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6a.2 - steps for hypothesis tests, the logic of hypothesis testing section .
A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.
How do we decide whether to reject the null hypothesis?
In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.
We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.
Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.
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Published on January 28, 2020 by Rebecca Bevans . Revised on June 22, 2023.
Statistical tests are used in hypothesis testing . They can be used to:
Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they determine whether the observed data fall outside of the range of values predicted by the null hypothesis.
If you already know what types of variables you’re dealing with, you can use the flowchart to choose the right statistical test for your data.
Statistical tests flowchart
What does a statistical test do, when to perform a statistical test, choosing a parametric test: regression, comparison, or correlation, choosing a nonparametric test, flowchart: choosing a statistical test, other interesting articles, frequently asked questions about statistical tests.
Statistical tests work by calculating a test statistic – a number that describes how much the relationship between variables in your test differs from the null hypothesis of no relationship.
It then calculates a p value (probability value). The p -value estimates how likely it is that you would see the difference described by the test statistic if the null hypothesis of no relationship were true.
If the value of the test statistic is more extreme than the statistic calculated from the null hypothesis, then you can infer a statistically significant relationship between the predictor and outcome variables.
If the value of the test statistic is less extreme than the one calculated from the null hypothesis, then you can infer no statistically significant relationship between the predictor and outcome variables.
You can perform statistical tests on data that have been collected in a statistically valid manner – either through an experiment , or through observations made using probability sampling methods .
For a statistical test to be valid , your sample size needs to be large enough to approximate the true distribution of the population being studied.
To determine which statistical test to use, you need to know:
Statistical tests make some common assumptions about the data they are testing:
If your data do not meet the assumptions of normality or homogeneity of variance, you may be able to perform a nonparametric statistical test , which allows you to make comparisons without any assumptions about the data distribution.
If your data do not meet the assumption of independence of observations, you may be able to use a test that accounts for structure in your data (repeated-measures tests or tests that include blocking variables).
The types of variables you have usually determine what type of statistical test you can use.
Quantitative variables represent amounts of things (e.g. the number of trees in a forest). Types of quantitative variables include:
Categorical variables represent groupings of things (e.g. the different tree species in a forest). Types of categorical variables include:
Choose the test that fits the types of predictor and outcome variables you have collected (if you are doing an experiment , these are the independent and dependent variables ). Consult the tables below to see which test best matches your variables.
Parametric tests usually have stricter requirements than nonparametric tests, and are able to make stronger inferences from the data. They can only be conducted with data that adheres to the common assumptions of statistical tests.
The most common types of parametric test include regression tests, comparison tests, and correlation tests.
Regression tests look for cause-and-effect relationships . They can be used to estimate the effect of one or more continuous variables on another variable.
Predictor variable | Outcome variable | Research question example | |
---|---|---|---|
What is the effect of income on longevity? | |||
What is the effect of income and minutes of exercise per day on longevity? | |||
Logistic regression | What is the effect of drug dosage on the survival of a test subject? |
Comparison tests look for differences among group means . They can be used to test the effect of a categorical variable on the mean value of some other characteristic.
T-tests are used when comparing the means of precisely two groups (e.g., the average heights of men and women). ANOVA and MANOVA tests are used when comparing the means of more than two groups (e.g., the average heights of children, teenagers, and adults).
Predictor variable | Outcome variable | Research question example | |
---|---|---|---|
Paired t-test | What is the effect of two different test prep programs on the average exam scores for students from the same class? | ||
Independent t-test | What is the difference in average exam scores for students from two different schools? | ||
ANOVA | What is the difference in average pain levels among post-surgical patients given three different painkillers? | ||
MANOVA | What is the effect of flower species on petal length, petal width, and stem length? |
Correlation tests check whether variables are related without hypothesizing a cause-and-effect relationship.
These can be used to test whether two variables you want to use in (for example) a multiple regression test are autocorrelated.
Variables | Research question example | |
---|---|---|
Pearson’s | How are latitude and temperature related? |
Non-parametric tests don’t make as many assumptions about the data, and are useful when one or more of the common statistical assumptions are violated. However, the inferences they make aren’t as strong as with parametric tests.
Predictor variable | Outcome variable | Use in place of… | |
---|---|---|---|
Spearman’s | |||
Pearson’s | |||
Sign test | One-sample -test | ||
Kruskal–Wallis | ANOVA | ||
ANOSIM | MANOVA | ||
Wilcoxon Rank-Sum test | Independent t-test | ||
Wilcoxon Signed-rank test | Paired t-test | ||
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This flowchart helps you choose among parametric tests. For nonparametric alternatives, check the table above.
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
Methodology
Research bias
Statistical tests commonly assume that:
If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences.
A test statistic is a number calculated by a statistical test . It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.
The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.
Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.
Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .
When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.
Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).
Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).
You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .
Discrete and continuous variables are two types of quantitative variables :
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About hypothesis testing.
Watch the video for a brief overview of hypothesis testing:
Can’t see the video? Click here to watch it on YouTube.
Contents (Click to skip to the section):
What is hypothesis testing.
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A hypothesis is an educated guess about something in the world around you. It should be testable, either by experiment or observation. For example:
It can really be anything at all as long as you can put it to the test.
If you are going to propose a hypothesis, it’s customary to write a statement. Your statement will look like this: “If I…(do this to an independent variable )….then (this will happen to the dependent variable ).” For example:
A good hypothesis statement should:
Hypothesis testing can be one of the most confusing aspects for students, mostly because before you can even perform a test, you have to know what your null hypothesis is. Often, those tricky word problems that you are faced with can be difficult to decipher. But it’s easier than you think; all you need to do is:
If you trace back the history of science, the null hypothesis is always the accepted fact. Simple examples of null hypotheses that are generally accepted as being true are:
You won’t be required to actually perform a real experiment or survey in elementary statistics (or even disprove a fact like “Pluto is a planet”!), so you’ll be given word problems from real-life situations. You’ll need to figure out what your hypothesis is from the problem. This can be a little trickier than just figuring out what the accepted fact is. With word problems, you are looking to find a fact that is nullifiable (i.e. something you can reject).
