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StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

StatPearls [Internet].

Hypothesis testing, p values, confidence intervals, and significance.

Jacob Shreffler ; Martin R. Huecker .

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Last Update: March 13, 2023 .

Definition/Introduction

Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting these findings, which may affect the adequate application of the data.

Issues of Concern

Without a foundational understanding of hypothesis testing, p values, confidence intervals, and the difference between statistical and clinical significance, it may affect healthcare providers' ability to make clinical decisions without relying purely on the research investigators deemed level of significance. Therefore, an overview of these concepts is provided to allow medical professionals to use their expertise to determine if results are reported sufficiently and if the study outcomes are clinically appropriate to be applied in healthcare practice.

Hypothesis Testing

Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. Any research problem or statement is grounded in a better understanding of relationships between two or more variables. For this article, we will use the following research question example:

Research Question: Is Drug 23 an effective treatment for Disease A?

Research questions do not directly imply specific guesses or predictions; we must formulate research hypotheses. A hypothesis is a predetermined declaration regarding the research question in which the investigator(s) makes a precise, educated guess about a study outcome. This is sometimes called the alternative hypothesis and ultimately allows the researcher to take a stance based on experience or insight from medical literature. An example of a hypothesis is below.

Research Hypothesis: Drug 23 will significantly reduce symptoms associated with Disease A compared to Drug 22.

The null hypothesis states that there is no statistical difference between groups based on the stated research hypothesis.

Researchers should be aware of journal recommendations when considering how to report p values, and manuscripts should remain internally consistent.

Regarding p values, as the number of individuals enrolled in a study (the sample size) increases, the likelihood of finding a statistically significant effect increases. With very large sample sizes, the p-value can be very low significant differences in the reduction of symptoms for Disease A between Drug 23 and Drug 22. The null hypothesis is deemed true until a study presents significant data to support rejecting the null hypothesis. Based on the results, the investigators will either reject the null hypothesis (if they found significant differences or associations) or fail to reject the null hypothesis (they could not provide proof that there were significant differences or associations).

To test a hypothesis, researchers obtain data on a representative sample to determine whether to reject or fail to reject a null hypothesis. In most research studies, it is not feasible to obtain data for an entire population. Using a sampling procedure allows for statistical inference, though this involves a certain possibility of error. [1] When determining whether to reject or fail to reject the null hypothesis, mistakes can be made: Type I and Type II errors. Though it is impossible to ensure that these errors have not occurred, researchers should limit the possibilities of these faults. [2]

Significance

Significance is a term to describe the substantive importance of medical research. Statistical significance is the likelihood of results due to chance. [3] Healthcare providers should always delineate statistical significance from clinical significance, a common error when reviewing biomedical research. [4] When conceptualizing findings reported as either significant or not significant, healthcare providers should not simply accept researchers' results or conclusions without considering the clinical significance. Healthcare professionals should consider the clinical importance of findings and understand both p values and confidence intervals so they do not have to rely on the researchers to determine the level of significance. [5] One criterion often used to determine statistical significance is the utilization of p values.

P values are used in research to determine whether the sample estimate is significantly different from a hypothesized value. The p-value is the probability that the observed effect within the study would have occurred by chance if, in reality, there was no true effect. Conventionally, data yielding a p<0.05 or p<0.01 is considered statistically significant. While some have debated that the 0.05 level should be lowered, it is still universally practiced. [6] Hypothesis testing allows us to determine the size of the effect.

An example of findings reported with p values are below:

Statement: Drug 23 reduced patients' symptoms compared to Drug 22. Patients who received Drug 23 (n=100) were 2.1 times less likely than patients who received Drug 22 (n = 100) to experience symptoms of Disease A, p<0.05.

Statement:Individuals who were prescribed Drug 23 experienced fewer symptoms (M = 1.3, SD = 0.7) compared to individuals who were prescribed Drug 22 (M = 5.3, SD = 1.9). This finding was statistically significant, p= 0.02.

For either statement, if the threshold had been set at 0.05, the null hypothesis (that there was no relationship) should be rejected, and we should conclude significant differences. Noticeably, as can be seen in the two statements above, some researchers will report findings with < or > and others will provide an exact p-value (0.000001) but never zero [6] . When examining research, readers should understand how p values are reported. The best practice is to report all p values for all variables within a study design, rather than only providing p values for variables with significant findings. [7] The inclusion of all p values provides evidence for study validity and limits suspicion for selective reporting/data mining.

