. It is much less than our critical value of 2.575. Thus, this time, we fail to reject the null hypothesis. Our conclusion would look something like:

Notice two things about the end of the conclusion. First, we wrote that p is greater than instead of p is less than, like we did in the previous two examples. This is because we failed to reject the null hypothesis. We don’t know exactly what the p- value is, but we know it must be larger than the α level we used to test our hypothesis. Second, we used 0.01 instead of the usual 0.05, because this time we tested at a different level. The number you compare to the p-value should always be the significance level you test at. Because we did not detect a statistically significant effect, we do not need to calculate an effect size. Note: some statisticians will suggest to always calculate effects size as a possibility of Type II error. Although insignificant, calculating d = (60.4-60)/10 = .04 which suggests no effect (and not a possibility of Type II error).

Review Considerations in Hypothesis Testing

Errors in hypothesis testing.

Keep in mind that rejecting the null hypothesis is not an all-or-nothing decision. The Type I error rate is affected by the α level: the lower the α level the lower the Type I error rate. It might seem that α is the probability of a Type I error. However, this is not correct. Instead, α is the probability of a Type I error given that the null hypothesis is true. If the null hypothesis is false, then it is impossible to make a Type I error. The second type of error that can be made in significance testing is failing to reject a false null hypothesis. This kind of error is called a Type II error.

Statistical Power

The statistical power of a research design is the probability of rejecting the null hypothesis given the sample size and expected relationship strength. Statistical power is the complement of the probability of committing a Type II error. Clearly, researchers should be interested in the power of their research designs if they want to avoid making Type II errors. In particular, they should make sure their research design has adequate power before collecting data. A common guideline is that a power of .80 is adequate. This means that there is an 80% chance of rejecting the null hypothesis for the expected relationship strength.

Given that statistical power depends primarily on relationship strength and sample size, there are essentially two steps you can take to increase statistical power: increase the strength of the relationship or increase the sample size. Increasing the strength of the relationship can sometimes be accomplished by using a stronger manipulation or by more carefully controlling extraneous variables to reduce the amount of noise in the data (e.g., by using a within-subjects design rather than a between-subjects design). The usual strategy, however, is to increase the sample size. For any expected relationship strength, there will always be some sample large enough to achieve adequate power.

Inferential statistics uses data from a sample of individuals to reach conclusions about the whole population. The degree to which our inferences are valid depends upon how we selected the sample (sampling technique) and the characteristics (parameters) of population data. Statistical analyses assume that sample(s) and population(s) meet certain conditions called statistical assumptions.

It is easy to check assumptions when using statistical software and it is important as a researcher to check for violations; if violations of statistical assumptions are not appropriately addressed then results may be interpreted incorrectly.

Learning Objectives

Having read the chapter, students should be able to:

Conduct a hypothesis test using a z-score statistics, locating critical region, and make a statistical decision including.
Explain the purpose of measuring effect size and power, and be able to compute Cohen’s d.

Exercises – Ch. 10

List the main steps for hypothesis testing with the z-statistic. When and why do you calculate an effect size?
z = 1.99, two-tailed test at α = 0.05
z = 1.99, two-tailed test at α = 0.01
z = 1.99, one-tailed test at α = 0.05
You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with μ = 78 and σ = 12. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: 82, 74, 62, 68, 79, 94, 90, 81, 80.
A study examines self-esteem and depression in teenagers. A sample of 25 teens with a low self-esteem are given the Beck Depression Inventory. The average score for the group is 20.9. For the general population, the average score is 18.3 with σ = 12. Use a two-tail test with α = 0.05 to examine whether teenagers with low self-esteem show significant differences in depression.
You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $12 (μ = 42, σ = 12). You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is $44.50 from tips. Test for a difference between this value and the population mean at the α = 0.05 level of significance.

Answers to Odd- Numbered Exercises – Ch. 10

1. List hypotheses. Determine critical region. Calculate z. Compare z to critical region. Draw Conclusion. We calculate an effect size when we find a statistically significant result to see if our result is practically meaningful or important

5. Step 1: H 0 : μ = 42 “My average tips does not differ from other servers”, H A : μ ≠ 42 “My average tips do differ from others”

Introduction to Statistics for Psychology Copyright © 2021 by Alisa Beyer is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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11 Independent-samples t-Test

Learning outcomes.

In this chapter, you will learn how to:

Identify when to use an independent-samples t -test
Conduct an independent-samples t -test hypothesis test
Evaluate effect size for an independent-samples t-test
Identify the assumptions for conducting an independent-samples t -test

We have seen how to compare a single mean against a given value. While single sample techniques are used in real research, most research requires the comparison of at least two sets of data. Now, we will learn how to compare two separate means from two different groups to see if there is a difference between them. The process of testing hypotheses about two means is exactly the same as it is for testing hypotheses about a single mean, and the logical structure of the formula is the same as well. However, we will be adding a few extra steps this time to account for the fact that our data are coming from different sources.

The research designs used to obtain two sets of data can be classified into two general categories. If there’s no logical or meaningful way to link individuals across groups, or if there is no overlap between the groups, then we say the groups are independent and use the independent samples t -test (AKA between-subjects), the subject of this chapter. If they came from two time points with the same people, you know you are working with repeated measures data (the measurement literally was repeated) and will use a repeated measures (sometimes called a dependent samples or paired samples) t -test, which we will discuss in the next chapter. It is very important to keep these two tests separate and understand the distinctions between them because they assess very different questions and require different approaches to the data. When in doubt, think about how the data were collected and where they came from.

Research Questions about Independent Means

Remember that our research questions are always about the populations of interest, though we conduct the studies on samples. Thus, an independent samples t -test is truly designed to compare populations. If we want to know if two populations differ and we do not know the mean of either population, we take a sample from each and then conduct an independent sample t -test. Many research ideas in the behavioral sciences and other areas of research are concerned with whether or not two means are the same or different. Logically, we therefore say that these research questions are concerned with group mean differences. That is, on average, do we Group A to be higher or lower on some variable than Group B. In any time of research design looking at group mean differences, there are some key criteria we must consider: the groups must be mutually exclusive (i.e., you can only be part of one group at any given time) and the groups have to be measured on the same variable (i.e., you can’t compare personality in one group to reaction time in another group because those values would not be the same anyway).

Figure 1. Collecting data from two different samples drawn from two different populations.

Let’s look at one of the most common and logical examples: testing a new medication. When a new medication is developed, the researchers who created it need to demonstrate that it effectively treats the symptoms they are trying to alleviate. The simplest design that will answer this question involves two groups: one group that receives the new medication (the “treatment” group) and one group that receives a placebo (the “control” group). Participants are randomly assigned to one of the two groups (remember that random assignment is the hallmark of a true experiment), and the researchers test the symptoms experienced by each person in each group after they received either the medication or the placebo. They then calculate the average symptoms in each group and compare them to see if the treatment group did better (i.e., had fewer or less severe symptoms) than the control group.

In this example, we had two groups (the independent variable): treatment and control. Membership in these two groups was mutually exclusive; each individual participant received either the experimental medication or the placebo. No one in the experiment received both, so there was no overlap between the two groups. Additionally, each group could be measured on the same variable: symptoms related to the disease or ailment being treated (the dependent variable). Because each group was measured on the same variable, the average scores in each group could be meaningfully compared. If the treatment was ineffective, we would expect that the average symptoms of someone receiving the treatment would be the same as the average symptoms of someone receiving the placebo (i.e., there is no difference between the groups). However, if the treatment was effective, we would expect fewer symptoms from the treatment group, leading to a lower group average.

Now, let’s look at an example using groups that already exist. A common, and perhaps salient, question is how students feel about their job prospects after graduation. Suppose that we have narrowed our potential choice of colleges down to two universities and, in the course of trying to decide between the two, we come across a survey that has data from each university on how students at those universities feel about their future job prospects. As with our last example, we have two groups: University A and University B, and each participant is in only one of the two groups. Because students at each university completed the same survey, they are measuring the same thing, so we can use an independent samples t -test to compare the average perceptions of students at each university to see if they are the same. If they are the same, then we should continue looking for other things about each university to help us decide on where to go. But, if they are different, we can use that information in favor of the university with higher job prospects.

As we can see, the grouping variable we use for an independent samples t -test can be a set of groups we create (as in the experimental medication example) or groups that already exist naturally (as in the university example). There are countless other examples of research questions relating to two group means, making the independent samples t -test one of the most widely used analyses around.

Hypotheses and Decision Criteria

The process of testing hypotheses using an independent samples t -test is the same as it was in the last two chapters, and it starts with stating our hypotheses and laying out the criteria we will use to test them.

The Null and Alternative Hypotheses

We still state the null and alternative hypotheses mathematically in terms of the population parameters and and then in words written out as statements answering the research questions as yes and no answers. Just like before, we need to decide if we are looking for directional or non-directional hypotheses. For a non-directional hypothesis test, our null hypothesis for an independent samples t -test is the same as all others: there is no difference. But instead of comparing a single 𝜇 to a numerical value, we are now comparing two populations, thus to 𝜇’s. The means of the two groups are the same under the non-directional null hypothesis. Mathematically, this takes on two equivalent forms:

𝐻 0 : 𝜇 1 = 𝜇 2

𝐻 0 : 𝜇 1 − 𝜇 2 = 0

Both of these formulations of the null hypothesis tell us exactly the same thing: that the numerical values of the means are the same in both groups. This is more clear in the first formulation, but the second formulation also makes sense (any number minus itself is always zero) and helps us out a little when we get to the math of the test statistic. Either one is acceptable and you only need to report one. The written out version of both of them is also the same:

H 0 : There is no difference between the means of the two groups

Our alternative hypotheses are also unchanged. We simply replace the equal sign (=) with the inequality sign (≠):

𝐻 𝐴 : 𝜇 1 ≠ 𝜇 2

𝐻 𝐴 : 𝜇 1 − 𝜇 2 ≠ 0

Whichever format you chose for the null hypothesis should be the one you use for the alternative hypothesis (be consistent), and the interpretation of them is always the same:

H A : There is a difference between the means of the two groups

Notice that we are now dealing with two means instead of just one, so it will be very important to keep track of which mean goes with which population and, by extension, which dataset and sample data. We use subscripts to differentiate between the populations, so make sure to keep track of which is which. If it is helpful, you can also use more descriptive subscripts. To use the experimental medication example:

H 0 : There is no difference between the means of the treatment and control groups

𝐻 0 : 𝜇 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 = 𝜇 𝑐𝑜𝑛𝑡𝑟𝑜𝑙

H A : There is a difference between the means of the treatment and control groups

𝐻 𝐴 : 𝜇 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 ≠ 𝜇 𝑐𝑜𝑛𝑡𝑟𝑜𝑙

Degrees of Freedom for the Critical Value

Once we have our hypotheses laid out, we can set our criteria to test them using the same three pieces of information as before: significance level (α), directionality (one-tailed: left or right, or two-tailed), and degrees of freedom, which for an independent samples t -test are calculated a bit differently than before.

Calculating degrees of freedom for Independent Samples t-test

[latex]df=(n_1-1)+(n_2-1)[/latex]

Where n 1 represents the sample size for group 1 and n 2 represents the sample size for group 2. We have two separate groups, each with their own possible sample mean, so each has their own possible set of degrees of freedom ( df = n -1).

This looks different than before, but it is just adding the individual degrees of freedom from each group (n – 1) together. Notice that the sample sizes, n , also get subscripts so we can tell them apart.

For an independent samples t -test, it is often the case that our two groups will have slightly different sample sizes, either due to chance or some characteristic of the groups themselves. Generally, this is not as issue, so long as one group is not massively larger than the other group. What is of greater concern is keeping track of which is which using the subscripts.

Computing the Test Statistic

The test statistic for our independent samples t -test takes on the same logical structure and format as our other t -tests: our observed effect minus our null hypothesis value, all divided by the standard error.

Calculating the test statistic for Independent Samples t-test

[latex]t_{test}=\frac{(M_1-M_2)-(𝜇_1-𝜇_2)}{s_{(M_1-M_2)}}[/latex]

M 1 is the sample mean for group 1 and M 2 is the sample mean for group 2. This looks like more work to calculate, but remember that our null hypothesis states that the quantity μ 1 – μ 2 = 0, so we can drop that out of the equation and are left with:

[latex]t_{test}=\frac{(M_1-M_2)}{s_{(M_1-M_2)}}[/latex]

Our standard error in the denominator is still estimated standard deviation (s) with a subscript denoting what it is the standard error of. Because we are dealing with the difference between two separate means, rather than a single mean, we put both means in the subscript. Calculating our standard error, as we will see next, is where the biggest differences between this t -test and other t -tests appears. However, once we do calculate it and use it in our test statistic, everything else goes back to normal. Our decision criteria is still comparing our obtained test statistic to our critical value, and our interpretation based on whether or not we reject the null hypothesis is unchanged as well.

Standard Error and Pooled Variance

Recall that the standard error is the average distance between any given sample mean and the center of its corresponding sampling distribution, and it is a function of the standard deviation of the population (either given or estimated) and the sample size. This definition and interpretation hold true for our independent samples t -test as well, but because we are working with two samples drawn from two populations, we have to first combine their estimates of standard deviation – or, more accurately, their estimates of variance – into a single value that we can then use to calculate our standard error.

The combined estimate of variance using the information from each sample is called the pooled variance and is denoted s p 2 ; the subscript p serves as a reminder indicating that it is the pooled variance. The term “pooled variance” is a literal name because we are simply pooling or combining the information on variance – the Sum of Squares and Degrees of Freedom – from both of our samples into a single number. The result is a weighted average of the observed sample variances, the weight for each being determined by the sample size, and will always fall between the two observed variances.

Calculating the Pooled Variance and Standard Error for Independent Samples t-test

[latex]s_p^2=\frac{SS_1+SS_2}{df_1+df_2}[/latex]

Notice that pooled variance requires that we calculate the sum of squares for each sample and the degrees of freedom for each sample. Make sure you add before you divide when calculating pooled variance!

Once we have our pooled variance calculated, we can drop it into the equation for our standard error:

[latex]s_{(M_1-M_1)}=\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}[/latex]

Unlike when calculating the pooled variance, here you want to make sure that you divide your fractions before you add!

Once again, although this formula may seem different than it was before, in reality it is just a different way of writing the same thing. Think back to the standard error options presented before, when our standard error was the square root of the just a single variance divided by a single sample size.

[latex]s_M=\sqrt{\frac{s^2}{n}}[/latex]

Looking at that, we can now see that, once again, we are simply adding together two pieces of information: no new logic or interpretation required. Once the standard error is calculated, it goes in the denominator of our test statistic, as shown above and as was the case in all previous chapters. Thus, the only additional step to calculating an independent samples t -statistic is computing the pooled variance. Let’s see an example in action.