A researcher thinks that if knee surgery patients go to physical therapy twice a week (instead of 3 times), their recovery period will be longer. Average recovery times for knee surgery patients is 8.2 weeks.
The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as: H 1 : μ > 8.2
Next, you’ll need to state the null hypothesis . That’s what will happen if the researcher is wrong . In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s: H 0 μ ≤ 8.2
Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy— the hard part is usually figuring out what your null hypothesis is in the first place.
The one sample z test isn’t used very often (because we rarely know the actual population standard deviation ). However, it’s a good idea to understand how it works as it’s one of the simplest tests you can perform in hypothesis testing. In English class you got to learn the basics (like grammar and spelling) before you could write a story; think of one sample z tests as the foundation for understanding more complex hypothesis testing. This page contains two hypothesis testing examples for one sample z-tests .
Watch the video for an example:
A principal at a certain school claims that the students in his school are above average intelligence. A random sample of thirty students IQ scores have a mean score of 112.5. Is there sufficient evidence to support the principal’s claim? The mean population IQ is 100 with a standard deviation of 15.
Step 1: State the Null hypothesis . The accepted fact is that the population mean is 100, so: H 0 : μ = 100.
Step 2: State the Alternate Hypothesis . The claim is that the students have above average IQ scores, so: H 1 : μ > 100. The fact that we are looking for scores “greater than” a certain point means that this is a one-tailed test.
Step 4: State the alpha level . If you aren’t given an alpha level , use 5% (0.05).
Step 5: Find the rejection region area (given by your alpha level above) from the z-table . An area of .05 is equal to a z-score of 1.645.
Step 6: If Step 6 is greater than Step 5, reject the null hypothesis. If it’s less than Step 5, you cannot reject the null hypothesis. In this case, it is more (4.56 > 1.645), so you can reject the null.
Watch the video for an example of a two-tailed z-test:
Blood glucose levels for obese patients have a mean of 100 with a standard deviation of 15. A researcher thinks that a diet high in raw cornstarch will have a positive or negative effect on blood glucose levels. A sample of 30 patients who have tried the raw cornstarch diet have a mean glucose level of 140. Test the hypothesis that the raw cornstarch had an effect.
*This process is made much easier if you use a TI-83 or Excel to calculate the z-score (the “critical value”). See:
You can use the TI 83 calculator for hypothesis testing, but the calculator won’t figure out the null and alternate hypotheses; that’s up to you to read the question and input it into the calculator.
Example problem : A sample of 200 people has a mean age of 21 with a population standard deviation (σ) of 5. Test the hypothesis that the population mean is 18.9 at α = 0.05.
Step 1: State the null hypothesis. In this case, the null hypothesis is that the population mean is 18.9, so we write: H 0 : μ = 18.9
Step 2: State the alternative hypothesis. We want to know if our sample, which has a mean of 21 instead of 18.9, really is different from the population, therefore our alternate hypothesis: H 1 : μ ≠ 18.9
Step 3: Press Stat then press the right arrow twice to select TESTS.
Step 4: Press 1 to select 1:Z-Test… . Press ENTER.
Step 5: Use the right arrow to select Stats .
Step 6: Enter the data from the problem: μ 0 : 18.9 σ: 5 x : 21 n: 200 μ: ≠μ 0
Step 7: Arrow down to Calculate and press ENTER. The calculator shows the p-value: p = 2.87 × 10 -9
This is smaller than our alpha value of .05. That means we should reject the null hypothesis .
Bayesian hypothesis testing helps to answer the question: Can the results from a test or survey be repeated? Why do we care if a test can be repeated? Let’s say twenty people in the same village came down with leukemia. A group of researchers find that cell-phone towers are to blame. However, a second study found that cell-phone towers had nothing to do with the cancer cluster in the village. In fact, they found that the cancers were completely random. If that sounds impossible, it actually can happen! Clusters of cancer can happen simply by chance . There could be many reasons why the first study was faulty. One of the main reasons could be that they just didn’t take into account that sometimes things happen randomly and we just don’t know why.
It’s good science to let people know if your study results are solid, or if they could have happened by chance. The usual way of doing this is to test your results with a p-value . A p value is a number that you get by running a hypothesis test on your data. A P value of 0.05 (5%) or less is usually enough to claim that your results are repeatable. However, there’s another way to test the validity of your results: Bayesian Hypothesis testing. This type of testing gives you another way to test the strength of your results.
Traditional testing (the type you probably came across in elementary stats or AP stats) is called Non-Bayesian. It is how often an outcome happens over repeated runs of the experiment. It’s an objective view of whether an experiment is repeatable. Bayesian hypothesis testing is a subjective view of the same thing. It takes into account how much faith you have in your results. In other words, would you wager money on the outcome of your experiment?
Traditional testing (Non Bayesian) requires you to repeat sampling over and over, while Bayesian testing does not. The main different between the two is in the first step of testing: stating a probability model. In Bayesian testing you add prior knowledge to this step. It also requires use of a posterior probability , which is the conditional probability given to a random event after all the evidence is considered.
Many researchers think that it is a better alternative to traditional testing, because it:
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Specific Tests:
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Establishing the parameter of interest, type of distribution to use, the test statistic, and p -value can help you figure out how to go about a hypothesis test. However, there are several other factors you should consider when interpreting the results.
Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. Remember that your assumption is just an assumption; it is not a fact, and it may or may not be true. But your sample data are real and are showing you a fact that seems to contradict your assumption.