While researchers have historically used p values, experts who find p values problematic encourage the use of confidence intervals. [8] . P-values alone do not allow us to understand the size or the extent of the differences or associations. [3] In March 2016, the American Statistical Association (ASA) released a statement on p values, noting that scientific decision-making and conclusions should not be based on a fixed p-value threshold (e.g., 0.05). They recommend focusing on the significance of results in the context of study design, quality of measurements, and validity of data. Ultimately, the ASA statement noted that in isolation, a p-value does not provide strong evidence. [9]

When conceptualizing clinical work, healthcare professionals should consider p values with a concurrent appraisal study design validity. For example, a p-value from a double-blinded randomized clinical trial (designed to minimize bias) should be weighted higher than one from a retrospective observational study [7] . The p-value debate has smoldered since the 1950s [10] , and replacement with confidence intervals has been suggested since the 1980s. [11]

Confidence Intervals

A confidence interval provides a range of values within given confidence (e.g., 95%), including the accurate value of the statistical constraint within a targeted population. [12] Most research uses a 95% CI, but investigators can set any level (e.g., 90% CI, 99% CI). [13] A CI provides a range with the lower bound and upper bound limits of a difference or association that would be plausible for a population. [14] Therefore, a CI of 95% indicates that if a study were to be carried out 100 times, the range would contain the true value in 95, [15] confidence intervals provide more evidence regarding the precision of an estimate compared to p-values. [6]

In consideration of the similar research example provided above, one could make the following statement with 95% CI:

Statement: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22; there was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

It is important to note that the width of the CI is affected by the standard error and the sample size; reducing a study sample number will result in less precision of the CI (increase the width). [14] A larger width indicates a smaller sample size or a larger variability. [16] A researcher would want to increase the precision of the CI. For example, a 95% CI of 1.43 – 1.47 is much more precise than the one provided in the example above. In research and clinical practice, CIs provide valuable information on whether the interval includes or excludes any clinically significant values. [14]

Null values are sometimes used for differences with CI (zero for differential comparisons and 1 for ratios). However, CIs provide more information than that. [15] Consider this example: A hospital implements a new protocol that reduced wait time for patients in the emergency department by an average of 25 minutes (95% CI: -2.5 – 41 minutes). Because the range crosses zero, implementing this protocol in different populations could result in longer wait times; however, the range is much higher on the positive side. Thus, while the p-value used to detect statistical significance for this may result in "not significant" findings, individuals should examine this range, consider the study design, and weigh whether or not it is still worth piloting in their workplace.

Similarly to p-values, 95% CIs cannot control for researchers' errors (e.g., study bias or improper data analysis). [14] In consideration of whether to report p-values or CIs, researchers should examine journal preferences. When in doubt, reporting both may be beneficial. [13] An example is below:

Reporting both: Individuals who were prescribed Drug 23 had no symptoms after three days, which was significantly faster than those prescribed Drug 22, p = 0.009. There was a mean difference between the two groups of days to the recovery of 4.2 days (95% CI: 1.9 – 7.8).

Clinical Significance

Recall that clinical significance and statistical significance are two different concepts. Healthcare providers should remember that a study with statistically significant differences and large sample size may be of no interest to clinicians, whereas a study with smaller sample size and statistically non-significant results could impact clinical practice. [14] Additionally, as previously mentioned, a non-significant finding may reflect the study design itself rather than relationships between variables.

Healthcare providers using evidence-based medicine to inform practice should use clinical judgment to determine the practical importance of studies through careful evaluation of the design, sample size, power, likelihood of type I and type II errors, data analysis, and reporting of statistical findings (p values, 95% CI or both). [4] Interestingly, some experts have called for "statistically significant" or "not significant" to be excluded from work as statistical significance never has and will never be equivalent to clinical significance. [17]

The decision on what is clinically significant can be challenging, depending on the providers' experience and especially the severity of the disease. Providers should use their knowledge and experiences to determine the meaningfulness of study results and make inferences based not only on significant or insignificant results by researchers but through their understanding of study limitations and practical implications.

Nursing, Allied Health, and Interprofessional Team Interventions

All physicians, nurses, pharmacists, and other healthcare professionals should strive to understand the concepts in this chapter. These individuals should maintain the ability to review and incorporate new literature for evidence-based and safe care.

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Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

Cite this Page Shreffler J, Huecker MR. Hypothesis Testing, P Values, Confidence Intervals, and Significance. [Updated 2023 Mar 13]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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Hypothesis Testing: A Complete Guide for Beginners

Statistical hypothesis testing is a key concept in statistics. It helps researchers, data analysts, and scientists make decisions based on data. Hypothesis testing allows you to determine whether your results are meaningful when analyzing experiments, surveys, or other data.

In this blog, we’ll explain statistical hypothesis testing from the basics to more advanced ideas, making it easy to understand even for 10th-grade students.

By the end of this blog, you’ll be able to understand hypothesis testing and how it’s used in research.

What is a Hypothesis?

Table of Contents

A hypothesis is a statement that can be tested. It’s like a guess you make after observing something, and you want to see if that guess holds when you collect more data.

For example:

“Eating more vegetables improves health.”
“Students who study regularly perform better in exams.”

These statements are testable because we can gather data to check if they are true or false.

What is Hypothesis Testing?

Hypothesis testing is a statistical process that helps us make decisions based on data. Suppose you collect data from an experiment or survey. Hypothesis testing helps you decide whether the results are significant or could have happened by chance.

For example, if you believe a new teaching method helps students score better, hypothesis testing can help you decide if the improvement is real or just a random fluctuation.