Example: Movies and Mood

We are interested in whether the type of movie someone sees at the theater affects their mood when they leave. We decide to ask people about their mood as they leave one of two movies: a comedy (group 1, n 1 = 35) or a horror film (group 2, n 2 = 29). Our data are coded so that higher scores indicate a more positive mood. We have good reason to believe that people leaving the comedy will be in a better mood, so we use a one-tailed test at α = 0.05 to test our hypothesis.

Research Question : Will those who watch a comedy movie have a better mood than those who watch a horror film?

Step 1: State the Hypotheses

As always, we start with hypotheses:

H 0 : Those who watch the comedy will not have a better mood than those who watch the horror film

H 0 : μ comedy < μ horror

(Note that the null hypothesis includes both no difference (the equal sign) and a decrease in mood.)

H A : Those who watch the comedy will have a better mood than those who watch the horror film

H A : μ comedy > μ horror

Notice that we used labels to make sure that our data stay clear when we do our calculations later in this hypothesis test.

Step 2: Find the Critical Values

Just like before, we will need critical values, which come from our t -table. In this example, we have a one-tailed test at α = 0.05 and expect a positive answer (because we expect the difference between the means to be greater than zero). Our degrees of freedom for our independent samples t -test is just the degrees of freedom from each group added together:

[latex]df=(n_{comedy}-1)+(n_{horror}-1)=(35-1)+(29-1)=34+28=62[/latex]

From our t -table, we find that our critical value is t crit = 1.671. Note that because 62 does not appear on the table, we use the next lowest value, which in this case is 60. Also, remember that since we are using a directional test, and looking for an increase or a positive change, that we keep the positive critical value. So, if we were to draw this out on a normal distribution, our critical region would be on the right side.

Step 3: Compute the Test Statistic

The data from our two groups are presented in the tables below. Table 1 shows the values for the Comedy group, and Table 2 shows the values for the Horror group. Values for both have already been placed in the Sum of Squares tables since we will need to use them for our further calculations. (See attached raw data set for additional practice.) Example Independent Samples t-test data (1)

Table 1. Raw scores and Sum of Squares for Group 1

Table 2. Raw scores and Sum of Squares for Group 2.

These values have all been calculated and take on the same interpretation as they have since chapter 3 – no new computations yet. Before we move on to the pooled variance that will allow us to calculate standard error, let’s compute our standard deviation for each group; even though we will not use them in our calculation of the test statistic, they are still important descriptors of our data.

[latex]s_{comedy}^2=\frac{5061.60}{34}=148.87[/latex]

[latex]s_{comedy}=\sqrt{\frac{5061.60}{34}}=12.20[/latex]

[latex]s_{horror}^2=\frac{3896.45}{28}=139.16[/latex]

[latex]s_{horror}=\sqrt{\frac{3896.45}{28}}=11.80[/latex]

Now we can move on to our new calculation, the pooled variance, which is the Sums of Squares that we calculated from our table and the degrees of freedom, which is n – 1, for each group.

[latex]s_p^2=\frac{SS_{comedy}+SS_{horror}}{df_{comedy}+df_{horror}}[/latex]

[latex]s_p^2=\frac{5061.60+3896.45}{34+28}=\frac{8958.05}{62}=144.48[/latex]

As you can see, if you follow the regular process of calculating standard deviation using the Sum of Squares table, finding the pooled variance is very easy.

Now we can use the pooled variance calculated above to calculate our standard error, the last step before we can find our test statistic.

[latex]s_{M_1-M_2}=\sqrt{\frac{s_p^2}{n_{comedy}}+\frac{s_p^2}{n_{horror}}}[/latex]

[latex]s_{M_1-M_2}=\sqrt{\frac{144.48}{35}+\frac{144.48}{29}}=\sqrt{4.13+4.98}=\sqrt{9.11}=3.02[/latex]

Finally, we can use our standard error and the means we calculated earlier to compute our test statistic. Because the null hypothesis value of μ 1 – μ 2 is 0, we will leave that portion out of the equation for simplicity.

[latex]t_{test}=\frac{M_{comedy}-M_{horror}}{s_{M_1-M_2}}[/latex]

[latex]t_{test}=\frac{24.00-16.50}{3.02}=\frac{7.50}{3.02}=2.48[/latex]

The process of calculating our test statistic t test = 2.48 followed the same sequence of steps as before: use raw data to compute the mean and sum of squares (this time for two groups instead of one), use the sum of squares and degrees of freedom to calculate standard error (this time using pooled variance instead of standard deviation), and use that standard error and the observed means to get t test .

Now we can move on to the final step of the hypothesis testing procedure.

Step 4: Make and Interpret the Decision

Our test statistic has a value of t test = 2.48, and in step 2 we found that the critical value is t crit = 1.671. 2.48 > 1.671, so we reject the null hypothesis. We can now write our decision up in APA style. Notice that we use our combined degrees of freedom in our inferential statistics in the statement below.

Reject H 0 . The average mood after watching a comedy movie ( M = 24.00, SD = 12.20) is significantly better than the average mood after watching a horror movie ( M = 16.50, SD = 11.80), t (62) = 2.48, p < .05.

However, we shouldn’t stop there. Since we found an effect, we want to make sure to calculate effect size next.

Effect Sizes and Confidence Intervals

We have seen in previous chapters that even a statistically significant effect needs to be interpreted along with an effect size to see if it is practically meaningful. We have also seen that our sample means, as a single estimate, are not perfect and would be better represented by a range of values that we call a confidence interval. As with all other topics, this is also true of our independent samples t -tests.

Our effect size for the independent samples t -test is still Cohen’s d , and it is still just our observed effect or difference divided by the standard deviation. Remember that standard deviation is just the square root of the variance, and because we work with pooled variance in our test statistic, we will use the square root of the pooled variance as our denominator in the formula for Cohen’s d .

Calculating Cohen’s d for Independent Samples t-test

[latex]d=\bigg |\frac{M_1-M_2}{\sqrt{s_p^2}}\bigg |[/latex]

For our example above, we can calculate the effect size to be:

[latex]d=\bigg |\frac{24.00-16.50}{\sqrt{144.48}}\bigg |=\bigg |\frac{7.50}{12.02}\bigg |=0.62[/latex]

We interpret this using the same guidelines as before, so we would consider this a medium effect. And, now we should add this to our sentence from above.

The average mood after watching a comedy movie ( M = 24.00, SD = 12.20) is significantly better than the average mood after watching a horror movie ( M = 16.50, SD = 11.80), t (62) = 2.48, p < .05, d = 0.62.

Confidence Intervals

Our confidence intervals also take on the same form and interpretation as they have in the past. The value we are interested in is the difference between the two means, so our single estimate is the value of one mean minus the other, or M 1 – M 2 . Just like before, this is our observed effect and is the same value as the one we place in the numerator of our test statistic. We calculate this value then place the margin of error – still our two-tailed critical value times our standard error – above and below it.

Calculating Confidence Interval for Independent Samples t-test

[latex]CI_{upper}=(M_1-M_2)+t_{crit}\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}[/latex]

[latex]CI_{lower}=(M_1-M_2)-t_{crit}\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}[/latex]

Because our hypothesis testing example used a one-tailed test, it would be inappropriate to calculate a confidence interval on those data (remember that we can only calculate a confidence interval for a two-tailed test because the interval extends in both directions). Let’s say we find summary statistics on the average life satisfaction of people from two different towns and want to create a confidence interval to see if the difference between the two might actually be zero.

Our sample data are M 1 =28.65, s 1 = 12.40, n 1 = 40 and M 2 = 25.40, s 2 = 15.68, n 2 = 42. At face value, it looks like the people from the first town have higher life satisfaction (28.65 vs. 25.40), but it will take a confidence interval (or complete hypothesis testing process) to see if that is true or just due to random chance.

First, we want to calculate the difference between our sample means, which is 28.65 – 25.40 = 3.25.

Next, we need a critical value from our t -table. If we want to test at the normal 95% level of confidence, then our sample sizes will yield degrees of freedom equal to (40-1) + (42-1) = 40 + 42 – 2 = 80. From our table, that gives us a critical value of t crit = 1.990.

Finally, we need our standard error. Recall that our standard error for an independent samples t -test uses pooled variance, which requires the Sum of Squares and degrees of freedom. Up to this point, we have calculated the Sum of Squares using raw data, but in this situation, we do not have access to it. So, what are we to do?

If we have summary data like standard deviation and sample size, it is very easy to calculate the pooled variance another version of the formula.

Calculating Pooled Variance from Sample Standard Deviations

[latex]s_p^2=\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}[/latex]

If s 1 = 12.40, then s 1 2 = 12.40*12.40 = 153.76, and if s 2 = 15.68, then s 2 2 = 15.68*16.68 = 245.86. With n 1 = 40 and n 2 = 42, we are all set to calculate the pooled variance.

[latex]s_p^2=\frac{(40-1)(153.76)+(42-1)(245.86)}{40+42-2}=\frac{(39)(153.76)+(41)(245.86)}{80}[/latex]

[latex]s_p^2=\frac{5996.64+10080.36}{80}=\frac{16077.00}{80}=200.96[/latex]

Now we can plug s p 2 = 200.96, n 1 = 40, and n 2 = 42nd, in to our standard error equation.

[latex]s_{M_1-M_2}=\sqrt{\frac{s_p^2}{n_1}+\frac{s_p^2}{n_2}}=\sqrt{\frac{200.96}{40}+\frac{200.96}{42}}=\sqrt{5.02+4.78}=\sqrt{9.89}=3.14[/latex]

All of these steps are just slightly different ways of using the same formula, numbers, and ideas we have worked with up to this point. Once we get out standard error, it’s time to build our confidence interval.

95% CI = 3.25 + 1.990(3.14)

95% CI upper = 3.25 + 1.990(3.14) = 3.25 + 6.25 = 9.50

95% CI lower = 3.25 – 1.990(3.14) = 3.25 – 6.25 = -3.00

CI 95 [-3.00, 9.50]

Our confidence interval, as always, represents a range of values that would be considered reasonable or plausible based on our observed data. In this instance, our interval (-3.00, 9.50) does contain zero. Thus, even though the means look a little bit different, it may very well be the case that the life satisfaction in both of these towns is the same. Proving otherwise would require more data.

Homogeneity of Variance

Before wrapping up the coverage of independent samples t -tests, there is one other important topic to cover. Using the pooled variance to calculate the test statistic relies on an assumption known as homogeneity of variance AKA equality of variances AKA homoscedasticity. In statistics, an assumption is some characteristic that we assume is true about our data, and our ability to use our inferential statistics accurately and correctly relies on these assumptions being true. If these assumptions are not true, then our analyses are at best ineffective (e.g., low power to detect effects) and at worst inappropriate (e.g., too many Type I errors). A detailed coverage of assumptions is beyond the scope of this textbook, but it is important to know that they exist for all analyses.

For the current hypothesis test, one important assumption is homogeneity of variance. This is fancy statistical talk for the idea that the true population variance for each group is the same and any difference in the observed sample variances is due to random chance. (If this sounds eerily similar to the idea of testing the null hypothesis that the true population means are equal, that’s because it is exactly the same!) This notion allows us to compute a single pooled variance that uses our easily calculated degrees of freedom. If the assumption is shown to not be true, then we have to use a very complicated formula to estimate the proper degrees of freedom. There are formal tests to assess whether or not this assumption is met, but we will not discuss them here., however most statistical programs will run these assumption checks for us.

Many statistical programs incorporate the test of homogeneity of variance automatically and can report the results of the analysis assuming it is true or assuming it has been violated. You can easily tell which is which by the degrees of freedom: the corrected degrees of freedom (which is used when the assumption of homogeneity of variance is violated) will have decimal places. Fortunately, the independent samples t -test is very robust to violations of this assumption (an analysis is “robust” if it works well even when its assumptions are not met), which is why we do not bother going through the tedious work of testing and estimating new degrees of freedom by hand.

Review: Statistical Power

There are three factors that can affect statistical power:

Sample size: Larger samples provide greater statistical power
Effect size: A given design will always have greater power to find a large effect than a small effect (because finding large effects is easier)
Type I error rate: There is a relationship between Type I error and power such that (all else being equal) decreasing Type I error will also decrease power.

We can see this through simulation. First let’s simulate a single experiment, in which we compare the means of two groups using a standard t-test. We will vary the size of the effect (specified in terms of Cohen’s d), the Type I error rate, and the sample size, and for each of these we will examine how the proportion of significant results (i.e. power) is affected. Figure 1 shows an example of how power changes as a function of these factors.

Results from power simulation, showing power as a function of sample size, with effect sizes shown as different colors, and alpha shown as line type. The standard criterion of 80 percent power is shown by the dotted black line.

Figure 1: Results from power simulation, showing power as a function of sample size, with effect sizes shown as different colors, and alpha shown as line type. The standard criterion of 80 percent power is shown by the dotted black line.

This simulation shows us that even with a sample size of 96, we will have relatively little power to find a small effect (d=0.2) with α=0.005. This means that a study designed to do this would be futile – that is, it is almost guaranteed to find nothing even if a true effect of that size exists.

There are at least two important reasons to care about statistical power. First, if you are a researcher, you probably don’t want to spend your time doing futile experiments. Running an underpowered study is essentially futile, because it means that there is a very low likelihood that one will find an effect, even if it exists. Second, it turns out that any positive findings that come from an underpowered study are more likely to be false compared to a well-powered study . Fortunately, there are tools available that allow us to determine the statistical power of an experiment. The most common use of these tools is in planning an experiment, when we would like to determine how large our sample needs to be in order to have sufficient power to find our effect of interest.

Assumptions of independent-samples t-tests

Assumptions are conditions that must be met in order for our hypothesis testing conclusion to be valid. [Important: If the assumptions are not met then our hypothesis testing conclusion is not likely to be valid. Testing errors can still occur even if the assumptions for the test are met.]

Recall that inferential statistics allow us to make inferences (decisions, estimates, predictions) about a population based on data collected from a sample. Recall also that an inference about a population is true only if the sample studied is representative of the population. A statement about a population based on a biased sample is not likely to be true.

Assumption 1: Individuals in the sample were selected randomly and independently, so the sample is highly likely to be representative of the larger population.

Random sampling ensures that each member of the population is equally likely to be selected.
An independent sample is one in which the selection of one member has no effect on the selection of any other.

Assumption 2: The distribution of sample mean differences is a normal, because we drew the samples from a population that was normally distributed or we drew large enough samples that the law of large numbers has worked in our favor.

This assumption is very important because we are estimating probabilities using the t- table, which provide accurate estimates of probabilities for events distributed normally.

Assumption 3: Sampled populations have equal variances or have homogeneity of variance as discussed above.

a research design that uses a separate group of participants for each treatment condition

the statistical assumption of equal variance, meaning that the average squared distance of a score from the mean is the same across all groups sampled in a study.