When you perform a hypothesis test, there are four possible outcomes depending on the actual truth (or falseness) of the null hypothesis H 0 and the decision to reject or not. The outcomes are summarized in the following table:
IS ACTUALLY | ||
---|---|---|
Action | ||
Correct outcome | Type II error | |
Type I error | Correct outcome |
The four possible outcomes in the table are:
Each of the errors occurs with a particular probability. The Greek letters α and β represent the probabilities.
α = probability of a type I error = P (type I error) = probability of rejecting the null hypothesis when the null hypothesis is true. These are also known as false positives. We know that α is often determined in advance, and α = 0.05 is often widely accepted. In that case, you are saying, “We are OK making this type of error in 5% of samples.” In fact, the p -value is the exact probability of a type I error based on what you observed.
β = probability of a type II error = P (type II error) = probability of not rejecting the null hypothesis when the null hypothesis is false. These are also known as false negatives.
The power of a test is 1 – β .
Ideally, α and β should be as small as possible because they are probabilities of errors but are rarely zero. We want a high power that is as close to one as well. Increasing the sample size can help us achieve these by reducing both α and β and therefore increasing the power of the test.
Suppose the null hypothesis, H 0 , is that Frank’s rock climbing equipment is safe.
Type I error: Frank thinks that his rock climbing equipment may not be safe when, in fact, it really is safe. Type II error: Frank thinks that his rock climbing equipment may be safe when, in fact, it is not safe.
α = probability that Frank thinks his rock climbing equipment may not be safe when, in fact, it really is safe. β = probability that Frank thinks his rock climbing equipment may be safe when, in fact, it is not safe.
Notice that, in this case, the error with the greater consequence is the type II error, in which Frank thinks his rock climbing equipment is safe, so he goes ahead and uses it.
Suppose the null hypothesis, H 0 , is that the blood cultures contain no traces of pathogen X . State the type I and type II errors.
When the sample size becomes larger, point estimates become more precise and any real differences in the mean and null value become easier to detect and recognize. Even a very small difference would likely be detected if we took a large enough sample. Sometimes, researchers will take such large samples that even the slightest difference is detected, even differences where there is no practical value. In such cases, we still say the difference is statistically significant , but it is not practically significant.
For example, an online experiment might identify that placing additional ads on a movie review website statistically significantly increases viewership of a TV show by 0.001%, but this increase might not have any practical value.
One role of a data scientist in conducting a study often includes planning the size of the study. The data scientist might first consult experts or scientific literature to learn what would be the smallest meaningful difference from the null value. She also would obtain other information, such as a very rough estimate of the true proportion p , so that she could roughly estimate the standard error. From here, she could suggest a sample size that is sufficiently large enough to detect the real difference if it is meaningful. While larger sample sizes may still be used, these calculations are especially helpful when considering costs or potential risks, such as possible health impacts to volunteers in a medical study.
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The decision is to reject the null hypothesis when, in fact, the null hypothesis is true
Erroneously rejecting a true null hypothesis or erroneously failing to reject a false null hypothesis
The probability of failing to reject a true hypothesis
Finding sufficient evidence that the observed effect is not just due to variability, often from rejecting the null hypothesis
Significant Statistics Copyright © 2024 by John Morgan Russell, OpenStaxCollege, OpenIntro is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.
Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.
A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.
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Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.
Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.
The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.
The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.
In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.
All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.
Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:
We will learn more about these test statistics in the upcoming section.
Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.
A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:
The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.
The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.
One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.
Right Tailed Hypothesis Testing
The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:
\(H_{0}\): The population parameter is ≤ some value
\(H_{1}\): The population parameter is > some value.
If the test statistic has a greater value than the critical value then the null hypothesis is rejected
Left Tailed Hypothesis Testing
The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:
\(H_{0}\): The population parameter is ≥ some value
\(H_{1}\): The population parameter is < some value.
The null hypothesis is rejected if the test statistic has a value lesser than the critical value.
In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:
\(H_{0}\): the population parameter = some value
\(H_{1}\): the population parameter ≠ some value
The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.
Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:
The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.
Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.
Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.
Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.
1 - \(\alpha\) = 1 - 0.05 = 0.95
0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.
Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.
z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15
z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56
Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.
Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.
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Important Notes on Hypothesis Testing
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What is hypothesis testing.
Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.
The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.
The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.
The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).
The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.
When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.
To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.
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Here is a list hypothesis testing exercises and solutions. Try to solve a question by yourself first before you look at the solution.
Question 1 In the population, the average IQ is 100 with a standard deviation of 15. A team of scientists want to test a new medication to see if it has either a positive or negative effect on intelligence, or not effect at all. A sample of 30 participants who have taken the medication has a mean of 140. Did the medication affect intelligence? View Solution to Question 1
A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency test. The professor wants the class to be able to score above 70 on the test. The six students get the following scores:62, 92, 75, 68, 83, 95. Can the professor have 90% confidence that the mean score for the class on the test would be above 70. Solution to Question 2
Question 3 In a packaging plant, a machine packs cartons with jars. It is supposed that a new machine would pack faster on the average than the machine currently used. To test the hypothesis, the time it takes each machine to pack ten cartons are recorded. The result in seconds is as follows.
42.1 | 42.7 |
41 | 43.6 |
41.3 | 43.8 |
41.8 | 43.3 |
42.4 | 42.5 |
42.8 | 43.5 |
43.2 | 43.1 |
42.3 | 41.7 |
41.8 | 44 |
42.7 | 44.1 |
Do the data provide sufficient evidence to conclude that, on the average, the new machine packs faster? Perform the required hypothesis test at the 5% level of significance. Solution to Question 3
Question 4 We want to compare the heights in inches of two groups of individuals. Here are the measurements: X: 175, 168, 168, 190, 156, 181, 182, 175, 174, 179 Y: 120, 180, 125, 188, 130, 190, 110, 185, 112, 188 Solution to Question 4
Question 5 A clinic provides a program to help their clients lose weight and asks a consumer agency to investigate the effectiveness of the program. The agency takes a sample of 15 people, weighing each person in the sample before the program begins and 3 months later. The results a tabulated below
Determine is the program is effective. Solution to Question 5
Question 6 A sample of 20 students were selected and given a diagnostic module prior to studying for a test. And then they were given the test again after completing the module. . The result of the students scores in the test before and after the test is tabulated below.