Null and Alternative Hypothesis

Hypothesis testing usually involves two competing hypotheses:

Example: “There is no difference in exam scores between students using the new method and those who don’t.”
Example: “Students using the new method perform better in exams than those who don’t.”

Key Terms in Hypothesis Testing

Before diving into the details, let’s understand some important terms used in hypothesis testing:

1. Test Statistic

The test statistic is a number calculated from your data that is compared against a known distribution (like the normal distribution) to test the null hypothesis. It tells you how much your sample data differs from what’s expected under the null hypothesis.

The p-value is the probability of observing the sample data or something more extreme, assuming the null hypothesis is true. A smaller p-value suggests that the null hypothesis is less likely to be true. In many studies, a p-value of 0.05 or less is considered statistically significant.

3. Significance Level (α)

The significance level is the threshold at which you decide to reject the null hypothesis. Commonly, this level is set at 5% (α = 0.05), meaning there’s a 5% chance of rejecting the null hypothesis even when it is true.

4. Critical Value

The critical value is the boundary that defines the region where we reject the null hypothesis. It is calculated based on the significance level and tells us how extreme the test statistic needs to be to reject the null hypothesis.

5. Type I and Type II Errors

Type I Error (False Positive): Rejecting the null hypothesis when it’s true.
Type II Error (False Negative): Failing to reject the null hypothesis when it’s false.

In simpler terms:

Type I error is like thinking something has changed when it hasn’t.
Type II error is like thinking nothing has changed when it actually has.

Types of Hypothesis Testing

1. one-tailed test.

A one-tailed test checks for an effect in a single direction. For example, if you are only interested in testing whether students who study 2 hours daily score higher than those who don’t, that’s a one-tailed test.

2. Two-Tailed Test

A two-tailed test checks for an effect in both directions. This means you’re testing if the scores are different , regardless of whether they are higher or lower. For example, “Do students who study 2 hours daily score differently than those who don’t?” That’s a two-tailed test.

Steps in Hypothesis Testing

Step 1: define hypotheses.

Start by defining the:

Null Hypothesis (H₀): The status quo or no change.
Alternative Hypothesis (H₁): The hypothesis you believe in, suggesting that something has changed.

Step 2: Set the Significance Level (α)

Next, set the significance level, typically 0.05 . This means you’re willing to accept a 5% risk of incorrectly rejecting the null hypothesis.

Step 3: Collect and Analyze Data

Conduct your experiment or survey and collect data. Then, analyze this data to calculate the test statistic. The formula you use depends on the type of test you’re conducting (e.g., Z-test, T-test).

Step 4: Calculate the P-value or Critical Value

Compare the test statistic to a standard distribution (such as the normal distribution). If you calculate a p-value , compare it to the significance level. If the p-value is less than the significance level, reject the null hypothesis.

Alternatively, you can compare your test statistic to a critical value from statistical tables to determine if you should reject the null hypothesis.

Step 5: Make a Decision

Based on your calculations:

If the p-value is less than the significance level (e.g., p < 0.05), reject the null hypothesis.
If the p-value is greater than the significance level, do not reject the null hypothesis.

Step 6: Interpret the Results

Finally, interpret the results in context. If you reject the null hypothesis, you have evidence to support the alternative hypothesis. If not, the data does not provide enough evidence to reject the null.

P-Value and Significance

The p-value is a key part of hypothesis testing. It tells us the likelihood of getting results as extreme as the observed data, assuming the null hypothesis is true. In simple terms:

A low p-value (≤ 0.05) suggests strong evidence against the null hypothesis, so you reject it.
A high p-value (> 0.05) means the data is consistent with the null hypothesis, and you don’t reject it.

Here’s a table to summarize:

Common Hypothesis Tests

There are different types of hypothesis tests depending on the data and what you are testing for.

Example of Hypothesis Testing

Let’s say a nutritionist claims that a new diet increases the average weight loss for people by 5 kg in a month.

Null Hypothesis (H₀): The average weight loss is not 5 kg (no difference).
Alternative Hypothesis (H₁): The average weight loss is greater than 5 kg.

Suppose we collect data from 30 people and find that the average weight loss is 5.5 kg. Now we follow these steps:

Significance level : Set α = 0.05 (5%).
Calculate the test statistic: Using the T-test formula.
Find the p-value : Calculate the p-value for the test statistic.
Make a decision : Compare the p-value to the significance level.

If the p-value is less than 0.05, we reject the null hypothesis and conclude that the new diet results in more than 5 kg of weight loss.

Statistical hypothesis testing is an essential method in statistics for making informed decisions based on data. By understanding the basics of null and alternative hypotheses, test statistics, p-values, and the steps in hypothesis testing, you can analyze experiments and surveys effectively.

Hypothesis testing is a powerful tool for everything from scientific research to everyday decisions, and mastering it can lead to better data analysis and decision-making.

Also Read: Step-by-step guide to hypothesis testing in statistics

What is the difference between the null hypothesis and the alternative hypothesis?