Introduction to Statistics for the Social Sciences Copyright © 2021 by Jennifer Ivie; Alicia MacKay is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Fisher, Neyman-Pearson or NHST? A tutorial for teaching data testing

Despite frequent calls for the overhaul of null hypothesis significance testing (NHST), this controversial procedure remains ubiquitous in behavioral, social and biomedical teaching and research. Little change seems possible once the procedure becomes well ingrained in the minds and current practice of researchers; thus, the optimal opportunity for such change is at the time the procedure is taught, be this at undergraduate or at postgraduate levels. This paper presents a tutorial for the teaching of data testing procedures, often referred to as hypothesis testing theories. The first procedure introduced is Fisher's approach to data testing—tests of significance; the second is Neyman-Pearson's approach—tests of acceptance; the final procedure is the incongruent combination of the previous two theories into the current approach—NSHT. For those researchers sticking with the latter, two compromise solutions on how to improve NHST conclude the tutorial.

Introduction

This paper introduces the classic approaches for testing research data: tests of significance, which Fisher helped develop and promote starting in 1925; tests of statistical hypotheses, developed by Neyman and Pearson ( 1928 ); and null hypothesis significance testing (NHST), first concocted by Lindquist ( 1940 ). This chronological arrangement is fortuitous insofar it introduces the simpler testing approach by Fisher first, then moves onto the more complex one by Neyman and Pearson, before tackling the incongruent hybrid approach represented by NHST (Gigerenzer, 2004 ; Hubbard, 2004 ). Other theories, such as Bayes's hypotheses testing (Lindley, 1965 ) and Wald's ( 1950 ) decision theory, are not object of this tutorial.

The main aim of the tutorial is to illustrate the bases of discord in the debate against NHST (Macdonald, 2002 ; Gigerenzer, 2004 ), which remains a problem not only yet unresolved but very much ubiquitous in current data testing (e.g., Franco et al., 2014 ) and teaching (e.g., Dancey and Reidy, 2014 ), especially in the biological sciences (Lovell, 2013 ; Ludbrook, 2013 ), social sciences (Frick, 1996 ), psychology (Nickerson, 2000 ; Gigerenzer, 2004 ) and education (Carver, 1978 , 1993 ).

This tutorial is appropriate for the teaching of data testing at undergraduate and postgraduate levels, and is best introduced when students are knowledgeable on important background information regarding research methods (such as random sampling) and inferential statistics (such as frequency distributions of means).

In order to improve understanding, statistical constructs that may bring about confusion between theories are labeled differently, attending to their function in preference to their historical use (Perezgonzalez, 2014 ). Descriptive notes (notes) and caution notes (caution) are provided to clarify matters whenever appropriate.

Fisher's approach to data testing

Ronald Aylmer Fisher was the main force behind tests of significance (Neyman, 1967 ) and can be considered the most influential figure in the current approach to testing research data (Hubbard, 2004 ). Although some steps in Fisher's approach may be worked out a priori (e.g., the setting of hypotheses and levels of significance), the approach is eminently inferential and all steps can be set up a posteriori, once the research data are ready to be analyzed (Fisher, 1955 ; Macdonald, 1997 ). Some of these steps can even be omitted in practice, as it is relatively easy for a reader to recreate them. Fisher's approach to data testing can be summarized in the five steps described below.

Step 1–Select an appropriate test . This step calls for selecting a test appropriate to, primarily, the research goal of interest (Fisher, 1932 ), although you may also need to consider other issues, such as the way your variables have been measured. For example, if your research goal is to assess differences in the number of people in two independent groups, you would choose a chi-square test (it requires variables measured at nominal levels); on the other hand, if your interest is to assess differences in the scores that the people in those two groups have reported on a questionnaire, you would choose a t -test (it requires variables measured at interval or ratio levels and a close-to-normal distribution of the groups' differences).

Step 2–Set up the null hypothesis (H 0 ) . The null hypothesis derives naturally from the test selected in the form of an exact statistical hypothesis (e.g., H 0 : M1–M2 = 0; Neyman and Pearson, 1933 ; Carver, 1978 ; Frick, 1996 ). Some parameters of this hypothesis, such as variance and degrees of freedom, are estimated from the sample, while other parameters, such as the distribution of frequencies under a particular distribution, are deduced theoretically. The statistical distribution so established thus represents the random variability that is theoretically expected for a statistical nil hypothesis (i.e., H 0 = 0) given a particular research sample (Fisher, 1954 , 1955 ; Bakan, 1966 ; Macdonald, 2002 ; Hubbard, 2004 ). It is called the null hypothesis because it stands to be nullified with research data (Gigerenzer, 2004 ).

Among things to consider when setting the null hypothesis is its directionality.

Directional and non-directional hypotheses . With some research projects, the direction of the results is expected (e.g., one group will perform better than the other). In these cases, a directional null hypothesis covering all remaining possible results can be set (e.g., H 0 : M1–M2 = 0). With other projects, however, the direction of the results is not predictable or of no research interest. In these cases, a non-directional hypothesis is most suitable (e.g., H 0 : M1–M2 = 0).

H 0 does not need to be a nil hypothesis, that is, one that always equals zero (Fisher, 1955 ; Gigerenzer, 2004 ). For example, H 0 could be that the group difference is not larger than certain value (Newman et al., 2001 ). More often than not, however, H 0 tends to be zero.

Setting up H 0 is one of the steps usually omitted if following the typical nil expectation (e.g., no correlation between variables, no differences in variance among groups, etc.). Even directional nil hypotheses are often omitted, instead specifying that one-tailed tests (see below) have been used in the analysis.

Step 3-Calculate the theoretical probability of the results under H 0 ( p ) . Once the corresponding theoretical distribution is established, the probability ( p -value) of any datum under the null hypothesis is also established, which is what statistics calculate (Fisher, 1955 , 1960 ; Bakan, 1966 ; Johnstone, 1987 ; Cortina and Dunlap, 1997 ; Hagen, 1997 ). Data closer to the mean of the distribution (Figure (Figure1) 1 ) have a greater probability of occurrence under the null distribution; that is, they appear more frequently and show a larger p -value (e.g., p = 0.46, or 46 times in a 100 trials). On the other hand, data located further away from the mean have a lower probability of occurrence under the null distribution; that is, they appear less often and, thus, show a smaller p -value (e.g., p = 0.003). Of interest to us is the probability of our research results under such null distribution (e.g., the probability of the difference in means between two research groups).

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Location of a t -value and its corresponding p -value on a theoretical t distribution with 30 degrees of freedom . The actual p -value conveys stronger evidence against H 0 than sig ≈0.05 and can be considered highly significant.

The p -value comprises the probability of the observed results and also of any other more extreme results (e.g., the probability of the actual difference between groups and any other difference more extreme than that). Thus, the p -value is a cumulative probability rather than an exact point probability: It covers the probability area extending from the observed results toward the tail of the distribution (Fisher, 1960 ; Carver, 1978 ; Frick, 1996 ; Hubbard, 2004 ).

P-values provide information about the theoretical probability of the observed and more extreme results under a null hypothesis assumed to be true (Fisher, 1960 ; Bakan, 1966 ), or, said otherwise, the probability of the data given a true hypothesis—P(D|H); (Carver, 1978 ; Hubbard, 2004 ). As H 0 is always true (i.e., it shows the theoretical random distribution of frequencies under certain parameters), it cannot, at the same time, be false nor falsifiable a posteriori. Basically, if at any point you say that H 0 is false, then you are also invalidating the whole test and its results. Furthermore, because H 0 is always true, it cannot be proved, either.

Step 4–Assess the statistical significance of the results . Fisher proposed tests of significance as a tool for identifying research results of interest, defined as those with a low probability of occurring as mere random variation of a null hypothesis. A research result with a low p -value may, thus, be taken as evidence against the null (i.e., as evidence that it may not explain those results satisfactorily; Fisher, 1960 ; Bakan, 1966 ; Johnstone, 1987 ; Macdonald, 2002 ). How small a result ought to be in order to be considered statistically significant is largely dependent on the researcher in question, and may vary from research to research (Fisher, 1960 ; Gigerenzer, 2004 ). The decision can also be left to the reader, so reporting exact p -values is very informative (Fisher, 1973 ; Macdonald, 1997 ; Gigerenzer, 2004 ).

Overall, however, the assessment of research results is largely made bound to a given level of significance, by comparing whether the research p -value is smaller than such level of significance or not (Fisher, 1954 , 1960 ; Johnstone, 1987 ):

If the p -value is approximately equal to or smaller than the level of significance, the result is considered statistically significant.
If the p -value is larger than the level of significance, the result is considered statistically non-significant.

Among things to consider when assessing the statistical significance of research results are the level of significance, and how it is affected by the directionality of the test and other corrections.

Level of significance (sig) . The level of significance is a theoretical p -value used as a point of reference to help identify statistically significant results (Figure (Figure1). 1 ). There is no need to set up a level of significance a priori nor for a particular level of significance to be used in all occasions, although levels of significance such as 5% (sig ≈0.05) or 1% (sig ≈0.01) may be used for convenience, especially with novel research projects (Fisher, 1960 ; Carver, 1978 ; Gigerenzer, 2004 ). This highlights an important property of Fisher's levels of significance: They do not need to be rigid (e.g., p -values such as 0.049 and 0.051 have about the same statistical significance around a convenient level of significance of 5%; Johnstone, 1987 ).

Another property of tests of significance is that the observed p -value is taken as evidence against the null hypothesis, so that the smaller the p -value the stronger the evidence it provides (Fisher, 1960 ; Spielman, 1978 ). This means that it is plausible to gradate the strength of such evidence with smaller levels of significance. For example, if using 5% (sig ≈0.05) as a convenient level for identifying results which are just significant, then 1% (sig ≈0.01) may be used as a convenient level for identifying highly significant results and 1‰ (sig ≈0.001) for identifying extremely significant results.

Setting up a level of significance is another step usually omitted. In such cases, you may assume the researcher is using conventional levels of significance.

If both H 0 and sig are made explicit, they could be joined in a single postulate, such as H 0 : M 1 –M 2 = 0, sig ≈0.05.

Notice that the p-value informs about the probability associated with a given test value (e.g., a t value). You could use this test value to decide about the significance of your results in a fashion similar to Neyman-Pearson's approach (see below). However, you get more information about the strength of the research evidence with p-values.

Although the p-value is the most informative statistic of a test of significance, in psychology (e.g., American Psychological Association, 2010 ) you also report the research value of the test—e.g., t (30) = 2.25, p = 0.016, 1-tailed. Albeit cumbersome and largely ignored by the reader, the research value of the test offers potentially useful information (e.g., about the valid sample size used with a test).

Caution: Be careful not to interpret Fisher's p-values as Neyman-Pearson's Type I errors (α, see below). Probability values in single research projects are not the same than probability values in the long run (Johnstone, 1987 ), something illustrated by Berger ( 2003 )—who reported that p = 0.05 often corresponds to α = 0.5 (or anywhere between α = 0.22 and α > 0.5)—and Cumming ( 2014 )—who simulates the “dance” of p-values in the long run, commented further in Perezgonzalez ( 2015 ).

One-tailed and two-tailed tests . With some tests (e.g., F -tests) research data can only be tested against one side of the null distribution (one-tailed tests), while other tests (e.g., t -tests) can test research data against both sides of the null distribution at the same time. With one-tailed tests you set the level of significance on the appropriate tail of the distribution. With two-tailed tests you cover both eventualities by dividing the level of significance between both tails (Fisher, 1960 ; Macdonald, 1997 ), which is commonly done by halving the total level of significance in two equal areas (thus covering, for example, the 2.5% most extreme positive differences and the 2.5% most extreme negative differences).

The tail of a test depends on the test in question, not on whether the null hypothesis is directional or non-directional. However, you can use two-tailed tests as one-tailed ones when testing data against directional hypotheses.

Correction of the level of significance for multiple tests . As we introduced earlier, a p -value can be interpreted in terms of its expected frequency of occurrence under the specific null distribution for a particular test (e.g., p = 0.02 describes a result that is expected to appear 2 times out of 100 under H 0 ). The same goes for theoretical p -values used as levels of significance. Thus, if more than one test is performed, this has the consequence of also increasing the probability of finding statistical significant results which are due to mere chance variation. In order to keep such probability at acceptable levels overall, the level of significance may be corrected downwards (Hagen, 1997 ). A popular correction is Bonferroni's, which reduces the level of significance proportionally to the number of tests carried out. For example, if your selected level of significance is 5% (sig ≈0.05) and you carry out two tests, then such level of significance is maintained overall by correcting the level of significance for each test down to 2.5% (sig ≈0.05/2 tests ≈0.025, or 2.5% per test).

Bonferroni's correction is popular but controversial, mainly because it is too conservative, more so as the number of multiple tests increases. There are other methods for controlling the probability of false results when doing multiple comparisons, including familywise error rate methods (e.g., Holland and Copenhaver, 1987 ), false discovery rate methods (e.g., Benjamini and Hochberg, 1995 ), resampling methods (jackknifing, bootstrapping— e.g., Efron, 1981 ), and permutation tests (i.e., exact tests—e.g., Gill, 2007 ).

Step 5–Interpret the statistical significance of the results . A significant result is literally interpreted as a dual statement: Either a rare result that occurs only with probability p (or lower) just happened, or the null hypothesis does not explain the research results satisfactorily (Fisher, 1955 ; Carver, 1978 ; Johnstone, 1987 ; Macdonald, 1997 ). Such literal interpretation is rarely encountered, however, and most common interpretations are in the line of “The null hypothesis did not seem to explain the research results well, thus we inferred that other processes—which we believe to be our experimental manipulation—exist that account for the results,” or “The research results were statistically significant, thus we inferred that the treatment used accounted for such difference.”

Non-significant results may be ignored (Fisher, 1960 ; Nunnally, 1960 ), although they can still provide useful information, such as whether results were in the expected direction and about their magnitude (Fisher, 1955 ). In fact, although always denying that the null hypothesis could ever be supported or established, Fisher conceded that non-significant results might be used for confirming or strengthening it (Fisher, 1955 ; Johnstone, 1987 ).

Statistically speaking, Fisher's approach only ascertains the probability of the research data under a null hypothesis. Doubting or denying such hypothesis given a low p-value does not necessarily “support” or “prove” that the opposite is true (e.g., that there is a difference or a correlation in the population). More importantly, it does not “support” or “prove” that whatever else has been done in the research (e.g., the treatment used) explains the results, either (Macdonald, 1997 ). For Fisher, a good control of the research design (Fisher, 1955 ; Johnstone, 1987 ; Cortina and Dunlap, 1997 ), especially random allocation, is paramount to make sensible inferences based on the results of tests of significance (Fisher, 1954 ; Neyman, 1967 ). He was also adamant that, given a significant result, further research was needed to establish that there has indeed been an effect due to the treatment used (Fisher, 1954 ; Johnstone, 1987 ; Macdonald, 2002 ). Finally, he considered significant results as mere data points and encouraged the use of meta-analysis for progressing further, combining significant and non-significant results from related research projects (Fisher, 1960 ; Neyman, 1967 ).