We want to see if there is significant improvement in the student’s performance due to this teaching method Solution to Question 6
Question 7 A study was performed to test wether cars get better mileage on premium gas than on regular gas. Each of 10 cars was first filled with regular or premium gas, decided by a coin toss, and the mileage for the tank was recorded. The mileage was recorded again for the same cars using other kind of gasoline. Determine wether cars get significantly better mileage with premium gas.
Mileage with regular gas: 16,20,21,22,23,22,27,25,27,28 Mileage with premium gas: 19, 22,24,24,25,25,26,26,28,32 Solution to Question 7
Question 8 An automatic cutter machine must cut steel strips of 1200 mm length. From a preliminary data, we checked that the lengths of the pieces produced by the machine can be considered as normal random variables with a 3mm standard deviation. We want to make sure that the machine is set correctly. Therefore 16 pieces of the products are randomly selected and weight. The figures were in mm: 1193,1196,1198,1195,1198,1199,1204,1193,1203,1201,1196,1200,1191,1196,1198,1191 Examine wether there is any significant deviation from the required size Solution to Question 8
Question 9 Blood pressure reading of ten patients before and after medication for reducing the blood pressure are as follows
Patient: 1,2,3,4,5,6,7,8,9,10 Before treatment: 86,84,78,90,92,77,89,90,90,86 After treatment: 80,80,92,79,92,82,88,89,92,83
Test the null hypothesis of no effect agains the alternate hypothesis that medication is effective. Execute it with Wilcoxon test Solution to Question 9
Question on ANOVA Sussan Sound predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound or no sound at all. She randomly divides 24 students into three groups of 8 each. All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with nose that changes volume periodically. Those in group 3 study with no sound at all. After studying, all students take a 10 point multiple choice test over the material. Their scores are tabulated below.
Group1: Constant sound: 7,4,6,8,6,6,2,9 Group 2: Random sound: 5,5,3,4,4,7,2,2 Group 3: No sound at all: 2,4,7,1,2,1,5,5 Solution to Question 10
Question 11 Using the following three groups of data, perform a one-way analysis of variance using α = 0.05.
51 | 23 | 56 |
45 | 43 | 76 |
33 | 23 | 74 |
45 | 43 | 87 |
67 | 45 | 56 |
Solution to Question 11
Question 12 In a packaging plant, a machine packs cartons with jars. It is supposed that a new machine would pack faster on the average than the machine currently used. To test the hypothesis, the time it takes each machine to pack ten cartons are recorded. The result in seconds is as follows.
New Machine: 42,41,41.3,41.8,42.4,42.8,43.2,42.3,41.8,42.7 Old Machine: 42.7,43.6,43.8,43.3,42.5,43.5,43.1,41.7,44,44.1
Perform an F-test to determine if the null hypothesis should be accepted. Solution to Question 12
Question 13 A random sample 500 U.S adults are questioned about their political affiliation and opinion on a tax reform bill. We need to test if the political affiliation and their opinon on a tax reform bill are dependent, at 5% level of significance. The observed contingency table is given below.
total | ||||
138 | 83 | 64 | 285 | |
64 | 67 | 84 | 215 | |
total | 202 | 150 | 148 | 500 |
Solution to Question 13
Question 14 Can a dice be considered regular which is showing the following frequency distribution during 1000 throws?
1 | 2 | 3 | 4 | 5 | 6 | |
182 | 154 | 162 | 175 | 151 | 176 |
Solution to Question 14
Solution to Question 15
Question 16 A newly developed muesli contains five types of seeds (A, B, C, D and E). The percentage of which is 35%, 25%, 20%, 10% and 10% according to the product information. In a randomly selected muesli, the following volume distribution was found.
Component | A | B | C | D | E |
Number of Pieces | 184 | 145 | 100 | 63 | 63 |
Lets us decide about the null hypothesis whether the composition of the sample corresponds to the distribution indicated on the packaging at alpha = 0.1 significance level. Solution to Question 16
Question 17 A research team investigated whether there was any significant correlation between the severity of a certain disease runoff and the age of the patients. During the study, data for n = 200 patients were collected and grouped according to the severity of the disease and the age of the patient. The table below shows the result
41 | 34 | 9 | ||
25 | 25 | 12 | ||
6 | 33 | 15 |
Let us decided about the correlation between the age of the patients and the severity of disease progression. Solution to Question 17
Question 18 A publisher is interested in determine which of three book cover is most attractive. He interviews 400 people in each of the three states (California, Illinois and New York), and asks each person which of the cover he or she prefers. The number of preference for each cover is as follows:
81 | 60 | 182 | 323 | |
78 | 93 | 95 | 266 | |
241 | 247 | 123 | 611 | |
400 | 400 | 400 | 1200 |
Do these data indicate that there are regional differences in people’s preferences concerning these covers? Use the 0.05 level of significance. Solution to Question 18
Question 19 Trees planted along the road were checked for which ones are healthy(H) or diseased (D) and the following arrangement of the trees were obtained:
H H H H D D D H H H H H H H D D H H D D D
Test at the = 0.05 significance wether this arrangement may be regarded as random
Solution to Question 19
Question 20 Suppose we flip a coin n = 15 times and come up with the following arrangements
H T T T H H T T T T H H T H H
(H = head, T = tail)
Test at the alpha = 0.05 significance level whether this arrangement may be regarded as random.
Solution to Question 20
You might also like, hypothesis testing question 19 – run test ( trees were planted…), how to perform fisher’s test (f-test) – example question 12, how to perform mann-witney u test(step by step) – hypothesis testing.