The null hypothesis (H₀) is the default assumption that there is no effect or no difference. It’s what we try to disprove. The alternative hypothesis (H₁) is what you want to prove. It suggests that there is a significant effect or difference.

What is the difference between a one-tailed test and a two-tailed test?

A one-tailed test looks for evidence of an effect in one direction (either greater or smaller). A two-tailed test checks for evidence of an effect in both directions (whether greater or smaller), making it a more conservative test.

Can we always reject the null hypothesis if the p-value is less than 0.05?

Yes, if the p-value is less than 0.05 , we typically reject the null hypothesis. However, this does not guarantee that the alternative hypothesis is true; it simply indicates that the data provide strong evidence against it.

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Hypothesis Testing – A Complete Guide with Examples

Published by Alvin Nicolas at August 14th, 2021 , Revised On October 26, 2023

In statistics, hypothesis testing is a critical tool. It allows us to make informed decisions about populations based on sample data. Whether you are a researcher trying to prove a scientific point, a marketer analysing A/B test results, or a manufacturer ensuring quality control, hypothesis testing plays a pivotal role. This guide aims to introduce you to the concept and walk you through real-world examples.

What is a Hypothesis and a Hypothesis Testing?

A hypothesis is considered a belief or assumption that has to be accepted, rejected, proved or disproved. In contrast, a research hypothesis is a research question for a researcher that has to be proven correct or incorrect through investigation.

What is Hypothesis Testing?

Hypothesis testing is a scientific method used for making a decision and drawing conclusions by using a statistical approach. It is used to suggest new ideas by testing theories to know whether or not the sample data supports research. A research hypothesis is a predictive statement that has to be tested using scientific methods that join an independent variable to a dependent variable.

Example: The academic performance of student A is better than student B

Characteristics of the Hypothesis to be Tested

A hypothesis should be:

Clear and precise
Capable of being tested
Able to relate to a variable
Stated in simple terms
Consistent with known facts
Limited in scope and specific
Tested in a limited timeframe
Explain the facts in detail

What is a Null Hypothesis and Alternative Hypothesis?

A null hypothesis is a hypothesis when there is no significant relationship between the dependent and the participants’ independent variables .

In simple words, it’s a hypothesis that has been put forth but hasn’t been proved as yet. A researcher aims to disprove the theory. The abbreviation “Ho” is used to denote a null hypothesis.

If you want to compare two methods and assume that both methods are equally good, this assumption is considered the null hypothesis.

Example: In an automobile trial, you feel that the new vehicle’s mileage is similar to the previous model of the car, on average. You can write it as: Ho: there is no difference between the mileage of both vehicles. If your findings don’t support your hypothesis and you get opposite results, this outcome will be considered an alternative hypothesis.

If you assume that one method is better than another method, then it’s considered an alternative hypothesis. The alternative hypothesis is the theory that a researcher seeks to prove and is typically denoted by H1 or HA.

If you support a null hypothesis, it means you’re not supporting the alternative hypothesis. Similarly, if you reject a null hypothesis, it means you are recommending the alternative hypothesis.

Example: In an automobile trial, you feel that the new vehicle’s mileage is better than the previous model of the vehicle. You can write it as; Ha: the two vehicles have different mileage. On average/ the fuel consumption of the new vehicle model is better than the previous model.

If a null hypothesis is rejected during the hypothesis test, even if it’s true, then it is considered as a type-I error. On the other hand, if you don’t dismiss a hypothesis, even if it’s false because you could not identify its falseness, it’s considered a type-II error.

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How to Conduct Hypothesis Testing?

Here is a step-by-step guide on how to conduct hypothesis testing.

Step 1: State the Null and Alternative Hypothesis

Once you develop a research hypothesis, it’s important to state it is as a Null hypothesis (Ho) and an Alternative hypothesis (Ha) to test it statistically.

A null hypothesis is a preferred choice as it provides the opportunity to test the theory. In contrast, you can accept the alternative hypothesis when the null hypothesis has been rejected.

Example: You want to identify a relationship between obesity of men and women and the modern living style. You develop a hypothesis that women, on average, gain weight quickly compared to men. Then you write it as: Ho: Women, on average, don’t gain weight quickly compared to men. Ha: Women, on average, gain weight quickly compared to men.

Step 2: Data Collection

Hypothesis testing follows the statistical method, and statistics are all about data. It’s challenging to gather complete information about a specific population you want to study. You need to gather the data obtained through a large number of samples from a specific population.

Example: Suppose you want to test the difference in the rate of obesity between men and women. You should include an equal number of men and women in your sample. Then investigate various aspects such as their lifestyle, eating patterns and profession, and any other variables that may influence average weight. You should also determine your study’s scope, whether it applies to a specific group of population or worldwide population. You can use available information from various places, countries, and regions.

Step 3: Select Appropriate Statistical Test

There are many types of statistical tests , but we discuss the most two common types below, such as One-sided and two-sided tests.

Note: Your choice of the type of test depends on the purpose of your study

One-sided Test

In the one-sided test, the values of rejecting a null hypothesis are located in one tail of the probability distribution. The set of values is less or higher than the critical value of the test. It is also called a one-tailed test of significance.