Highlights of Fisher's approach

Flexibility . Because most of the work is done a posteriori, Fisher's approach is quite flexible, allowing for any number of tests to be carried out and, therefore, any number of null hypotheses to be tested (a correction of the level of significance may be appropriate, though—Macdonald, 1997 ).

Better suited for ad-hoc research projects . Given above flexibility, Fisher's approach is well suited for single, ad-hoc , research projects (Neyman, 1956 ; Johnstone, 1987 ), as well as for exploratory research (Frick, 1996 ; Macdonald, 1997 ; Gigerenzer, 2004 ).

Inferential . Fisher's procedure is largely inferential, from the sample to the population of reference, albeit of limited reach, mainly restricted to populations that share parameters similar to those estimated from the sample (Fisher, 1954 , 1955 ; Macdonald, 2002 ; Hubbard, 2004 ).

No power analysis . Neyman ( 1967 ) and Kruskal and Savage (Kruskal, 1980 ) were surprised that Fisher did not explicitly attend to the power of a test. Fisher talked about sensitiveness, a similar concept, and how it could be increased by increasing sample size (Fisher, 1960 ). However, he never created a mathematical procedure for controlling sensitiveness in a predictable manner (Macdonald, 1997 ; Hubbard, 2004 ).

No alternative hypothesis . One of the main critiques to Fisher's approach is the lack of an explicit alternative hypothesis (Macdonald, 2002 ; Gigerenzer, 2004 ; Hubbard, 2004 ), because there is no point in rejecting a null hypothesis without an alternative explanation being available (Pearson, 1990 ). However, Fisher considered alternative hypotheses implicitly—these being the negation of the null hypotheses—so much so that for him the main task of the researcher—and a definition of a research project well done—was to systematically reject with enough evidence the corresponding null hypothesis (Fisher, 1960 ).

Neyman-Pearson's approach to data testing

Jerzy Neyman and Egon Sharpe Pearson tried to improve Fisher's procedure (Fisher, 1955 ; Pearson, 1955 ; Jones and Tukey, 2000 ; Macdonald, 2002 ) and ended up developing an alternative approach to data testing. Neyman-Pearson's approach is more mathematical than Fisher's and does much of its work a priori, at the planning stage of the research project (Fisher, 1955 ; Macdonald, 1997 ; Gigerenzer, 2004 ; Hubbard, 2004 ). It also introduces a number of constructs, some of which are similar to those of Fisher. Overall, Neyman-Pearson's approach to data testing can be considered tests of acceptance (Fisher, 1955 ; Pearson, 1955 ; Spielman, 1978 ; Perezgonzalez, 2014 ), summarized in the following eight main steps.

A priori steps

Step 1–Set up the expected effect size in the population . The main conceptual innovation of Neyman-Pearson's approach was the consideration of explicit alternative hypotheses when testing research data (Neyman and Pearson, 1928 , 1933 ; Neyman, 1956 ; Macdonald, 2002 ; Gigerenzer, 2004 ; Hubbard, 2004 ). In their simplest postulate, the alternative hypothesis represents a second population that sits alongside the population of the main hypothesis on the same continuum of values. These two groups differ by some degree: the effect size (Cohen, 1988 ; Macdonald, 1997 ).

Although the effect size was a new concept introduced by Neyman and Pearson, in psychology it was popularized by Cohen ( 1988 ). For example, Cohen's conventions for capturing differences between groups—d (Figure (Figure2)—were 2 )—were based on the degree of visibility of such differences in the population: the smaller the effect size, the more difficult to appreciate such differences; the larger the effect size, the easier to appreciate such differences. Thus, effect sizes also double as a measure of importance in the real world (Nunnally, 1960 ; Cohen, 1988 ; Frick, 1996 ).

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A conventional large difference—Cohen's d = 0.8 —between two normally distributed populations, as a fraction of one standard deviation .

When testing data about samples, however, statistics do not work with unknown population distributions but with distributions of samples, which have narrower standard errors. In these cases, the effect size can still be defined as above because the means of the populations remain unaffected, but the sampling distributions would appear separated rather than overlapping (Figure (Figure3). 3 ). Because we rarely know the parameters of populations, it is their equivalent effect size measures in the context of sampling distributions which are of interest.

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Sampling distributions ( N = 50 each) of the populations in Figure Figure2 2 . MES ( d = 0.8), assuming β = 0.20, is d = 0.32 (i.e., the expected difference in the population ranges between d = 0.32 and infinity).

As we shall see below, the alternative hypothesis is the one that provides information about the effect size to be expected. However, because this hypothesis is not tested, Neyman-Pearson's procedure largely ignores its distribution except for a small percentage of it, which is called “beta” (β; Gigerenzer, 2004 ). Therefore, it is easier to understand Neyman-Pearson's procedure if we peg the effect size to beta and call it the expected minimum effect size (MES; Figure Figure3). 3 ). This helps us conceptualize better how Neyman-Pearson's procedure works (Schmidt, 1996 ): The minimum effect size effectively represents that part of the main hypothesis that is not going to be rejected by the test ( i.e., MES captures values of no research interest which you want to leave under H M ; Cortina and Dunlap, 1997 ; Hagen, 1997 ; Macdonald, 2002 ). (Worry not, as there is no need to perform any further calculations: The population effect size is the one to use, for example, for estimating research power.)

A particularity of Neyman-Pearson's approach is that the two hypotheses are assumed to represent defined populations, the research sample being an instance of either of them (i.e., they are populations of samples generated by repetition of a common random process—Neyman and Pearson, 1928 ; Pearson, 1955 ; Hagen, 1997 ; Hubbard, 2004 ). This is unlike Fisher's population, which can be considered more theoretical, generated ad-hoc so as for providing the appropriate random distribution for the research sample at hand (i.e., a population of samples similar to the research sample—Fisher, 1955 ; Johnstone, 1987 ).

Step 2–Select an optimal test . As we shall see below, another of Neyman-Pearson's contributions was the construct of the power of a test. A spin-off of this contribution is that it has been possible to establish which tests are most powerful (for example, parametric tests are more powerful than non-parametric tests, and one-tailed tests are more powerful than two-tailed tests), and under which conditions (for example, increasing sample size increases power). For Neyman and Pearson, thus, you are better off choosing the most powerful test for your research project (Neyman, 1942 , 1956 ).

Step 3–Set up the main hypothesis (H M ) . Neyman-Pearson's approach considers, at least, two competing hypotheses, although it only tests data under one of them. The hypothesis which is the most important for the research (i.e., the one you do not want to reject too often) is the one tested (Neyman and Pearson, 1928 ; Neyman, 1942 ; Spielman, 1973 ). This hypothesis is better off written so as for incorporating the minimum expected effect size within its postulate (e.g., H M : M 1 –M 2 = 0 ± MES), so that it is clear that values within such minimum threshold are considered reasonably probable under the main hypothesis, while values outside such minimum threshold are considered as more probable under the alternative hypothesis (Figure (Figure4 4 ).

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Neyman-Pearson's approach tests data under H M using the rejection region delimited by α . H A contributes MES and β. Differences of research interest will be equal or larger than MES and will fall within this rejection region.

Neyman-Pearson's H M is very similar to Fisher's H 0 . Indeed, Neyman and Pearson also called it the null hypothesis and often postulated it in a similar manner (e.g., as H M : M 1 –M 2 = 0). However, this similarity is merely superficial on three accounts: H M needs to be considered at the design stage (H 0 is rarely made explicit); it is implicitly designed to incorporate any value below the MES—i.e., the a priori power analysis of a test aims to capture such minimum difference (effect sizes are not part of Fisher's approach); and it is but one of two competing explanations for the research results (H 0 is the only hypothesis, to be nullified with evidence).

The main aspect to consider when setting the main hypothesis is the Type I error you want to control for during the research.

Type I error . A Type I error (or error of the first class) is made every time the main hypothesis is wrongly rejected (thus, every time the alternative hypothesis is wrongly accepted). Because the hypothesis under test is your main hypothesis, this is an error that you want to minimize as much as possible in your lifetime research (Neyman and Pearson, 1928 , 1933 ; Neyman, 1942 ; Macdonald, 1997 ).

A Type I error is possible under Fisher's approach, as it is similar to the error made when rejecting H 0 (Carver, 1978 ). However, this similarity is merely superficial on two accounts: Neyman and Pearson considered it an error whose relevance only manifests itself in the long run because it is not possible to know whether such an error has been made in any particular trial (Fisher's approach is eminently ad-hoc, so the risk of a long-run Type I error is of little relevance); therefore, it is an error that needs to be considered and minimized at the design stage of the research project in order to ensure good power—you cannot minimize this error a posteriori (with Fisher's approach, the potential impact of errors on individual projects is better controlled by correcting the level of significance as appropriate, for example, with a Bonferroni correction).

Alpha (α) . Alpha is the probability of committing a Type I error in the long run (Gigerenzer, 2004 ). Neyman and Pearson often worked with convenient alpha levels such as 5% (α = 0.05) and 1% (α = 0.01), although different levels can also be set. The main hypothesis can, thus, be written so as for incorporating the alpha level in its postulate (e.g., H M : M 1 –M 2 = 0 ± MES, α = 0.05), to be read as the probability level at which the main hypothesis will be rejected in favor of the alternative hypothesis.

Neyman-Pearson's α looks very similar to Fisher's sig. Indeed, Neyman and Pearson also called it the significance level of the test and used the same conventional cut-off points (5, 1%). However, this similarity is merely superficial on three accounts: α needs to be set a priori (not necessarily so under Fisher's approach); Neyman-Pearson's approach is not a test of significance (they are not interested in the strength of the evidence against H M ) but a test of acceptance (deciding whether to accept H A instead of H M ); and α does not admit gradation—i.e., you may choose, for example, either α = 0.05 or α = 0.01, but not both, for the same test (while with Fisher's approach you can have different levels of more extreme significance).

The critical region (CR test ) and critical value (CV test , Test crit ) of a test . The alpha level helps draw a critical region, or rejection region (Figure (Figure4), 4 ), on the probability distribution of the main hypothesis (Neyman and Pearson, 1928 ). Any research value that falls outside this critical region will be taken as reasonably probable under the main hypothesis, and any research result that falls within the critical region will be taken as most probable under the alternative hypothesis. The alpha level, thus, also helps identify the location of the critical value of such test, the boundary for deciding between hypotheses. Thus, once the critical value is known—see below—, the main hypothesis can also be written so as for incorporating such critical value, if so desired (e.g., H M : M 1 –M 2 = 0 ± MES, α = 0.05, CV t = 2.38).

Neyman-Pearson's critical region is very similar to the equivalent critical region you would obtain by using Fisher's sig as a cut-off point on a null distribution. However, this similarity is rather unimportant on three accounts: it is based on a critical value which delimits the region to reject H M in favor of H A , irrespective of the actual observed value of the test (Fisher, on the contrary, is more interested in the actual p-value of the research result); it is fixed a priori and, thus, rigid and immobile (Fisher's level of significance can be flexible—Macdonald, 2002 ); and it is non-gradable (with Fisher's approach, you may delimit several more extreme critical regions as areas of stronger evidence).

Step 4–Set up the alternative hypothesis (H A ) . One of the main innovations of Neyman-Pearson's approach was the consideration of alternative hypotheses (Neyman and Pearson, 1928 , 1933 ; Neyman, 1956 ). Unfortunately, the alternative hypothesis is often postulated in an unspecified manner (e.g., as H A : M 1 –M 2 ≠ 0), even by Neyman and Pearson themselves (Macdonald, 1997 ; Jones and Tukey, 2000 ). In practice, a fully specified alternative hypothesis (e.g., its mean and variance) is not necessary because this hypothesis only provides partial information to the testing of the main hypothesis (a.k.a., the effect size and β). Therefore, the alternative hypothesis is better written so as for incorporating the minimum effect size within its postulate (e.g., H A : M 1 –M 2 ≠ 0 ± MES). This way it is clear that values beyond such minimum effect size are the ones considered of research importance.

Neyman-Pearson's H A is often postulated as the negation of a nil hypothesis (H A : M 1 –M 2 ≠ 0), which is coherent with a simple postulate of H M (H M : M 1 –M 2 = 0). These simplified postulates are not accurate and are easily confused with Fisher's approach to data testing—H M resembles Fisher's H 0 , and H A resembles a mere negation of H 0 . However, merely negating H 0 does not make its negation a valid alternative hypothesis—otherwise Fisher would have put forward such alternative hypothesis, something which he was vehemently against (Hubbard, 2004 ). As discussed earlier, Neyman-Pearson's approach introduces the construct of effect size into their testing approach; thus, incorporating such construct in the specification of both H M and H A makes them more accurate, and less confusing, than their simplified versions.

Among things to consider when setting the alternative hypothesis are the expected effect size in the population (see above) and the Type II error you are prepared to commit.

Type II error . A Type II error (or error of the second class) is made every time the main hypothesis is wrongly retained (thus, every time H A is wrongly rejected). Making a Type II error is less critical than making a Type I error, yet you still want to minimize the probability of making this error once you have decided which alpha level to use (Neyman and Pearson, 1933 ; Neyman, 1942 ; Macdonald, 2002 ).

Beta (β) . Beta is the probability of committing a Type II error in the long run and is, therefore, a parameter of the alternative hypothesis (Figure (Figure4, 4 , Neyman, 1956 ). You want to make beta as small as possible, although not smaller than alpha (if β needed to be smaller than α, then H A should be your main hypothesis, instead!). Neyman and Pearson proposed 20% (β = 0.20) as an upper ceiling for beta, and the value of alpha (β = α) as its lower floor (Neyman, 1953 ). For symmetry with the main hypothesis, the alternative hypothesis can, thus, be written so as for incorporating the beta level in its postulate (e.g., H A : M 1 –M 2 ≠ 0 ± MES, β = 0.20).

Step 5–Calculate the sample size (N) required for good power (1–β) . Neyman-Pearson's approach is eminently a priori in order to ensure that the research to be done has good power (Neyman, 1942 , 1956 ; Pearson, 1955 ; Macdonald, 2002 ). Power is the probability of correctly rejecting the main hypothesis in favor of the alternative hypothesis (i.e., of correctly accepting H A ). It is the mathematical opposite of the Type II error (thus, 1–β; Macdonald, 1997 ; Hubbard, 2004 ). Power depends on the type of test selected (e.g., parametric tests and one-tailed tests increase power), as well as on the expected effect size (larger ES's increase power), alpha (larger α's increase power) and beta (smaller β 's increase power). A priori power is ensured by calculating the correct sample size given those parameters (Spielman, 1973 ). Because power is the opposite of beta, the lower floor for good power is, thus, 80% (1–β = 0.80), and its upper ceiling is 1–alpha (1–β = 1–α).