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Below are given the gain in weights (in lbs.) of pigs fed on two diet A and B Dieta 25 32 30 34 24 14 32 24 30 31 35 25 – – DietB 44 34 22 10 47 31 40 30 32 35 18 21 35 29
Hypothesis testing allows us to make data-driven decisions by testing assertions about populations. It is the backbone behind scientific research, business analytics, financial modeling, and more.
This comprehensive guide aims to solidify your understanding with:
So let‘s get comfortable with making statements, gathering evidence, and letting the data speak!
Hypothesis testing is structured around making a claim in the form of competing hypotheses, gathering data, performing statistical tests, and making decisions about which hypothesis the evidence supports.
Here are some key terms about hypotheses and the testing process:
Null Hypothesis ($H_0$): The default statement about a population parameter. Generally asserts that there is no statistical significance between two data sets or that a sample parameter equals some claimed population parameter value. The statement being tested that is either rejected or supported.
Alternative Hypothesis ($H_1$): The statement that sample observations indicate statistically significant effect or difference from what the null hypothesis states. $H_1$ and $H_0$ are mutually exclusive, meaning if statistical tests support rejecting $H_0$, then you conclude $H_1$ has strong evidence.
Significance Level ($\alpha$): The probability of incorrectly rejecting a true null hypothesis, known as making a Type I error. Common significance levels are 90%, 95%, and 99%. The lower significance level, the more strict the criteria is for rejecting $H_0$.
Test Statistic: Summary calculations of sample data including mean, proportion, correlation coefficient, etc. Used to determine statistical significance and improbability under $H_0$.
P-value: Probability of obtaining sample results at least as extreme as the test statistic, assuming $H_0$ is true. Small p-values indicate strong statistical evidence against the null hypothesis.
Type I Error: Incorrectly rejecting a true null hypothesis
Type II Error : Failing to reject a false null hypothesis
These terms set the stage for the overall process:
1. Make Hypotheses
Define the null ($H_0$) and alternative hypothesis ($H_1$).
2. Set Significance Level
Typical significance levels are 90%, 95%, and 99%. Higher significance means more strict burden of proof for rejecting $H_0$.
3. Collect Data
Gather sample and population data related to the hypotheses under examination.
4. Determine Test Statistic
Calculate relevant test statistics like p-value, z-score, t-statistic, etc along with degrees of freedom.
5. Compare to Significance Level
If the test statistic falls in the critical region based on the significance, reject $H_0$, otherwise fail to reject $H_0$.
6. Draw Conclusions
Make determinations about hypotheses given the statistical evidence and context of the situation.
Now that you know the process and objectives, let’s apply this to some concrete examples.
We‘ll demonstrate hypothesis testing using Numpy, Scipy, Pandas and simulated data sets. Specifically, we‘ll conduct and interpret:
These represent some of the most widely used methods for determining statistical significance between groups.
We‘ll plot the data distributions to check normality assumptions where applicable. And determine if evidence exists to reject the null hypotheses across several scenarios.
Two sample t-tests determine whether the mean of a numerical variable differs significantly across two independent groups. It assumes observations follow approximate normal distributions within each group, but not that variances are equal.
Let‘s test for differences in reported salaries at hypothetical Company X vs Company Y:
$H_0$ : Average reported salaries are equal at Company X and Company Y
$H_1$ : Average reported salaries differ between Company X and Company Y
First we‘ll simulate salary samples for each company based on random normal distributions, set a 95% confidence level, run the t-test using NumPy, then interpret.
The t-statistic of 9.35 shows the difference between group means is nearly 9.5 standard errors. The very small p-value rejects the idea the salaries are equal across a randomly sampled population of employees.
Since the test returned a p-value lower than the significance level, we reject $H_0$, meaning evidence supports $H_1$ that average reported salaries differ between these hypothetical companies.
While an independent groups t-test analyzes mean differences between distinct groups, a paired t-test looks for significant effects pre vs post some treatment within the same set of subjects. This helps isolate causal impacts by removing effects from confounding individual differences.
Let‘s analyze Amazon purchase data to determine if spending increases during the holiday months of November and December.
$H_0$ : Average monthly spending is equal pre-holiday and during the holiday season
$H_1$ : Average monthly spending increases during the holiday season
We‘ll import transaction data using Pandas, add seasonal categories, then run and interpret the paired t-test.
Since the p-value is below the 0.05 significance level, we reject $H_0$. The output shows statistically significant evidence at 95% confidence that average spending increases during November-December relative to January-October.
Visualizing the monthly trend helps confirm the spike during the holiday months.
A single sample z-test allows testing whether a sample mean differs significantly from a population mean. It requires knowing the population standard deviation.
Let‘s test if recently surveyed shoppers differ significantly in their reported ages from the overall customer base:
$H_0$ : Sample mean age equals population mean age of 39
$H_1$ : Sample mean age does not equal population mean of 39
Here the absolute z-score over 2 and p-value under 0.05 indicates statistically significant evidence that recently surveyed shopper ages differ from the overall population parameter.
Chi-squared tests help determine independence between categorical variables. The test statistic measures deviations between observed and expected outcome frequencies across groups to determine magnitude of relationship.
Let‘s test if credit card application approvals are independent across income groups using simulated data:
$H_0$ : Credit card approvals are independent of income level
$H_1$ : Credit approvals and income level are related
Since the p-value is greater than the 0.05 significance level, we fail to reject $H_0$. There is not sufficient statistical evidence to conclude that credit card approval rates differ by income categories.
Analysis of variance (ANOVA) hypothesis tests determine if mean differences exist across more than two groups. ANOVA expands upon t-tests for multiple group comparisons.
Let‘s test if average debt obligations vary depending on highest education level attained.
$H_0$ : Average debt obligations are equal across education levels
$H_1$ : Average debt obligations differ based on education level
We‘ll simulate ordered education and debt data for visualization via box plots and then run ANOVA.