Example: If you want to test that all mangoes in a basket are ripe. You can write it as: Ho: All mangoes in the basket, on average, are ripe. If you find all ripe mangoes in the basket, the null hypothesis you developed will be true.

Two-sided Test

In the two-sided test, the values of rejecting a null hypothesis are located on both tails of the probability distribution. The set of values is less or higher than the first critical value of the test and higher than the second critical value test. It is also called a two-tailed test of significance.

Example: Nothing can be explicitly said whether all mangoes are ripe in the basket. If you reject the null hypothesis (Ho: All mangoes in the basket, on average, are ripe), then it means all mangoes in the basket are not likely to be ripe. A few mangoes could be raw as well.

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Step 4: Select the Level of Significance

When you reject a null hypothesis, even if it’s true during a statistical hypothesis, it is considered the significance level . It is the probability of a type one error. The significance should be as minimum as possible to avoid the type-I error, which is considered severe and should be avoided.

If the significance level is minimum, then it prevents the researchers from false claims.

The significance level is denoted by P, and it has given the value of 0.05 (P=0.05)

If the P-Value is less than 0.05, then the difference will be significant. If the P-value is higher than 0.05, then the difference is non-significant.

Example: Suppose you apply a one-sided test to test whether women gain weight quickly compared to men. You get to know about the average weight between men and women and the factors promoting weight gain.

Step 5: Find out Whether the Null Hypothesis is Rejected or Supported

After conducting a statistical test, you should identify whether your null hypothesis is rejected or accepted based on the test results. It would help if you observed the P-value for this.

Example: If you find the P-value of your test is less than 0.5/5%, then you need to reject your null hypothesis (Ho: Women, on average, don’t gain weight quickly compared to men). On the other hand, if a null hypothesis is rejected, then it means the alternative hypothesis might be true (Ha: Women, on average, gain weight quickly compared to men. If you find your test’s P-value is above 0.5/5%, then it means your null hypothesis is true.

Step 6: Present the Outcomes of your Study

The final step is to present the outcomes of your study . You need to ensure whether you have met the objectives of your research or not.

In the discussion section and conclusion , you can present your findings by using supporting evidence and conclude whether your null hypothesis was rejected or supported.

In the result section, you can summarise your study’s outcomes, including the average difference and P-value of the two groups.

If we talk about the findings, our study your results will be as follows:

Example: In the study of identifying whether women gain weight quickly compared to men, we found the P-value is less than 0.5. Hence, we can reject the null hypothesis (Ho: Women, on average, don’t gain weight quickly than men) and conclude that women may likely gain weight quickly than men.

Did you know in your academic paper you should not mention whether you have accepted or rejected the null hypothesis?

Always remember that you either conclude to reject Ho in favor of Haor do not reject Ho . It would help if you never rejected Ha or even accept Ha .

Suppose your null hypothesis is rejected in the hypothesis testing. If you conclude reject Ho in favor of Haor do not reject Ho, then it doesn’t mean that the null hypothesis is true. It only means that there is a lack of evidence against Ho in favour of Ha. If your null hypothesis is not true, then the alternative hypothesis is likely to be true.

Example: We found that the P-value is less than 0.5. Hence, we can conclude reject Ho in favour of Ha (Ho: Women, on average, don’t gain weight quickly than men) reject Ho in favour of Ha. However, rejected in favour of Ha means (Ha: women may likely to gain weight quickly than men)

Frequently Asked Questions

What are the 3 types of hypothesis test.

The 3 types of hypothesis tests are:

One-Sample Test : Compare sample data to a known population value.
Two-Sample Test : Compare means between two sample groups.
ANOVA : Analyze variance among multiple groups to determine significant differences.

What is a hypothesis?

A hypothesis is a proposed explanation or prediction about a phenomenon, often based on observations. It serves as a starting point for research or experimentation, providing a testable statement that can either be supported or refuted through data and analysis. In essence, it’s an educated guess that drives scientific inquiry.

What are null hypothesis?

A null hypothesis (often denoted as H0) suggests that there is no effect or difference in a study or experiment. It represents a default position or status quo. Statistical tests evaluate data to determine if there’s enough evidence to reject this null hypothesis.

What is the probability value?

The probability value, or p-value, is a measure used in statistics to determine the significance of an observed effect. It indicates the probability of obtaining the observed results, or more extreme, if the null hypothesis were true. A small p-value (typically <0.05) suggests evidence against the null hypothesis, warranting its rejection.

What is p value?

The p-value is a fundamental concept in statistical hypothesis testing. It represents the probability of observing a test statistic as extreme, or more so, than the one calculated from sample data, assuming the null hypothesis is true. A low p-value suggests evidence against the null, possibly justifying its rejection.

What is a t test?

A t-test is a statistical test used to compare the means of two groups. It determines if observed differences between the groups are statistically significant or if they likely occurred by chance. Commonly applied in research, there are different t-tests, including independent, paired, and one-sample, tailored to various data scenarios.