H A does not need to be tested under Neyman-Pearson's approach, only H M (Neyman and Pearson, 1928 , 1933 ; Neyman, 1942 ; Pearson, 1955 ; Spielman, 1973 ). Therefore, the procedure looks similar to Fisher's and, under similar circumstances (e.g., when using the same test and sample size), it will lead to the same results. The main difference between procedures is that Neyman-Pearson's H A provides explicit information to the test; that is, information about ES and β. If this information is not taken into account for designing a research project with adequate power, then, by default, you are carrying out a test under Fisher's approach.

Caution: For Neyman and Pearson, there is little justification in carrying out research projects with low power. When a research project has low power, Type II errors are too big, so it is less probable to reject H M in favor of H A , while, at the same time, it makes unreasonable to accept H M as the best explanation for the research results. If you face a research project with low a priori power, try the best compromise between its parameters (such as increasing α, relaxing β, settling for a larger ES, or using one-tailed tests; Neyman and Pearson, 1933 ). If all fails, consider Fisher's approach, instead.

Step 6–Calculate the critical value of the test (CV test , or Test crit ) . Some of above parameters (test, α and N) can be used for calculating the critical value of the test; that is, the value to be used as the cut-off point for deciding between hypotheses (Figure (Figure5, 5 , Neyman and Pearson, 1933 ).

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Neyman-Pearson's test in action: CV test is the point for deciding between hypotheses; it coincides with the cut-off points underlying α, β, and MES .

A posteriori steps

Step 7–Calculate the test value for the research (RV test ) . In order to carry out the test, some unknown parameters of the populations are estimated from the sample (e.g., variance), while other parameters are deduced theoretically (e.g., the distribution of frequencies under a particular statistical distribution). The statistical distribution so established thus represents the random variability that is theoretically expected for a statistical main hypothesis given a particular research sample, and provides information about the values expected at different locations under such distribution.

By applying the corresponding formula, the research value of the test (RV test ) is obtained. This value is closer to zero the closer the research data is to the mean of the main hypothesis; it gets larger the further away the research data is from the mean of the main hypothesis.

P-values can also be used for testing data when using Neyman-Pearson's approach, as testing data under H M is similar to testing data under Fisher's H 0 (Fisher, 1955 ). It implies calculating the theoretical probability of the research data under the distribution of H M — P ( D | H M ). Just be mindful that p-values go in the opposite way than RVs, with larger p-values being closer to H M and smaller p-values being further away from it.

Caution: Because of above equivalence, you may use p-values instead of CV test with Neyman-Pearson's approach. However, p-values need to be considered mere proxies under this approach and, thus, have no evidential properties whatsoever (Frick, 1996 ; Gigerenzer, 2004 ). For example, if working with a priori α = 0.05, p = 0.01 would lead you to reject H M at α = 0.05; however, it would be incorrect to reject it at α = 0.01 (i.e., α cannot be adjusted a posteriori), and it would be incorrect to conclude that you reject H M strongly (i.e., α cannot be gradated). If confused, you are better off sticking to CV test , and using p-values only with Fisher's approach.

Step 8–Decide in favor of either the main or the alternative hypothesis . Neyman-Pearson's approach is rather mechanical once the a priori steps have been satisfied (Neyman and Pearson, 1933 ; Neyman, 1942 , 1956 ; Spielman, 1978 ; Macdonald, 2002 ). Thus, the analysis is carried out as per the optimal test selected and the interpretation of results is informed by the mathematics of the test, following on the a priori pattern set up for deciding between hypotheses:

If the observed result falls within the critical region, reject the main hypothesis and accept the alternative hypothesis.
If the observed result falls outside the critical region and the test has good power, accept the main hypothesis.
If the observed result falls outside the critical region and the test has low power, conclude nothing. (Ideally, you would not carry out research with low power—Neyman, 1955 ).

Neyman-Pearson's approach leads to a decision between hypotheses (Neyman and Pearson, 1933 ; Spielman, 1978 ). In principle, this decision should be between rejecting H M or retaining H M (assuming good power), as the test is carried out on H M only (Neyman, 1942 ). In practice, it does not really make much difference whether you accept H M or H A , as appropriate (Macdonald, 1997 ). In fact, accepting either H M or H A is beneficial as it prevents confusion with Fisher's approach, which can only reject H 0 (Perezgonzalez, 2014 ).

Reporting the observed research test value is relevant under Neyman-Pearson's approach, as it serves to compare the observed value against the a priori critical value—e.g., t (64) = 3.31, 1-tailed > CV t = 2.38, thus accept H A . When using a p-value as a proxy for CV test , simply strip any evidential value off p—e.g., t (64) = 3.31, p < α, 1-tailed.

Neyman-Pearson's hypotheses are also assumed to be true. H M represents the probability distribution of the data given a true hypothesis—P(D|H M ), while H A represents the distribution of the data under an alternative true hypothesis—P(D|H A ), even when it is never tested. This means that H M and H A cannot be, at the same time false, nor proved or falsified a posteriori. The only way forward is to act as if the conclusion reached by the test was true—subject to a probability α or β of making a Type I or Type II error, respectively (Neyman and Pearson, 1933 ; Cortina and Dunlap, 1997 ).

Highlights of Neyman-Pearson's approach

More powerful . Neyman-Pearson's approach is more powerful than Fisher's for testing data in the long run (Williams et al., 2006 ). However, repeated sampling is rare in research (Fisher, 1955 ).

Better suited for repeated sampling projects . Because of above, Neyman-Pearson's approach is well-suited for repeated sampling research using the same population and tests, such as industrial quality control or large scale diagnostic testing (Fisher, 1955 ; Spielman, 1973 ).

Deductive . The approach is deductive and rather mechanical once the a priori steps have been set up (Neyman and Pearson, 1933 ; Neyman, 1942 ; Fisher, 1955 ).

Less flexible than Fisher's approach . Because most of the work is done a priori, this approach is less flexible for accommodating tests not thought of beforehand and for doing exploratory research (Macdonald, 2002 ).

Defaults easily to Fisher's approach . As this approach looks superficially similar to Fisher's, it is easy to confuse both and forget what makes Neyman-Pearson's approach unique (Lehman, 1993 ). If the information provided by the alternative hypothesis—ES and β —is not taken into account for designing research with good power, data analysis defaults to Fisher's test of significance.

Null hypothesis significance testing

NHST is the most common procedure used for testing data nowadays, albeit under the false assumption of testing substantive hypotheses (Carver, 1978 ; Nickerson, 2000 ; Hubbard, 2004 ; Hager, 2013 ). NHST is, in reality, an amalgamation of Fisher's and Neyman-Pearson's theories, offered as a seamless approach to testing (Macdonald, 2002 ; Gigerenzer, 2004 ). It is not a clearly defined amalgamation either and, depending on the author describing it or on the researcher using it, it may veer more toward Fisher's approach (e.g., American Psychological Association, 2010 ; Nunnally, 1960 ; Wilkinson and the Task Force on Statistical Inference, 1999 ; Krueger, 2001 ) or toward Neyman-Pearson's approach (e.g., Cohen, 1988 ; Rosnow and Rosenthal, 1989 ; Frick, 1996 ; Schmidt, 1996 ; Cortina and Dunlap, 1997 ; Wainer, 1999 ; Nickerson, 2000 ; Kline, 2004 ).

Unfortunately, if we compare Fisher's and Neyman-Pearson's approaches vis-à-vis, we find that they are incompatible in most accounts (Table (Table1). 1 ). Overall, however, most amalgamations follow Neyman-Pearson procedurally but Fisher philosophically (Spielman, 1978 ; Johnstone, 1986 ; Cortina and Dunlap, 1997 ; Hubbard, 2004 ).

Equivalence of constructs in Fisher's and Neyman-Pearson's theories, and amalgamation of constructs under NHST .

NHST is not only ubiquitous but very well ingrained in the minds and current practice of most researchers, journal editors and publishers (Spielman, 1978 ; Gigerenzer, 2004 ; Hubbard, 2004 ), especially in the biological sciences (Lovell, 2013 ; Ludbrook, 2013 ), social sciences (Frick, 1996 ), psychology (Nickerson, 2000 ; Gigerenzer, 2004 ) and education (Carver, 1978 , 1993 ). Indeed, most statistics textbooks for those disciplines still teach NHST rather than the two approaches of Fisher and of Neyman and Pearson as separate and rather incompatible theories (e.g., Dancey and Reidy, 2014 ). NHST has also the (false) allure of being presented as a procedure for testing substantive hypotheses (Macdonald, 2002 ; Gigerenzer, 2004 ).

In the situations in which they are most often used by researchers, and assuming the corresponding parameters are also the same, both Fisher's and Neyman-Pearson's theories work with the same statistical tools and produce the same statistical results; therefore, by extension, NHST also works with the same statistical tools and produces the same results—in practice, however, both approaches start from different starting points and lead to different outcomes (Fisher, 1955 ; Spielman, 1978 ; Berger, 2003 ). In a nutshell, the differences between Fisher's and Neyman-Pearson's theories are mostly about research philosophy and about how to interpret results (Fisher, 1955 ).

The most coherent plan of action is, of course, to follow the theory which is most appropriate for purpose, be this Fisher's or Neyman-Pearson's. It is also possible to use both for achieving different goals within the same research project (e.g., Neyman-Pearson's for tests thought of a priori, and Fisher's for exploring the data further, a posteriori), pending that those goals are not mixed up.

However, the apparent parsimony of NHST and its power to withstand threats to its predominance are also understandable. Thus, I propose two practical solutions to improve NHST: the first a compromise to improve Fisher-leaning NHST, the second a compromise to improve Neyman-Pearson-leaning NHST. A computer program such as G * Power can be used for implementing the recommendations made for both.

Improving fisher-leaning NHST

Fisher's is the closest approach to NHST; it is also the philosophy underlying common statistics packages, such as SPSS. Furthermore, because using Neyman-Pearson's concepts within NHST may be irrelevant or inelegant but hardly damaging, it requires little re-engineering. A clear improvement to NHST comes from incorporating Neyman-Pearson's constructs of effect size and of a priori sample estimation for adequate power. Estimating effect sizes (both a priori and a posteriori) ensures that researchers consider importance over mere statistical significance. A priori estimation of sample size for good power also ensures that the research has enough sensitiveness for capturing the expected effect size (Huberty, 1987 ; Macdonald, 2002 ).

Improving Neyman-Pearson-leaning NHST

NHST is particularly damaging for Neyman-Pearson's approach, simply because the latter defaults to Fisher's if important constructs are not used correctly. An importantly damaging issue is the assimilation of p -values as evidence of Type I errors and the subsequent correction of alphas to match such p -values (roving α's, Goodman, 1993 ; Hubbard, 2004 ). The best compromise for improving NHST under these circumstances is to compensate a posteriori roving alphas with a posteriori roving betas (or, if so preferred, with a posteriori roving power). Basically, if you are adjusting alpha a posteriori (roving α) to reflect both the strength of evidence (sig) and the long-run Type I error (α), you should also adjust the long-run probability of making a Type II error (roving β). Report both roving alphas and roving betas for each test, and take them into account when interpreting your research results.

NHST is very controversial, even if the controversy is not well known. A sample of helpful readings on this controversy are Christensen ( 2005 ); Hubbard ( 2004 ); Gigerenzer ( 2004 ); Goodman ( 1999 ), Louçã ( 2008 , http://www.iseg.utl.pt/departamentos/economia/wp/wp022008deuece.pdf ), Halpin and Stam ( 2006 ); Huberty ( 1993 ); Johnstone ( 1986 ), and Orlitzky ( 2012 ).

Data testing procedures represented a historical advancement for the so-called “softer” sciences, starting in biology but quickly spreading to psychology, the social sciences and education. These disciplines benefited from the principles of experimental design, the rejection of subjective probabilities and the application of statistics to small samples that Sir Ronald Fisher started popularizing in 1922 (Lehmann, 2011 ), under the umbrella of his tests of significance (e.g., Fisher, 1954 ). Two mathematical contemporaries, Jerzy Neyman and Egon Sharpe Pearson, attempted to improve Fisher's procedure and ended up developing a new theory, one for deciding between competing hypotheses (Neyman and Pearson, 1928 ), more suitable to quality control and large scale diagnostic testing (Spielman, 1973 ). Both theories had enough similarities to be easily confused (Perezgonzalez, 2014 ), especially by those less epistemologically inclined; a confusion fiercely opposed by the original authors (e.g., Fisher, 1955 )—and ever since (e.g., Nickerson, 2000 ; Lehmann, 2011 ; Hager, 2013 )—but something that irreversibly happened under the label of null hypothesis significance testing. NHST is an incompatible amalgamation of the theories of Fisher and of Neyman and Pearson (Gigerenzer, 2004 ). Curiously, it is an amalgamation that is technically reassuring despite it being, philosophically, pseudoscience. More interestingly, the numerous critiques raised against it for the past 80 years have not only failed to debunk NHST from the researcher's statistical toolbox, they have also failed to be widely known, to find their way into statistics manuals, to be edited out of journal submission requirements, and to be flagged up by peer-reviewers (e.g., Gigerenzer, 2004 ). NHST effectively negates the benefits that could be gained from Fisher's and from Neyman-Pearson's theories; it also slows scientific progress (Savage, 1957 ; Carver, 1978 , 1993 ) and may be fostering pseudoscience. The best option would be to ditch NHST altogether and revert to the theories of Fisher and of Neyman-Pearson as—and when—appropriate. For everything else, there are alternative tools, among them exploratory data analysis (Tukey, 1977 ), effect sizes (Cohen, 1988 ), confidence intervals (Neyman, 1935 ), meta-analysis (Rosenthal, 1984 ), Bayesian applications (Dienes, 2014 ) and, chiefly, honest critical thinking (Fisher, 1960 ).

Conflict of interest statement

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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directional and non-directional hypothesis in survey

Directional vs Non-Directional Hypothesis – Collect Feedback More Effectively

To conduct a perfect survey, you should know the basics of good research . That’s why in Startquestion we would like to share with you our knowledge about basic terms connected to online surveys and feedback gathering . Knowing the basis you can create surveys and conduct research in more effective ways and thanks to this get meaningful feedback from your customers, employees, and users. That’s enough for the introduction – let’s get to work. This time we will tell you about the hypothesis .

What is a Hypothesis?

A Hypothesis can be described as a theoretical statement built upon some evidence so that it can be tested as if it is true or false. In other words, a hypothesis is a speculation or an idea, based on insufficient evidence that allows it further analysis and experimentation.

The purpose of a hypothetical statement is to work like a prediction based on studied research and to provide some estimated results before it ha happens in a real position. There can be more than one hypothesis statement involved in a research study, where you need to question and explore different aspects of a proposed research topic. Before putting your research into directional vs non-directional hypotheses, let’s have some basic knowledge.

Most often, a hypothesis describes a relation between two or more variables. It includes:

An Independent variable – One that is controlled by the researcher

Dependent Variable – The variable that the researcher observes in association with the Independent variable.

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How to write an effective Hypothesis?

To write an effective hypothesis follow these essential steps.