The ANOVA output shows an F-statistic of 91.59 that along with a tiny p-value leads to rejecting $H_0$. We conclude there are statistically significant differences in average debt obligations based on highest degree attained.
The box plots visualize these distributions and means vary across four education attainment groups.
Hypothesis testing forms the backbone of data-driven decision making across science, research, business, public policy and more by allowing practitioners to draw statistically-validated conclusions.
Here is a sample of hypotheses commonly tested:
Pharmaceuticals
Politics & Social Sciences
This represents just a sample of the wide ranging real-world applications. Properly formulated hypotheses, statistical testing methodology, reproducible analysis, and unbiased interpretation helps ensure valid reliable findings.
However, hypothesis testing does still come with some limitations worth addressing.
While hypothesis testing empowers huge breakthroughs across disciplines, the methodology does come with some inherent restrictions:
Over-reliance on p-values
P-values help benchmark statistical significance, but should not be over-interpreted. A large p-value does not necessarily mean the null hypothesis is 100% true for the entire population. And small p-values do not directly prove causality as confounding factors always exist.
Significance also does not indicate practical real-world effect size. Statistical power calculations should inform necessary sample sizes to detect desired effects.
Errors from Multiple Tests
Running many hypothesis tests by chance produces some false positives due to randomness. Analysts should account for this by adjusting significance levels, pre-registering testing plans, replicating findings, and relying more on meta-analyses.
Poor Experimental Design
Bad data, biased samples, unspecified variables, and lack of controls can completely undermine results. Findings can only be reasonably extended to populations reflected by the test samples.
Garbage in, garbage out definitely applies to statistical analysis!
Assumption Violations
Most common statistical tests make assumptions about normality, homogeneity of variance, independent samples, underlying variable relationships. Violating these premises invalidates reliability.
Transformations, bootstrapping, or non-parametric methods can help navigate issues for sound methodology.
Lack of Reproducibility
The replication crisis impacting scientific research highlights issues around lack of reproducibility, especially involving human participants and high complexity systems. Randomized controlled experiments with strong statistical power provide much more reliable evidence.
While hypothesis testing methodology is rigorously developed, applying concepts correctly proves challenging even among academics and experts!
We‘ve covered core concepts, Python implementations, real-world use cases, and inherent limitations around hypothesis testing. What should you master next?
Parametric vs Non-parametric
Learn assumptions and application differences between parametric statistics like z-tests and t-tests that assume normal distributions versus non-parametric analogs like Wilcoxon signed-rank tests and Mann-Whitney U tests.
Effect Size and Power
Look beyond just p-values to determine practical effect magnitude using indexes like Cohen‘s D. And ensure appropriate sample sizes to detect effects using prospective power analysis.
Alternatives to NHST
Evaluate Bayesian inference models and likelihood ratios that move beyond binary reject/fail-to-reject null hypothesis outcomes toward more integrated evidence.
Tiered Testing Framework
Construct reusable classes encapsulating data processing, visualizations, assumption checking, and statistical tests for maintainable analysis code.
Big Data Integration
Connect statistical analysis to big data pipelines pulling from databases, data lakes and APIs at scale. Productionize analytics.
I hope this end-to-end look at hypothesis testing methodology, Python programming demonstrations, real-world grounding, inherent restrictions and next level considerations provides a launchpad for practically applying core statistics! Please subscribe using the form below for more data science tutorials.
Dr. Alex Mitchell is a dedicated coding instructor with a deep passion for teaching and a wealth of experience in computer science education. As a university professor, Dr. Mitchell has played a pivotal role in shaping the coding skills of countless students, helping them navigate the intricate world of programming languages and software development.
Beyond the classroom, Dr. Mitchell is an active contributor to the freeCodeCamp community, where he regularly shares his expertise through tutorials, code examples, and practical insights. His teaching repertoire includes a wide range of languages and frameworks, such as Python, JavaScript, Next.js, and React, which he presents in an accessible and engaging manner.
Dr. Mitchell’s approach to teaching blends academic rigor with real-world applications, ensuring that his students not only understand the theory but also how to apply it effectively. His commitment to education and his ability to simplify complex topics have made him a respected figure in both the university and online learning communities.
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In statistical hypothesis testing, a test statistic is a crucial tool used to determine the validity of the hypothesis about a population parameter. This article delves into the calculation of test statistics exploring its importance in hypothesis testing and its application in real-world scenarios. Understanding how to compute and interpret test statistics is essential for students and professionals in various fields including data analysis, research and quality control.
Table of Content
Types of test statistic, z-statistic, t-statistic, chi-square statistic, f-statistic, examples with solutions, example for z-statistic, example for t-statistic, example for chi-square statistic, example for f-statistic.
A test statistic is a value calculated from sample data during a hypothesis test. It is used to decide whether to reject the null hypothesis. The test statistic measures how far the sample data is from what we would expect under the null hypothesis. Depending on the type of test (e.g., t-test, chi-square test, etc.), the test statistic is compared to a critical value or used to calculate a p-value, which helps in determining the statistical significance of the results.
In simpler terms, think of a test statistic as a number that tells us how much the sample data stands out from what we expect if there’s no real effect or difference. If this number is big enough, we might conclude that something interesting is happening in the data.
There are many types of test statistic:
When the sample size is large and population variance is known, we can use z-statistic.
Formula for Z-Statistic is:
[Tex]Z = \frac{\bar{X} – \mu}{\frac{\sigma}{\sqrt{n}}} [/Tex]
Read More about Z-test .
When the sample size is small [Tex] n \leq 30 [/Tex] or population variance is unknown, we can use t-statistic.
Formula for t-statistic is:
[Tex]T = \frac{\bar{X} – \mu}{\frac{s}{\sqrt{n}}} [/Tex]
Read More about t-test .
For categorical data to test the independence of the two variables or goodness of fit, we can use chi-square statistic.