When to reject null hypothesis?

Reject the null hypothesis when the test statistic falls into a predefined rejection region or when the p-value is less than the chosen significance level (commonly 0.05). This suggests that the observed data is unlikely under the null hypothesis, indicating evidence for the alternative hypothesis. Always consider the study’s context.

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What is Hypothesis Testing? Types and Methods

Soumyaa Rawat
Jul 23, 2021

Hypothesis Testing

Hypothesis testing is the act of testing a hypothesis or a supposition in relation to a statistical parameter. Analysts implement hypothesis testing in order to test if a hypothesis is plausible or not.

In data science and statistics , hypothesis testing is an important step as it involves the verification of an assumption that could help develop a statistical parameter. For instance, a researcher establishes a hypothesis assuming that the average of all odd numbers is an even number.

In order to find the plausibility of this hypothesis, the researcher will have to test the hypothesis using hypothesis testing methods. Unlike a hypothesis that is ‘supposed’ to stand true on the basis of little or no evidence, hypothesis testing is required to have plausible evidence in order to establish that a statistical hypothesis is true.

Perhaps this is where statistics play an important role. A number of components are involved in this process. But before understanding the process involved in hypothesis testing in research methodology, we shall first understand the types of hypotheses that are involved in the process. Let us get started!

Types of Hypotheses

In data sampling, different types of hypothesis are involved in finding whether the tested samples test positive for a hypothesis or not. In this segment, we shall discover the different types of hypotheses and understand the role they play in hypothesis testing.

Alternative Hypothesis

Alternative Hypothesis (H1) or the research hypothesis states that there is a relationship between two variables (where one variable affects the other). The alternative hypothesis is the main driving force for hypothesis testing.

It implies that the two variables are related to each other and the relationship that exists between them is not due to chance or coincidence.

When the process of hypothesis testing is carried out, the alternative hypothesis is the main subject of the testing process. The analyst intends to test the alternative hypothesis and verifies its plausibility.

Null Hypothesis

The Null Hypothesis (H0) aims to nullify the alternative hypothesis by implying that there exists no relation between two variables in statistics. It states that the effect of one variable on the other is solely due to chance and no empirical cause lies behind it.

The null hypothesis is established alongside the alternative hypothesis and is recognized as important as the latter. In hypothesis testing, the null hypothesis has a major role to play as it influences the testing against the alternative hypothesis.

(Must read: What is ANOVA test? )

Non-Directional Hypothesis

The Non-directional hypothesis states that the relation between two variables has no direction.

Simply put, it asserts that there exists a relation between two variables, but does not recognize the direction of effect, whether variable A affects variable B or vice versa.

Directional Hypothesis

The Directional hypothesis, on the other hand, asserts the direction of effect of the relationship that exists between two variables.

Herein, the hypothesis clearly states that variable A affects variable B, or vice versa.

Statistical Hypothesis

A statistical hypothesis is a hypothesis that can be verified to be plausible on the basis of statistics.

By using data sampling and statistical knowledge, one can determine the plausibility of a statistical hypothesis and find out if it stands true or not.

(Related blog: z-test vs t-test )

Performing Hypothesis Testing

Now that we have understood the types of hypotheses and the role they play in hypothesis testing, let us now move on to understand the process in a better manner.

In hypothesis testing, a researcher is first required to establish two hypotheses - alternative hypothesis and null hypothesis in order to begin with the procedure.

To establish these two hypotheses, one is required to study data samples, find a plausible pattern among the samples, and pen down a statistical hypothesis that they wish to test.

A random population of samples can be drawn, to begin with hypothesis testing. Among the two hypotheses, alternative and null, only one can be verified to be true. Perhaps the presence of both hypotheses is required to make the process successful.

At the end of the hypothesis testing procedure, either of the hypotheses will be rejected and the other one will be supported. Even though one of the two hypotheses turns out to be true, no hypothesis can ever be verified 100%.

(Read also: Types of data sampling techniques )

Therefore, a hypothesis can only be supported based on the statistical samples and verified data. Here is a step-by-step guide for hypothesis testing.

Establish the hypotheses

First things first, one is required to establish two hypotheses - alternative and null, that will set the foundation for hypothesis testing.

These hypotheses initiate the testing process that involves the researcher working on data samples in order to either support the alternative hypothesis or the null hypothesis.

Generate a testing plan

Once the hypotheses have been formulated, it is now time to generate a testing plan. A testing plan or an analysis plan involves the accumulation of data samples, determining which statistic is to be considered and laying out the sample size.

All these factors are very important while one is working on hypothesis testing.

Analyze data samples

As soon as a testing plan is ready, it is time to move on to the analysis part. Analysis of data samples involves configuring statistical values of samples, drawing them together, and deriving a pattern out of these samples.

While analyzing the data samples, a researcher needs to determine a set of things -

Significance Level - The level of significance in hypothesis testing indicates if a statistical result could have significance if the null hypothesis stands to be true.