Inquire a Question

The very first step in writing an effective hypothesis is raising a question. Outline the research question very carefully keeping your research purpose in mind. Build it in a precise and targeted way. Here you must be clear about the research question vs hypothesis. A research question is the very beginning point of writing an effective hypothesis.

Do Literature Review

Once you are done with constructing your research question, you can start the literature review. A literature review is a collection of preliminary research studies done on the same or relevant topics. There is a diversified range of literature reviews. The most common ones are academic journals but it is not confined to that. It can be anything including your research, data collection, and observation.

At this point, you can build a conceptual framework. It can be defined as a visual representation of the estimated relationship between two variables subjected to research.

Frame an Answer

After a collection of literature reviews, you can find ways how to answer the question. Expect this stage as a point where you will be able to make a stand upon what you believe might have the exact outcome of your research. You must formulate this answer statement clearly and concisely.

Build a Hypothesis

At this point, you can firmly build your hypothesis. By now, you knew the answer to your question so make a hypothesis that includes:

Applicable Variables
Particular Group being Studied (Who/What)
Probable Outcome of the Experiment

Remember, your hypothesis is a calculated assumption, it has to be constructed as a sentence, not a question. This is where research question vs hypothesis starts making sense.

Refine a Hypothesis

Make necessary amendments to the constructed hypothesis keeping in mind that it has to be targeted and provable. Moreover, you might encounter certain circumstances where you will be studying the difference between one or more groups. It can be correlational research. In such instances, you must have to testify the relationships that you believe you will find in the subject variables and through this research.

Build Null Hypothesis

Certain research studies require some statistical investigation to perform a data collection. Whenever applying any scientific method to construct a hypothesis, you must have adequate knowledge of the Null Hypothesis and an Alternative hypothesis.

Null Hypothesis:

A null Hypothesis denotes that there is no statistical relationship between the subject variables. It is applicable for a single group of variables or two groups of variables. A Null Hypothesis is denoted as an H0. This is the type of hypothesis that the researcher tries to invalidate. Some of the examples of null hypotheses are:

– Hyperactivity is not associated with eating sugar.

– All roses have an equal amount of petals.

– A person’s preference for a dress is not linked to its color.

Alternative Hypothesis:

An alternative hypothesis is a statement that is simply inverse or opposite of the null hypothesis and denoted as H1. Simply saying, it is an alternative statement for the null hypothesis. The same examples will go this way as an alternative hypothesis:

– Hyperactivity is associated with eating sugar.

– All roses do not have an equal amount of petals.

– A person’s preference for a dress is linked to its color.

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Types of Hypothesis

Apart from null and alternative hypotheses, research hypotheses can be categorized into different types. Let’s have a look at them:

Simple Hypothesis:

This type of hypothesis is used to state a relationship between a particular independent variable and only a dependent variable.

Complex Hypothesis:

A statement that states the relationship between two or more independent variables and two or more dependent variables, is termed a complex hypothesis.

Associative and Causal Hypothesis:

This type of hypothesis involves predicting that there is a point of interdependency between two variables. It says that any kind of change in one variable will cause a change in the other one. Similarly, a casual hypothesis says that a change in the dependent variable is due to some variations in the independent variable.

Directional vs non-directional hypothesis

Directional hypothesis:.

A hypothesis that is built upon a certain directional relationship between two variables and constructed upon an already existing theory, is called a directional hypothesis. To understand more about what is directional hypothesis here is an example, Girls perform better than boys (‘better than’ shows the direction predicted)

Non-directional Hypothesis:

It involves an open-ended non-directional hypothesis that predicts that the independent variable will influence the dependent variable; however, the nature or direction of a relationship between two subject variables is not defined or clear.

For Example, there will be a difference in the performance of girls & boys (Not defining what kind of difference)

As a professional, we suggest you apply a non-directional alternative hypothesis when you are not sure of the direction of the relationship. Maybe you’re observing potential gender differences on some psychological test, but you don’t know whether men or women would have the higher ratio. Normally, this would say that you are lacking practical knowledge about the proposed variables. A directional test should be more common for tests.

Author: Ula Kamburov-Niepewna

Updated: 18 November 2022

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Two Tailed Hypothesis

In the vast realm of scientific inquiry, the two-tailed hypothesis holds a special place, serving as a compass for researchers exploring possibilities in two opposing directions. Instead of predicting a specific direction of the relationship between variables, it remains open to outcomes on both ends of the spectrum. Understanding how to craft such a hypothesis, enriched with insights and nuances, can elevate the robustness of one’s research. Delve into its world, discover thesis statement examples, learn the art of its formulation, and grasp tips to master its intricacies.

What is Two Tailed Hypothesis? – Definition

A two-tailed hypothesis, also known as a non-directional hypothesis , is a type of hypothesis used in statistical testing that predicts a relationship between variables without specifying the direction of the relationship. In other words, it tests for the possibility of the relationship in both directions. This approach is used when a researcher believes there might be a difference due to the experiment but doesn’t have enough preliminary evidence or basis to predict a specific direction of that difference.

What is an example of a Two Tailed hypothesis statement?

Let’s consider a study on the impact of a new teaching method on student performance:

Hypothesis Statement : The new teaching method will have an effect on student performance.

Notice that the hypothesis doesn’t specify whether the effect will be positive or negative (i.e., whether student performance will improve or decline). It’s open to both possibilities, making it a two-tailed hypothesis.

Two Tailed Hypothesis Statement Examples

The two-tailed hypothesis, an essential tool in research, doesn’t predict a specific directional outcome between variables. Instead, it posits that an effect exists, without specifying its nature. This approach offers flexibility, as it remains open to both positive and negative outcomes. Below are various examples from diverse fields to shed light on this versatile research method. You may also be interested to browse through our other one-tailed hypothesis .

Sleep and Cognitive Ability : Sleep duration affects cognitive performance in adults.
Dietary Fiber and Digestion : Consumption of dietary fiber influences digestion rates.
Exercise and Stress Levels : Engaging in physical activity impacts stress levels.
Vitamin C and Immunity : Intake of Vitamin C has an effect on immunity strength.
Noise Levels and Concentration : Ambient noise levels influence individual concentration ability.
Artificial Sweeteners and Appetite : Consumption of artificial sweeteners affects appetite.
UV Light and Skin Health : Exposure to UV light influences skin health.
Coffee Intake and Sleep Quality : Consuming coffee has an effect on sleep quality.
Air Pollution and Respiratory Issues : Levels of air pollution impact respiratory health.
Meditation and Blood Pressure : Practicing meditation affects blood pressure readings.
Pet Ownership and Loneliness : Having a pet influences feelings of loneliness.
Green Spaces and Mental Wellbeing : Exposure to green spaces impacts mental health.
Music Tempo and Heart Rate : Listening to music of varying tempos affects heart rate.
Chocolate Consumption and Mood : Eating chocolate has an effect on mood.
Social Media Usage and Self-Esteem : The frequency of social media usage influences self-esteem.
E-reading and Eye Strain : Using e-readers affects eye strain levels.
Vegan Diets and Energy Levels : Following a vegan diet influences daily energy levels.
Carbonated Drinks and Tooth Decay : Consumption of carbonated drinks has an effect on tooth decay rates.
Distance Learning and Student Engagement : Engaging in distance learning impacts student involvement.
Organic Foods and Health Perceptions : Consuming organic foods influences perceptions of health.
Urban Living and Stress Levels : Living in urban environments affects stress levels.
Plant-Based Diets and Cholesterol : Adopting a plant-based diet impacts cholesterol levels.
Virtual Reality Training and Skill Acquisition : Using virtual reality for training influences the rate of skill acquisition.
Video Game Play and Hand-Eye Coordination : Playing video games has an effect on hand-eye coordination.
Aromatherapy and Sleep Quality : Using aromatherapy impacts the quality of sleep.
Bilingualism and Cognitive Flexibility : Being bilingual affects cognitive flexibility.
Microplastics and Marine Health : The presence of microplastics in oceans influences marine organism health.
Yoga Practice and Joint Health : Engaging in yoga has an effect on joint health.
Processed Foods and Metabolism : Consuming processed foods impacts metabolic rates.
Home Schooling and Social Skills : Being homeschooled influences the development of social skills.
Smartphone Usage and Attention Span : Regular smartphone use affects attention spans.
E-commerce and Consumer Trust : Engaging with e-commerce platforms influences levels of consumer trust.
Work-from-Home and Productivity : The practice of working from home has an effect on productivity levels.
Classical Music and Plant Growth : Exposing plants to classical music impacts their growth rate.
Public Transport and Community Engagement : Using public transport influences community engagement levels.
Digital Note-taking and Memory Retention : Taking notes digitally affects memory retention.
Acoustic Music and Relaxation : Listening to acoustic music impacts feelings of relaxation.
GMO Foods and Public Perception : Consuming GMO foods influences public perception of food safety.
LED Lights and Eye Comfort : Using LED lights affects visual comfort.
Fast Fashion and Consumer Satisfaction : Engaging with fast fashion brands influences consumer satisfaction levels.
Diverse Teams and Innovation : Working in diverse teams impacts the level of innovation.
Local Produce and Nutritional Value : Consuming local produce affects its nutritional value.
Podcasts and Language Acquisition : Listening to podcasts influences the speed of language acquisition.
Augmented Reality and Learning Efficiency : Using augmented reality in education has an effect on learning efficiency.
Museums and Historical Interest : Visiting museums impacts interest in history.
E-books vs. Physical Books and Reading Retention : The type of book, whether e-book or physical, affects memory retention from reading.
Biophilic Design and Worker Well-being : Implementing biophilic designs in office spaces influences worker well-being.
Recycled Products and Consumer Preference : Using recycled materials in products impacts consumer preferences.
Interactive Learning and Critical Thinking : Engaging in interactive learning environments affects the development of critical thinking skills.
High-Intensity Training and Muscle Growth : Participating in high-intensity training has an effect on muscle growth rate.
Pet Therapy and Anxiety Levels : Engaging with therapy animals influences anxiety levels.
3D Printing and Manufacturing Efficiency : Implementing 3D printing in manufacturing affects production efficiency.
Electric Cars and Public Adoption Rates : Introducing more electric cars impacts the rate of public adoption.
Ancient Architectural Study and Modern Design Inspiration : Studying ancient architecture influences modern design inspirations.
Natural Lighting and Productivity : The amount of natural lighting in a workspace affects worker productivity.
Streaming Platforms and Traditional TV Viewing : The rise of streaming platforms has an effect on traditional TV viewing habits.
Handwritten Notes and Conceptual Understanding : Taking notes by hand influences the depth of conceptual understanding.
Urban Farming and Community Engagement : Implementing urban farming practices impacts levels of community engagement.
Influencer Marketing and Brand Loyalty : Collaborating with influencers affects brand loyalty among consumers.
Online Workshops and Skill Enhancement : Participating in online workshops influences skill enhancement.
Virtual Reality and Empathy Development : Using virtual reality experiences influences the development of empathy.
Gardening and Mental Well-being : Engaging in gardening activities affects overall mental well-being.
Drones and Wildlife Observation : The use of drones impacts the accuracy of wildlife observations.
Artificial Intelligence and Job Markets : The introduction of artificial intelligence in industries has an effect on job availability.
Online Reviews and Purchase Decisions : Reading online reviews influences purchase decisions for consumers.
Blockchain Technology and Financial Security : Implementing blockchain technology affects financial transaction security.
Minimalism and Life Satisfaction : Adopting a minimalist lifestyle influences levels of life satisfaction.
Microlearning and Long-term Retention : Engaging in microlearning practices impacts long-term information retention.
Virtual Teams and Communication Efficiency : Operating in virtual teams has an effect on the efficiency of communication.
Plant Music and Growth Rates : Exposing plants to specific music frequencies influences their growth rates.
Green Building Practices and Energy Consumption : Implementing green building designs affects overall energy consumption.
Fermented Foods and Gut Health : Consuming fermented foods impacts gut health.
Digital Art Platforms and Creative Expression : Using digital art platforms influences levels of creative expression.
Aquatic Therapy and Physical Rehabilitation : Engaging in aquatic therapy has an effect on the rate of physical rehabilitation.
Solar Energy and Utility Bills : Adopting solar energy solutions influences monthly utility bills.
Immersive Theatre and Audience Engagement : Experiencing immersive theatre performances affects audience engagement levels.
Podcast Popularity and Radio Listening Habits : The rise in podcast popularity impacts traditional radio listening habits.
Vertical Farming and Crop Yield : Implementing vertical farming techniques has an effect on crop yields.
DIY Culture and Craftsmanship Appreciation : The rise of DIY culture influences public appreciation for craftsmanship.
Crowdsourcing and Solution Innovation : Utilizing crowdsourcing methods affects the innovativeness of solutions derived.
Urban Beekeeping and Local Biodiversity : Introducing urban beekeeping practices impacts local biodiversity levels.
Digital Nomad Lifestyle and Work-Life Balance : Adopting a digital nomad lifestyle affects perceptions of work-life balance.
Virtual Tours and Tourism Interest : Offering virtual tours of destinations influences interest in real-life visits.
Neurofeedback Training and Cognitive Abilities : Engaging in neurofeedback training has an effect on various cognitive abilities.
Sensory Gardens and Stress Reduction : Visiting sensory gardens impacts levels of stress reduction.
Subscription Box Services and Consumer Spending : The popularity of subscription box services influences overall consumer spending patterns.
Makerspaces and Community Collaboration : Introducing makerspaces in communities affects collaboration levels among members.
Remote Work and Company Loyalty : Adopting long-term remote work policies impacts employee loyalty towards the company.
Upcycling and Environmental Awareness : Engaging in upcycling activities influences levels of environmental awareness.
Mixed Reality in Education and Engagement : Implementing mixed reality tools in education affects student engagement.
Microtransactions in Gaming and Player Commitment : The presence of microtransactions in video games impacts player commitment and longevity.
Floating Architecture and Sustainable Living : Adopting floating architectural solutions influences perceptions of sustainable living.
Edible Packaging and Waste Reduction : Introducing edible packaging in markets has an effect on overall waste reduction.
Space Tourism and Interest in Astronomy : The advent of space tourism influences the general public’s interest in astronomy.
Urban Green Roofs and Air Quality : Implementing green roofs in urban settings impacts the local air quality.
Smart Mirrors and Fitness Consistency : Using smart mirrors for workouts affects consistency in fitness routines.
Open Source Software and Technological Innovation : Promoting open-source software has an effect on the rate of technological innovation.
Microgreens and Nutrient Intake : Consuming microgreens influences nutrient intake.
Aquaponics and Sustainable Farming : Implementing aquaponic systems impacts perceptions of sustainable farming.
Esports Popularity and Physical Sport Engagement : The rise of esports affects engagement in traditional physical sports.