Formula for chi-square statistic is:
[Tex]\chi^2 = \sum \frac{(O_i – E_i)^2}{E_i} [/Tex]
Read More about Chi-square test .
For comparing variances between the two or more groups often used in the ANOVA, we can use f-statistic.
Formula for f-statistic is:
[Tex]F = \frac{\text{Variance between groups}}{\text{Variance within groups}} [/Tex]
Problem: A manufacturer claims that the mean weight of their product is 200 grams. A sample of 30 products has a mean weight of 198 grams with the known population standard deviation of the 5 grams. The Test the claim at a 0.05 significance level.
Hypotheses: Null Hypothesis [Tex]H_0: \mu = 200[/Tex] Alternative Hypothesis [Tex]H_1: \mu \neq 200[/Tex] Test Statistic: [Tex]Z = \frac{\bar{X} – \mu}{\frac{\sigma}{\sqrt{n}}} = \frac{198 – 200}{\frac{5}{\sqrt{30}}} \approx -2.19 [/Tex] Critical Value: For a two-tailed test at [Tex] \alpha = 0.05 [/Tex] critical values are [Tex] \pm 1.96[/Tex] . Decision: Since -2.19 < -1.96 reject the null hypothesis.
Problem: A researcher wants to the test if the average test score of the class differs from the 75. A sample of the 15 students has an average score of 78 with the sample standard deviation of 10. The Test the hypothesis at the 0.01 significance level.
Hypotheses: Null Hypothesis [Tex]H_0: \mu = 75[/Tex] Alternative Hypothesis [Tex]H_1: \mu \neq 75[/Tex] Test Statistic: [Tex]T = \frac{\bar{X} – \mu}{\frac{s}{\sqrt{n}}} = \frac{78 – 75}{\frac{10}{\sqrt{15}}} \approx 2.32 [/Tex] Critical Value: For a two-tailed test with the df = 14 and [Tex]\alpha = 0.01[/Tex] critical values are [Tex] \pm 2.977[/Tex] . Decision: Since 2.32 < 2.977 do not reject the null hypothesis.
Problem: A survey of 100 people found the following preferences for the types of movies: Action (30), Comedy (20), Drama (25) and Horror (25). Test if the preferences are equally distributed at the 0.05 significance level.
Hypotheses: Null Hypothesis [Tex]H_0[/Tex] : Preferences are equally distributed. Alternative Hypothesis [Tex]H_1[/Tex] : Preferences are not equally distributed. Expected Frequencies: All categories should have 25 expected frequency. Test Statistic: [Tex]\chi^2 = \sum \frac{(O_i – E_i)^2}{E_i} = \frac{(30 – 25)^2}{25} + \frac{(20 – 25)^2}{25} + \frac{(25 – 25)^2}{25} + \frac{(25 – 25)^2}{25} = 4 + 1 + 0 + 0 = 5[/Tex] Critical Value: For [Tex]df = 3 [/Tex] and [Tex]\alpha = 0.05[/Tex] critical value is 7.815. Decision: Since 5 < 7.815 do not reject the null hypothesis.
Problem: Two different types of fertilizers were tested to the compare their effects on the plant growth. The variance in plant height for the Fertilizer A is 16 and for Fertilizer B is 25. Test if the variances are equal at the 0.05 significance level.
Hypotheses: Null Hypothesis [Tex]H_0: \sigma_1^2 = \sigma_2^2[/Tex] Alternative Hypothesis [Tex]H_1: \sigma_1^2 \neq \sigma_2^2[/Tex] Test Statistic: [Tex]F = \frac{\text{Variance of Fertilizer B}}{\text{Variance of Fertilizer A}} = \frac{25}{16} = 1.56 [/Tex] Critical Value: For [Tex]df_1 = 1[/Tex] and [Tex]df_2 = 1[/Tex] critical value is 18.51. Decision: Since 1.56 < 18.51 do not reject the null hypothesis.
Question 1: A sample of 50 students has an average height of 165 cm. The population standard deviation is 8 cm. Test if the sample mean is significantly different from the 170 cm at a 0.01 significance level.
Question 2: An online retailer claims that 40% of their customers are repeat buyers. A survey of 200 customers shows that 85 are repeat buyers. Test this claim at a 0.05 significance level.
Question 3: A factory claims that the average lifespan of its light bulbs is 1200 hours. A sample of 20 bulbs has an average lifespan of 1180 hours with the standard deviation of the 50 hours. Test the factory’s claim at a 0.05 significance level.
Question 4: A researcher wants to test if there is a significant difference in the mean scores of two different teaching methods. Method A has a mean score of 85 with a standard deviation of 10 and Method B has a mean score of 80 with the standard deviation of 12. Assume the sample size for both the methods is 25. Test the hypothesis at the 0.05 significance level.
Question 5: A company wants to test if their new product’s defect rate is less than 5%. A sample of 150 products shows that 6 are defective. Test the claim at a 0.01 significance level.
Question 6: We have two independent samples with the following the statistics: Sample 1 (n=15, mean=25, variance=9) and Sample 2 (n=20, mean=22, variance=16). Test if the variances are equal at a 0.05 significance level.
Question 7: A drug manufacturer wants to test if the average recovery time with their new drug is less than the historical average of 30 days. A sample of 12 patients has an average recovery time of 28 days with the standard deviation of 4 days. Test the claim at a 0.05 significance level.
Question 8: In a study of customer satisfaction the variance of the satisfaction scores in two different regions is compared. Region 1 has a variance of 25 and Region 2 has a variance of the 36. The Test if the variances are equal at a 0.05 significance level.
Question 9: An agricultural experiment compares the effects of the two fertilizers on the crop yield. The Fertilizer A yields a mean of 50 kg/acre with the standard deviation of 5 kg/acre and Fertilizer B yields a mean of 55 kg/acre with a standard deviation of the 6 kg/acre. If the sample sizes are both 20 test if the mean yields are significantly different at a 0.05 significance level.