Testing Method - The testing method involves a type of sampling-distribution and a test statistic that leads to hypothesis testing. There are a number of testing methods that can assist in the analysis of data samples.

Test statistic - Test statistic is a numerical summary of a data set that can be used to perform hypothesis testing.

P-value - The P-value interpretation is the probability of finding a sample statistic to be as extreme as the test statistic, indicating the plausibility of the null hypothesis.

Infer the results

The analysis of data samples leads to the inference of results that establishes whether the alternative hypothesis stands true or not. When the P-value is less than the significance level, the null hypothesis is rejected and the alternative hypothesis turns out to be plausible.

Methods of Hypothesis Testing

As we have already looked into different aspects of hypothesis testing, we shall now look into the different methods of hypothesis testing. All in all, there are 2 most common types of hypothesis testing methods. They are as follows -

Frequentist Hypothesis Testing

The frequentist hypothesis or the traditional approach to hypothesis testing is a hypothesis testing method that aims on making assumptions by considering current data.

The supposed truths and assumptions are based on the current data and a set of 2 hypotheses are formulated. A very popular subtype of the frequentist approach is the Null Hypothesis Significance Testing (NHST).

The NHST approach (involving the null and alternative hypothesis) has been one of the most sought-after methods of hypothesis testing in the field of statistics ever since its inception in the mid-1950s.

Bayesian Hypothesis Testing

A much unconventional and modern method of hypothesis testing, the Bayesian Hypothesis Testing claims to test a particular hypothesis in accordance with the past data samples, known as prior probability, and current data that lead to the plausibility of a hypothesis.

The result obtained indicates the posterior probability of the hypothesis. In this method, the researcher relies on ‘prior probability and posterior probability’ to conduct hypothesis testing on hand.

On the basis of this prior probability, the Bayesian approach tests a hypothesis to be true or false. The Bayes factor, a major component of this method, indicates the likelihood ratio among the null hypothesis and the alternative hypothesis.

The Bayes factor is the indicator of the plausibility of either of the two hypotheses that are established for hypothesis testing.

(Also read - Introduction to Bayesian Statistics )

To conclude, hypothesis testing, a way to verify the plausibility of a supposed assumption can be done through different methods - the Bayesian approach or the Frequentist approach.

Although the Bayesian approach relies on the prior probability of data samples, the frequentist approach assumes without a probability. A number of elements involved in hypothesis testing are - significance level, p-level, test statistic, and method of hypothesis testing.

(Also read: Introduction to probability distributions )

A significant way to determine whether a hypothesis stands true or not is to verify the data samples and identify the plausible hypothesis among the null hypothesis and alternative hypothesis.

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Accepted 2019 Mar 28; Issue date 2019 Jul.

Key points.

Hypothesis tests are used to assess whether a difference between two samples represents a real difference between the populations from which the samples were taken.

A null hypothesis of ‘no difference’ is taken as a starting point, and we calculate the probability that both sets of data came from the same population. This probability is expressed as a p -value.

When the null hypothesis is false, p- values tend to be small. When the null hypothesis is true, any p- value is equally likely.

Learning objectives.

By reading this article, you should be able to:

Explain why hypothesis testing is used.

Use a table to determine which hypothesis test should be used for a particular situation.

Interpret a p- value.

A hypothesis test is a procedure used in statistics to assess whether a particular viewpoint is likely to be true. They follow a strict protocol, and they generate a ‘ p- value’, on the basis of which a decision is made about the truth of the hypothesis under investigation. All of the routine statistical ‘tests’ used in research— t- tests, χ 2 tests, Mann–Whitney tests, etc.—are all hypothesis tests, and in spite of their differences they are all used in essentially the same way. But why do we use them at all?

Comparing the heights of two individuals is easy: we can measure their height in a standardised way and compare them. When we want to compare the heights of two small well-defined groups (for example two groups of children), we need to use a summary statistic that we can calculate for each group. Such summaries (means, medians, etc.) form the basis of descriptive statistics, and are well described elsewhere. 1 However, a problem arises when we try to compare very large groups or populations: it may be impractical or even impossible to take a measurement from everyone in the population, and by the time you do so, the population itself will have changed. A similar problem arises when we try to describe the effects of drugs—for example by how much on average does a particular vasopressor increase MAP?

To solve this problem, we use random samples to estimate values for populations. By convention, the values we calculate from samples are referred to as statistics and denoted by Latin letters ( x ¯ for sample mean; SD for sample standard deviation) while the unknown population values are called parameters , and denoted by Greek letters (μ for population mean, σ for population standard deviation).

Inferential statistics describes the methods we use to estimate population parameters from random samples; how we can quantify the level of inaccuracy in a sample statistic; and how we can go on to use these estimates to compare populations.