Two Tailed Hypothesis Statement Examples in Research

In academic research, a two-tailed hypothesis is versatile, not pointing to a specific direction of effect but remaining open to outcomes on both ends of the spectrum. Such hypothesis aim to determine if a particular variable affects another, without specifying how. Here are examples tailored to research scenarios.

Interdisciplinary Collaboration and Innovation : Engaging in interdisciplinary collaborations impacts the degree of innovation in research findings.
Open Access Journals and Citation Rates : Publishing in open-access journals influences the citation rates of the papers.
Research Grants and Publication Quality : Receiving larger research grants affects the quality of resulting publications.
Laboratory Environment and Data Accuracy : The physical conditions of a research laboratory impact the accuracy of experimental data.
Peer Review Process and Research Integrity : The stringency of the peer review process influences the overall integrity of published research.
Researcher Mobility and Knowledge Transfer : The mobility of researchers between institutions affects the rate of knowledge transfer.
Interdisciplinary Conferences and Networking Opportunities : Attending interdisciplinary conferences impacts the depth and breadth of networking opportunities.
Qualitative Methods and Research Depth : Incorporating qualitative methods in research affects the depth of findings.
Data Visualization Tools and Research Comprehension : Utilizing advanced data visualization tools influences the comprehension of complex research data.
Collaborative Tools and Research Efficiency : The adoption of modern collaborative tools impacts research efficiency and productivity.

Two Tailed Testing Hypothesis Statement Examples

In hypothesis testing , a two-tailed test examines the possibility of a relationship in both directions. Unlike one-tailed tests, it doesn’t anticipate a specific direction of the relationship. The following are examples that encapsulate this approach within varied testing scenarios.

Load Testing and Website Speed : Conducting load testing on a website influences its loading speed.
A/B Testing and Conversion Rates : Implementing A/B testing affects the conversion rates of a webpage.
Drug Efficacy Testing and Patient Recovery : Testing a new drug’s efficacy impacts patient recovery rates.
Usability Testing and User Engagement : Conducting usability testing on an app influences user engagement metrics.
Genetic Testing and Disease Prediction : Utilizing genetic testing affects the accuracy of disease prediction.
Water Quality Testing and Contaminant Levels : Performing water quality tests influences our understanding of contaminant levels.
Battery Life Testing and Device Longevity : Conducting battery life tests impacts claims about device longevity.
Product Safety Testing and Recall Rates : Implementing rigorous product safety tests affects the rate of product recalls.
Emissions Testing and Pollution Control : Undertaking emissions testing on vehicles influences pollution control measures.
Material Strength Testing and Product Durability : Testing the strength of materials affects predictions about product durability.

How do you know if a hypothesis is two-tailed?

To determine if a hypothesis is two-tailed, you must look at the nature of the prediction. A two-tailed hypothesis is neutral concerning the direction of the predicted relationship or difference between groups. It simply predicts a difference or relationship without specifying whether it will be positive, negative, greater, or lesser. The hypothesis tests for effects in both directions.

What is one-tailed and two-tailed Hypothesis test with example?

In hypothesis testing, the choice between a one-tailed and a two-tailed test is determined by the nature of the research question.

One-tailed hypothesis: This tests for a specific direction of the effect. It predicts the direction of the relationship or difference between groups. For example, a one-tailed hypothesis might state: “The new drug will reduce symptoms more effectively than the standard treatment.”

Two-tailed hypothesis: This doesn’t specify the direction. It predicts that there will be a difference, but it doesn’t forecast whether the difference will be positive or negative. For example, a two-tailed hypothesis might state: “The new drug will have a different effect on symptoms compared to the standard treatment.”

What is a two-tailed hypothesis in psychology?

In psychology, a two-tailed hypothesis is frequently used when researchers are exploring new areas or relationships without a strong prior basis to predict the direction of findings. For instance, a psychologist might use a two-tailed hypothesis to explore whether a new therapeutic method has different outcomes than a traditional method, without predicting whether the outcomes will be better or worse.

What does a two-tailed alternative hypothesis look like?

A two-tailed alternative hypothesis is generally framed to show that a parameter is simply different from a certain value, without specifying the direction of the difference. Using mathematical notation, for a population mean (μ) and a proposed value (k), the two-tailed hypothesis would look like: H1: μ ≠ k.

How do you write a Two-Tailed hypothesis statement? – A Step by Step Guide

Identify the Variables: Start by identifying the independent and dependent variables you want to study.
Formulate a Relationship: Consider the potential relationship between these variables without setting a direction.
Avoid Directional Language: Words like “increase”, “decrease”, “more than”, or “less than” should be avoided as they point to a one-tailed hypothesis.
Keep it Simple: The statement should be clear, concise, and to the point.
Use Neutral Language: For instance, words like “affects”, “influences”, or “has an impact on” can be used to indicate a relationship without specifying a direction.
Finalize the Statement: Once the relationship is clear in your mind, form a coherent sentence that describes the relationship between your variables.

Tips for Writing Two Tailed Hypothesis

Start Broad: Given that you’re not seeking a specific direction, it’s okay to start with a broad idea.
Be Objective: Avoid letting any biases or expectations shape your hypothesis.
Stay Informed: Familiarize yourself with existing research on the topic to ensure your hypothesis is novel and not inadvertently directional.
Seek Feedback: Share your hypothesis with colleagues or mentors to ensure it’s indeed non-directional.
Revisit and Refine: As with any research process, be open to revisiting and refining your hypothesis as you delve deeper into the literature or collect preliminary data.

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Z-8: Two-Sample and Directional Hypothesis Testing

This lesson describes some refinements to the hypothesis testing approach that was introduced in the previous lesson. The truth of the matter is that the previous lesson was somewhat oversimplified in order to focus on the concept and general steps in the hypothesis testing procedure. With that background, we can now get into some of the finer points of hypothesis testing.

EdD, Assistant Professor Clinical Laboratory Science Program, University of Louisville Louisville, Kentucky October 1999

Refinements in calculations, refinements in procedure, a two-tailed test for our mice, a one-tailed test for our mice.

Critical t-values for one-tailed and two-tailed tests
About the Author

Two sample case

The "two sample case" is a special case in which the difference between two sample means is examined. This sounds like what we did in the last lesson, but we actually looked at the difference between an observed or sample group mean and a control group mean, which was treated as if it were a population mean (rather than an observed or sample mean). There are some fine points that need to be considered about the calculations and the procedure.

Mathematically, there are some differences in the formula that should be used. Recall the form of the equation for calculating a t-value:

where Xbar is the mean of the experimental group, and µ is a population parameter or the mean of the population. In the last lesson, we substituted the control group mean XbarB for µ. However, the control group was actually a sample from the population of all such mice (in this example) and following suit, the experimental group was also just a sample from its population. Phrasing the situation this way, there are now two sets of differences that must be considered: the difference between sample means or XbarA-XbarB and the difference between population means from which these samples came or µA-µB. Those expressions should be substituted into the t calc formula to give the proper mathematical form:

In this new equation, the difference between means (Xbar A - XbarB) is called bias and is important in determining test accuracy, so even though this discussion is getting more complicated, the statistic that we are deriving is very important.
When the square root is taken, the result is called the standard deviation of the difference between means , SDd , which is shown by the formula:
The proper equation for calculating the t-value for the two-sample case then becomes:

Remember the steps for testing a hypothesis are: (1) State the hypotheses; (2) Set the criterion for rejection of Ho; (3) Compute the test statistic; (4) Decide about Ho.

The null hypothesis can be stated as: Ho: µA = µB or µA - µB = 0. But it may be more revealing to say Ho: (XbarA-XbarB) - (µA - µB) = 0. The difference between the sample means minus the difference between the population means equals zero.
The alternative hypothesis can be stated as: Ha: µA is not equal to µB or Ha: (XbarA-XbarB) - (µA - µB) is not equal to 0, i.e., the means of the two groups are not equal.
The criteria for rejection or the alpha level is 0.05.
The test statistic is computed as:
Since it is hypothesized that the two methods are comparable and the difference between the means of the two populations is zero (µA - µB = 0), the calculation can be simplified as follows:
If this t calc is greater than 1.96 then the null hypothesis of no difference can be overturned (p0.05).
Even though we have used several different mathematical formulae, the interpretations are the same as before.

Directional hypothesis testing vs non-directional testing

Remember the example of testing the effect of antibiotics on mice in Lesson 7. The point of the study was to find out if the mice who were treated with the antibiotic would outlive those who were not treated (i.e., the control group). Are you surprised that the researcher did not hypothesize that the control group might outlive the treatment group? Would it make any difference in how the hypothesis testing were carried out? These questions raise the issue of directional testing, or one-tailed vs two-tailed tests.

The issue of two-side vs one-side tests becomes important when selecting the critical t-value. In the earlier discussion of this example, the alpha level was set to 0.05, but that 0.05 was actually divided equally between the left and right tails of the distribution curve. The condition being tested is that group A has a "different" life span as compared to group B, which represents a two-tailed test as illustrated in Figure 8-2.

Critical t-values for one-tailed and two tailed tests

This can be confusing and it should be helpful to think about having one or two gates on the curve. For example, for an alpha level of 0.05 and a two-tail test, there are two gates - one at -1.96 and one at +1.96. If you "walk" out of either of those gates, then you have demonstrated significance at p=0.05 or less. For a one-tail test, there is only one gate at +1.65. If you "walk" out of this gate, you have demonstrated significance at p=0.05 or less.

To reiterate, if you are standing right at the gate (1.96) for a two-tail test, then you have just barely met the p=0.05 requirement. However, if you are standing at the 1.96 point when running a one-tail test, then you have already exceeded the 1.65 gate and the probability must be even more significant, say p=0.025. It's important to find the critical t-value that is correct for the intended directional nature of the test.

About the author: Madelon F. Zady

Madelon F. Zady is an Assistant Professor at the University of Louisville, School of Allied Health Sciences Clinical Laboratory Science program and has over 30 years experience in teaching. She holds BS, MAT and EdD degrees from the University of Louisville, has taken other advanced course work from the School of Medicine and School of Education, and also advanced courses in statistics. She is a registered MT(ASCP) and a credentialed CLS(NCA) and has worked part-time as a bench technologist for 14 years. She is a member of the: American Society for Clinical Laboratory Science, Kentucky State Society for Clinical Laboratory Science, American Educational Research Association, and the National Science Teachers Association. Her teaching areas are clinical chemistry and statistics. Her research areas are metacognition and learning theory.

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Non-Directional Hypothesis

Non-directional hypotheses, also known as null hypotheses, serve as a foundational concept in scientific research across various fields, from psychology and biology to economics and sociology. These hypotheses are essential for hypothesis testing, where researchers aim to determine whether observed data supports or contradicts their predictions. In contrast to directional hypotheses, which specify the expected direction of a relationship , non-directional hypotheses do not make explicit predictions about the direction of the relationship between variables. Instead, they focus on testing for the mere presence or absence of a relationship .

Table of Contents

Understanding Non-Directional Hypotheses

A hypothesis is a statement that outlines a testable prediction about the relationship between variables in a research study. Hypotheses are fundamental to the scientific method, guiding the design of experiments and the analysis of data. While hypotheses can take various forms, including directional hypotheses, non-directional hypotheses, or null hypotheses, serve a specific purpose in hypothesis testing.

Non-directional hypotheses , often referred to as null hypotheses, do not make explicit predictions about the direction of the relationship between variables. Instead, they state that there is no significant relationship , difference, or effect. Researchers use non-directional hypotheses when they do not have a specific expectation regarding the outcome of their study or when they aim to test for the mere presence or absence of an effect.

In summary, non-directional hypotheses:

Do not predict the direction of the relationship between variables.
State that there is no significant relationship , difference, or effect.
Are used when researchers do not have specific expectations or when they want to test for the presence or absence of an effect.

The Structure of Non-Directional Hypotheses

Non-directional hypotheses typically consist of two main components: the independent variable (IV) and the dependent variable (DV). The IV is the variable that researchers manipulate or examine for its effect on the DV, which is the variable that researchers measure or observe. Here is the general structure of a non-directional hypothesis:

“There is no significant relationship /difference/effect between [IV] and [DV].”

“There is no” indicates the absence of a predicted effect.
“Significant” implies that the absence of the effect will be tested statistically.
“[IV]” represents the independent variable.
“and” connects the independent variable to the dependent variable.
“[DV]” represents the dependent variable.

Let’s break down the structure with a few examples:

“There is no significant relationship between the amount of rainfall and crop yield.”
“There is no significant difference in blood pressure between participants who take Drug A and those who take a placebo.”
“There is no significant effect of gender on test performance .”

Significance and Advantages of Non-Directional Hypotheses

Non-directional hypotheses offer several advantages in scientific research:

1. Objectivity:

Non-directional hypotheses are objective and neutral statements that do not impose specific expectations on the outcomes of a study. This objectivity is particularly important when researchers have limited prior knowledge about the variables being studied.

2. Flexibility:

Non-directional hypotheses are versatile and can be used in a wide range of research scenarios. They are not constrained by the need to predict a specific direction of the relationship , making them suitable for exploratory or preliminary research.

3. Rigorous Testing:

Non-directional hypotheses undergo rigorous statistical testing to determine whether the observed results are statistically significant. This testing provides a rigorous assessment of the presence or absence of effects.

4. Comparison with Alternative Hypotheses:

Non-directional hypotheses can be compared with alternative hypotheses, including directional hypotheses, to evaluate which hypothesis best fits the observed data. This process helps researchers refine their theories and hypotheses.

5. Hypothesis Testing:

Non-directional hypotheses are central to hypothesis testing, a fundamental aspect of the scientific method. Hypothesis testing enables researchers to draw conclusions about the relationships between variables based on empirical evidence.

Formulating Non-Directional Hypotheses

Formulating effective non-directional hypotheses requires a systematic approach. Here are the steps to formulate a non-directional hypothesis:

1. Identify the Variables:

Begin by identifying the independent and dependent variables in your study. These variables should be clearly defined and measurable.

2. Clarify the Research Question:

Clearly state the research question that you aim to answer. The research question should focus on the relationship , difference, or effect you want to investigate.

3. Determine the Absence of Effect:

Consider what it would mean for there to be no significant relationship , difference, or effect between the variables. This helps in formulating the negation that characterizes non-directional hypotheses.

4. Craft the Hypothesis:

Use the structure mentioned earlier to craft your non-directional hypothesis. Ensure that it is clear and concise, stating that there is no significant relationship , difference, or effect between the variables.

5. Ensure Testability:

Ensure that your non-directional hypothesis is testable. This means that you should be able to collect data and analyze it to determine whether the observed results support or contradict the hypothesis.

6. Revise and Refine:

Review your non-directional hypothesis and refine it as needed. Seek feedback from colleagues or mentors to improve its clarity and specificity.

Examples of Non-Directional Hypotheses

Non-directional hypotheses are used in various fields of research to test for the mere presence or absence of effects. Here are some examples from different scientific disciplines:

1. Psychology:

Research Question: Does exposure to a new teaching method affect students’ test performance ?
Non-Directional Hypothesis: “There is no significant difference in test performance between students exposed to the new teaching method and those not exposed to it.”