Question 10: A company tests whether the average time to assemble a product is different from expected 45 minutes. The sample of 25 assembly times has a mean of the 47 minutes with the standard deviation of 3 minutes. Test the company’s claim at a 0.05 significance level.
What is a test statistic.
A test statistic is a standardized value used to the test a hypothesis about a population parameter.
The Use a Z-test for the large samples or known population variance and a T-test for the small samples or unknown variance.
The significance level is the probability of the rejecting the null hypothesis when it is actually true commonly set at 0.05 or 0.01.
The Compare the chi-square statistic to the critical values from the chi-square distribution to the determine if there is a significant difference between the observed and expected frequencies.
The F-statistic is used to the compare the variances between the different groups to the determine if there are significant differences among group means.
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Step 5: Present your findings. The results of hypothesis testing will be presented in the results and discussion sections of your research paper, dissertation or thesis.. In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p-value).
Hypothesis testing involves five key steps, each critical to validating a research hypothesis using statistical methods: Formulate the Hypotheses: Write your research hypotheses as a null hypothesis (H 0) and an alternative hypothesis (H A). Data Collection: Gather data specifically aimed at testing the hypothesis.
In statistics, hypothesis tests are used to test whether or not some hypothesis about a population parameter is true. To perform a hypothesis test in the real world, researchers will obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:. Null Hypothesis (H 0): The sample data occurs purely from chance.
If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t* were less than -1.6939 (determined using statistical software or a t-table):s-3-3. Since the biologist's test statistic, t* = -4.60, is less than -1.6939, the biologist rejects the null hypothesis.
Example 3: Public Opinion About President Step 1. Determine the null and alternative hypotheses. Null hypothesis: There is no clear winning opinion on this issue; the proportions who would answer yes or no are each 0.50. Alternative hypothesis: Fewer than 0.50, or 50%, of the population would answer yes to this question.
Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing.
S.3 Hypothesis Testing. In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail. The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data).
In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it's biased. By rolling it a few times and analyzing the outcomes, you'd be engaging in the essence of hypothesis testing. Think of hypothesis testing as the ...
The null hypothesis (H0) answers "No, there's no effect in the population.". The alternative hypothesis (Ha) answers "Yes, there is an effect in the population.". The null and alternative are always claims about the population. That's because the goal of hypothesis testing is to make inferences about a population based on a sample.
Answer. Exercise 9.5.11 9.5. 11. A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. She performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo.
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. There are two types of statistical hypotheses: The null hypothesis, denoted as H 0, is the hypothesis that the sample data occurs purely from chance. The alternative ...
A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. The teacher performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.
Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence.
The null hypothesis has the same parameter and number with an equal sign. H0: μ = $30, 000 HA: μ> $30, 000. b. x = number od students who like math. p = proportion of students who like math. The guess is that p < 0.10 and that is the alternative hypothesis. H0: p = 0.10 HA: p <0.10. c. x = age of students in this class.
Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ...
T-tests are used when comparing the means of precisely two groups (e.g., the average heights of men and women). ANOVA and MANOVA tests are used when comparing the means of more than two groups (e.g., the average heights of children, teenagers, and adults). Predictor variable.
Step 2: State the Alternate Hypothesis. The claim is that the students have above average IQ scores, so: H 1: μ > 100. The fact that we are looking for scores "greater than" a certain point means that this is a one-tailed test. Step 3: Draw a picture to help you visualize the problem. Step 4: State the alpha level.
An Introduction to Statistics class in Davies County, KY conducted a hypothesis test at the local high school (a medium sized-approximately 1,200 students-small city demographic) to determine if the local high school's percentage was lower. One hundred fifty students were chosen at random and surveyed.
A statistical hypothesis test may return a value called p or the p-value. This is a quantity that we can use to interpret or quantify the result of the test and either reject or fail to reject the null hypothesis. This is done by comparing the p-value to a threshold value chosen beforehand called the significance level.
Statistical Significance vs. Practical Significance. When the sample size becomes larger, point estimates become more precise and any real differences in the mean and null value become easier to detect and recognize. Even a very small difference would likely be detected if we took a large enough sample.
Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid. A null hypothesis and an alternative ...
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View Solution to Question 1. Question 2. A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency test. The professor wants the class to be able to score above 70 on the test. The six students get the following scores:62, 92, 75 ...
Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing.
Fundamentals of Hypothesis Testing. Hypothesis testing is structured around making a claim in the form of competing hypotheses, gathering data, performing statistical tests, and making decisions about which hypothesis the evidence supports. Here are some key terms about hypotheses and the testing process:
A regression analysis between sales (y in $1000) and advertising (x in dollars) resulted in the following equation: y-hat = 30,000 + 4x\\The above equation implies that an increase of 1 in advertising is associated with increase of 4000 in sales A standard normal distribution is one that has zero mean and variance = 1 Regression analysis is a statistical procedure for developing a mathematical ...
In statistical hypothesis testing, a test statistic is a crucial tool used to determine the validity of the hypothesis about a population parameter. ... Question 6: We have two independent samples with the following the statistics: Sample 1 (n=15, mean=25, variance=9) and Sample 2 (n=20, mean=22, variance=16). Test if the variances are equal at ...
Not necessarily, and so we can only conclude that the p-value is one part of what we may need for evidence for or against a hypothesis (hint: part of the solution is to consider effect size — introduced in Chapter 9.2 — and the statistical power of the test, see Ch 11). What follows was covered by Goodman (1988) and others.
Bootstrapping depends heavily on the estimator used and, though simple, naive use of bootstrapping will not always yield asymptotically valid results and can lead to inconsistency. [17] Although bootstrapping is (under some conditions) asymptotically consistent, it does not provide general finite-sample guarantees.The result may depend on the representative sample.