2. Biology:

Research Question: Is there a relationship between the presence of a specific gene variant and the risk of a certain disease?
Non-Directional Hypothesis: “There is no significant relationship between the presence of the gene variant and the risk of the disease.”

3. Economics:

Research Question: Does the introduction of a new tax policy lead to changes in consumer spending behavior?
Non-Directional Hypothesis: “There is no significant relationship between the introduction of the new tax policy and changes in consumer spending behavior.”

4. Sociology:

Research Question: Is there an association between parental involvement and students’ academic achievement?
Non-Directional Hypothesis: “There is no significant relationship between parental involvement and students’ academic achievement.”

5. Environmental Science:

Research Question: Does the level of air pollution impact respiratory health in urban areas?
Non-Directional Hypothesis: “There is no significant relationship between the level of air pollution and respiratory health in urban areas.”

Null Hypothesis Testing

Non-directional hypotheses, also known as null hypotheses, are subjected to rigorous testing to determine whether the observed results support or contradict the hypothesis. The process of testing the null hypothesis involves collecting data, performing statistical analyses, and assessing whether the observed differences or effects are

statistically significant.

In hypothesis testing, researchers aim to reject the null hypothesis if the observed results are highly unlikely to occur by chance. If the null hypothesis is rejected, it suggests that there is a significant relationship , difference, or effect between the variables. If the null hypothesis is not rejected, it indicates that there is no significant evidence to support the presence of the hypothesized effect.

The significance level (often denoted as α) is predetermined by researchers and represents the threshold for determining statistical significance. Common significance levels include 0.05 and 0.01, indicating that researchers are willing to accept a 5% or 1% chance of making a Type I error (incorrectly rejecting a true null hypothesis).

Non-directional hypotheses, also known as null hypotheses, are fundamental to hypothesis testing and the scientific method. They allow researchers to test for the mere presence or absence of relationships, differences, or effects between variables, without making specific predictions about the direction of these relationships.

The objectivity, versatility, and rigor of null hypothesis testing make it an essential tool in scientific research across diverse disciplines. By formulating and testing non-directional hypotheses, researchers contribute to the systematic accumulation of knowledge, refine their theories, and draw meaningful conclusions about the relationships between variables.

Whether in the natural sciences, social sciences, or humanities, non-directional hypotheses remain a cornerstone of empirical research, enabling researchers to explore, question, and expand our understanding of the complex phenomena that shape our world.

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Directional Hypothesis: Definition and 10 Examples

directional hypothesis examples and definition, explained below

A directional hypothesis refers to a type of hypothesis used in statistical testing that predicts a particular direction of the expected relationship between two variables.

In simpler terms, a directional hypothesis is an educated, specific guess about the direction of an outcome—whether an increase, decrease, or a proclaimed difference in variable sets.

For example, in a study investigating the effects of sleep deprivation on cognitive performance, a directional hypothesis might state that as sleep deprivation (Independent Variable) increases, cognitive performance (Dependent Variable) decreases (Killgore, 2010). Such a hypothesis offers a clear, directional relationship whereby a specific increase or decrease is anticipated.

Global warming provides another notable example of a directional hypothesis. A researcher might hypothesize that as carbon dioxide (CO2) levels increase, global temperatures also increase (Thompson, 2010). In this instance, the hypothesis clearly articulates an upward trend for both variables.

In any given circumstance, it’s imperative that a directional hypothesis is grounded on solid evidence. For instance, the CO2 and global temperature relationship is based on substantial scientific evidence, and not on a random guess or mere speculation (Florides & Christodoulides, 2009).

Directional vs Non-Directional vs Null Hypotheses

A directional hypothesis is generally contrasted to a non-directional hypothesis. Here’s how they compare:

Directional hypothesis: A directional hypothesis provides a perspective of the expected relationship between variables, predicting the direction of that relationship (either positive, negative, or a specific difference).
Non-directional hypothesis: A non-directional hypothesis denotes the possibility of a relationship between two variables ( the independent and dependent variables ), although this hypothesis does not venture a prediction as to the direction of this relationship (Ali & Bhaskar, 2016). For example, a non-directional hypothesis might state that there exists a relationship between a person’s diet (independent variable) and their mood (dependent variable), without indicating whether improvement in diet enhances mood positively or negatively. Overall, the choice between a directional or non-directional hypothesis depends on the known or anticipated link between the variables under consideration in research studies.

Another very important type of hypothesis that we need to know about is a null hypothesis :

Null hypothesis : The null hypothesis stands as a universality—the hypothesis that there is no observed effect in the population under study, meaning there is no association between variables (or that the differences are down to chance). For instance, a null hypothesis could be constructed around the idea that changing diet (independent variable) has no discernible effect on a person’s mood (dependent variable) (Yan & Su, 2016). This proposition is the one that we aim to disprove in an experiment.

While directional and non-directional hypotheses involve some integrated expectations about the outcomes (either distinct direction or a vague relationship), a null hypothesis operates on the premise of negating such relationships or effects.

The null hypotheses is typically proposed to be negated or disproved by statistical tests, paving way for the acceptance of an alternate hypothesis (either directional or non-directional).

Directional Hypothesis Examples

1. exercise and heart health.

Research suggests that as regular physical exercise (independent variable) increases, the risk of heart disease (dependent variable) decreases (Jakicic, Davis, Rogers, King, Marcus, Helsel, Rickman, Wahed, Belle, 2016). In this example, a directional hypothesis anticipates that the more individuals maintain routine workouts, the lesser would be their odds of developing heart-related disorders. This assumption is based on the underlying fact that routine exercise can help reduce harmful cholesterol levels, regulate blood pressure, and bring about overall health benefits. Thus, a direction – a decrease in heart disease – is expected in relation with an increase in exercise.

2. Screen Time and Sleep Quality

Another classic instance of a directional hypothesis can be seen in the relationship between the independent variable, screen time (especially before bed), and the dependent variable, sleep quality. This hypothesis predicts that as screen time before bed increases, sleep quality decreases (Chang, Aeschbach, Duffy, Czeisler, 2015). The reasoning behind this hypothesis is the disruptive effect of artificial light (especially blue light from screens) on melatonin production, a hormone needed to regulate sleep. As individuals spend more time exposed to screens before bed, it is predictably hypothesized that their sleep quality worsens.

3. Job Satisfaction and Employee Turnover

A typical scenario in organizational behavior research posits that as job satisfaction (independent variable) increases, the rate of employee turnover (dependent variable) decreases (Cheng, Jiang, & Riley, 2017). This directional hypothesis emphasizes that an increased level of job satisfaction would lead to a reduced rate of employees leaving the company. The theoretical basis for this hypothesis is that satisfied employees often tend to be more committed to the organization and are less likely to seek employment elsewhere, thus reducing turnover rates.

4. Healthy Eating and Body Weight

Healthy eating, as the independent variable, is commonly thought to influence body weight, the dependent variable, in a positive way. For example, the hypothesis might state that as consumption of healthy foods increases, an individual’s body weight decreases (Framson, Kristal, Schenk, Littman, Zeliadt, & Benitez, 2009). This projection is based on the premise that healthier foods, such as fruits and vegetables, are generally lower in calories than junk food, assisting in weight management.

5. Sun Exposure and Skin Health

The association between sun exposure (independent variable) and skin health (dependent variable) allows for a definitive hypothesis declaring that as sun exposure increases, the risk of skin damage or skin cancer increases (Whiteman, Whiteman, & Green, 2001). The premise aligns with the understanding that overexposure to the sun’s ultraviolet rays can deteriorate skin health, leading to conditions like sunburn or, in extreme cases, skin cancer.

6. Study Hours and Academic Performance

A regularly assessed relationship in academia suggests that as the number of study hours (independent variable) rises, so too does academic performance (dependent variable) (Nonis, Hudson, Logan, Ford, 2013). The hypothesis proposes a positive correlation , with an increase in study time expected to contribute to enhanced academic outcomes.

7. Screen Time and Eye Strain

It’s commonly hypothesized that as screen time (independent variable) increases, the likelihood of experiencing eye strain (dependent variable) also increases (Sheppard & Wolffsohn, 2018). This is based on the idea that prolonged engagement with digital screens—computers, tablets, or mobile phones—can cause discomfort or fatigue in the eyes, attributing to symptoms of eye strain.

8. Physical Activity and Stress Levels

In the sphere of mental health, it’s often proposed that as physical activity (independent variable) increases, levels of stress (dependent variable) decrease (Stonerock, Hoffman, Smith, Blumenthal, 2015). Regular exercise is known to stimulate the production of endorphins, the body’s natural mood elevators, helping to alleviate stress.

9. Water Consumption and Kidney Health

A common health-related hypothesis might predict that as water consumption (independent variable) increases, the risk of kidney stones (dependent variable) decreases (Curhan, Willett, Knight, & Stampfer, 2004). Here, an increase in water intake is inferred to reduce the risk of kidney stones by diluting the substances that lead to stone formation.

10. Traffic Noise and Sleep Quality

In urban planning research, it’s often supposed that as traffic noise (independent variable) increases, sleep quality (dependent variable) decreases (Muzet, 2007). Increased noise levels, particularly during the night, can result in sleep disruptions, thus, leading to poor sleep quality.

11. Sugar Consumption and Dental Health

In the field of dental health, an example might be stating as one’s sugar consumption (independent variable) increases, dental health (dependent variable) decreases (Sheiham, & James, 2014). This stems from the fact that sugar is a major factor in tooth decay, and increased consumption of sugary foods or drinks leads to a decline in dental health due to the high likelihood of cavities.

See 15 More Examples of Hypotheses Here

A directional hypothesis plays a critical role in research, paving the way for specific predicted outcomes based on the relationship between two variables. These hypotheses clearly illuminate the expected direction—the increase or decrease—of an effect. From predicting the impacts of healthy eating on body weight to forecasting the influence of screen time on sleep quality, directional hypotheses allow for targeted and strategic examination of phenomena. In essence, directional hypotheses provide the crucial path for inquiry, shaping the trajectory of research studies and ultimately aiding in the generation of insightful, relevant findings.

Ali, S., & Bhaskar, S. (2016). Basic statistical tools in research and data analysis. Indian Journal of Anaesthesia, 60 (9), 662-669. doi: https://doi.org/10.4103%2F0019-5049.190623

Chang, A. M., Aeschbach, D., Duffy, J. F., & Czeisler, C. A. (2015). Evening use of light-emitting eReaders negatively affects sleep, circadian timing, and next-morning alertness. Proceeding of the National Academy of Sciences, 112 (4), 1232-1237. doi: https://doi.org/10.1073/pnas.1418490112

Cheng, G. H. L., Jiang, D., & Riley, J. H. (2017). Organizational commitment and intrinsic motivation of regular and contractual primary school teachers in China. New Psychology, 19 (3), 316-326. Doi: https://doi.org/10.4103%2F2249-4863.184631

Curhan, G. C., Willett, W. C., Knight, E. L., & Stampfer, M. J. (2004). Dietary factors and the risk of incident kidney stones in younger women: Nurses’ Health Study II. Archives of Internal Medicine, 164 (8), 885–891.

Florides, G. A., & Christodoulides, P. (2009). Global warming and carbon dioxide through sciences. Environment international , 35 (2), 390-401. doi: https://doi.org/10.1016/j.envint.2008.07.007

Framson, C., Kristal, A. R., Schenk, J. M., Littman, A. J., Zeliadt, S., & Benitez, D. (2009). Development and validation of the mindful eating questionnaire. Journal of the American Dietetic Association, 109 (8), 1439-1444. doi: https://doi.org/10.1016/j.jada.2009.05.006

Jakicic, J. M., Davis, K. K., Rogers, R. J., King, W. C., Marcus, M. D., Helsel, D., … & Belle, S. H. (2016). Effect of wearable technology combined with a lifestyle intervention on long-term weight loss: The IDEA randomized clinical trial. JAMA, 316 (11), 1161-1171.

Khan, S., & Iqbal, N. (2013). Study of the relationship between study habits and academic achievement of students: A case of SPSS model. Higher Education Studies, 3 (1), 14-26.

Killgore, W. D. (2010). Effects of sleep deprivation on cognition. Progress in brain research , 185 , 105-129. doi: https://doi.org/10.1016/B978-0-444-53702-7.00007-5

Marczinski, C. A., & Fillmore, M. T. (2014). Dissociative antagonistic effects of caffeine on alcohol-induced impairment of behavioral control. Experimental and Clinical Psychopharmacology, 22 (4), 298–311. doi: https://psycnet.apa.org/doi/10.1037/1064-1297.11.3.228

Muzet, A. (2007). Environmental Noise, Sleep and Health. Sleep Medicine Reviews, 11 (2), 135-142. doi: https://doi.org/10.1016/j.smrv.2006.09.001

Nonis, S. A., Hudson, G. I., Logan, L. B., & Ford, C. W. (2013). Influence of perceived control over time on college students’ stress and stress-related outcomes. Research in Higher Education, 54 (5), 536-552. doi: https://doi.org/10.1023/A:1018753706925

Sheiham, A., & James, W. P. (2014). A new understanding of the relationship between sugars, dental caries and fluoride use: implications for limits on sugars consumption. Public health nutrition, 17 (10), 2176-2184. Doi: https://doi.org/10.1017/S136898001400113X

Sheppard, A. L., & Wolffsohn, J. S. (2018). Digital eye strain: prevalence, measurement and amelioration. BMJ open ophthalmology , 3 (1), e000146. doi: http://dx.doi.org/10.1136/bmjophth-2018-000146

Stonerock, G. L., Hoffman, B. M., Smith, P. J., & Blumenthal, J. A. (2015). Exercise as Treatment for Anxiety: Systematic Review and Analysis. Annals of Behavioral Medicine, 49 (4), 542–556. doi: https://doi.org/10.1007/s12160-014-9685-9

Thompson, L. G. (2010). Climate change: The evidence and our options. The Behavior Analyst , 33 , 153-170. Doi: https://doi.org/10.1007/BF03392211

Whiteman, D. C., Whiteman, C. A., & Green, A. C. (2001). Childhood sun exposure as a risk factor for melanoma: a systematic review of epidemiologic studies. Cancer Causes & Control, 12 (1), 69-82. doi: https://doi.org/10.1023/A:1008980919928

Yan, X., & Su, X. (2009). Linear regression analysis: theory and computing . New Jersey: World Scientific.

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Directional and non-directional hypothesis: A Comprehensive Guide

Understanding Directional Hypothesis

Definition of directional hypothesis

The role of directional hypothesis in research

Examples of research questions for directional hypothesis

Definition of non-directional hypothesis

Advantages and utility of non-directional hypothesis

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8.1.2.2.1 - Minitab: 1 Proportion z Test, Raw Data

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8.1.2.2.2 - Minitab: 1 Sample Proportion z test, Summary Data

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