Sampling error

There are many reasons why a sample may give an inaccurate picture of the population it represents: it may be biased, it may not be big enough, and it may not be truly random. However, even if we have been careful to avoid these pitfalls, there is an inherent difference between the sample and the population at large. To illustrate this, let us imagine that the actual average height of males in London is 174 cm. If I were to sample 100 male Londoners and take a mean of their heights, I would be very unlikely to get exactly 174 cm. Furthermore, if somebody else were to perform the same exercise, it would be unlikely that they would get the same answer as I did. The sample mean is different each time it is taken, and the way it differs from the actual mean of the population is described by the standard error of the mean (standard error, or SEM ). The standard error is larger if there is a lot of variation in the population, and becomes smaller as the sample size increases. It is calculated thus:

where SD is the sample standard deviation, and n is the sample size.

As errors are normally distributed, we can use this to estimate a 95% confidence interval on our sample mean as follows:

We can interpret this as meaning ‘We are 95% confident that the actual mean is within this range.’

Some confusion arises at this point between the SD and the standard error. The SD is a measure of variation in the sample. The range x ¯ ± ( 1.96 × SD ) will normally contain 95% of all your data. It can be used to illustrate the spread of the data and shows what values are likely. In contrast, standard error tells you about the precision of the mean and is used to calculate confidence intervals.

One straightforward way to compare two samples is to use confidence intervals. If we calculate the mean height of two groups and find that the 95% confidence intervals do not overlap, this can be taken as evidence of a difference between the two means. This method of statistical inference is reasonably intuitive and can be used in many situations. 2 Many journals, however, prefer to report inferential statistics using p -values.

Inference testing using a null hypothesis

In 1925, the British statistician R.A. Fisher described a technique for comparing groups using a null hypothesis , a method which has dominated statistical comparison ever since. The technique itself is rather straightforward, but often gets lost in the mechanics of how it is done. To illustrate, imagine we want to compare the HR of two different groups of people. We take a random sample from each group, which we call our data. Then:

Assume that both samples came from the same group. This is our ‘null hypothesis’.

Calculate the probability that an experiment would give us these data, assuming that the null hypothesis is true. We express this probability as a p- value, a number between 0 and 1, where 0 is ‘impossible’ and 1 is ‘certain’.

If the probability of the data is low, we reject the null hypothesis and conclude that there must be a difference between the two groups.

Formally, we can define a p- value as ‘the probability of finding the observed result or a more extreme result, if the null hypothesis were true.’ Standard practice is to set a cut-off at p <0.05 (this cut-off is termed the alpha value). If the null hypothesis were true, a result such as this would only occur 5% of the time or less; this in turn would indicate that the null hypothesis itself is unlikely. Fisher described the process as follows: ‘Set a low standard of significance at the 5 per cent point, and ignore entirely all results which fail to reach this level. A scientific fact should be regarded as experimentally established only if a properly designed experiment rarely fails to give this level of significance.’ 3 This probably remains the most succinct description of the procedure.

A question which often arises at this point is ‘Why do we use a null hypothesis?’ The simple answer is that it is easy: we can readily describe what we would expect of our data under a null hypothesis, we know how data would behave, and we can readily work out the probability of getting the result that we did. It therefore makes a very simple starting point for our probability assessment. All probabilities require a set of starting conditions, in much the same way that measuring the distance to London needs a starting point. The null hypothesis can be thought of as an easy place to put the start of your ruler.

If a null hypothesis is rejected, an alternate hypothesis must be adopted in its place. The null and alternate hypotheses must be mutually exclusive, but must also between them describe all situations. If a null hypothesis is ‘no difference exists’ then the alternate should be simply ‘a difference exists’.

Hypothesis testing in practice

The components of a hypothesis test can be readily described using the acronym GOST: identify the Groups you wish to compare; define the Outcome to be measured; collect and Summarise the data; then evaluate the likelihood of the null hypothesis, using a Test statistic .

When considering groups, think first about how many. Is there just one group being compared against an audit standard, or are you comparing one group with another? Some studies may wish to compare more than two groups. Another situation may involve a single group measured at different points in time, for example before or after a particular treatment. In this situation each participant is compared with themselves, and this is often referred to as a ‘paired’ or a ‘repeated measures’ design. It is possible to combine these types of groups—for example a researcher may measure arterial BP on a number of different occasions in five different groups of patients. Such studies can be difficult, both to analyse and interpret.

In other studies we may want to see how a continuous variable (such as age or height) affects the outcomes. These techniques involve regression analysis, and are beyond the scope of this article.

The outcome measures are the data being collected. This may be a continuous measure, such as temperature or BMI, or it may be a categorical measure, such as ASA status or surgical specialty. Often, inexperienced researchers will strive to collect lots of outcome measures in an attempt to find something that differs between the groups of interest; if this is done, a ‘primary outcome measure’ should be identified before the research begins. In addition, the results of any hypothesis tests will need to be corrected for multiple measures.

The summary and the test statistic will be defined by the type of data that have been collected. The test statistic is calculated then transformed into a p- value using tables or software. It is worth looking at two common tests in a little more detail: the χ 2 test, and the t -test.

Categorical data: the χ 2 test

The χ 2 test of independence is a test for comparing categorical outcomes in two or more groups. For example, a number of trials have compared surgical site infections in patients who have been given different concentrations of oxygen perioperatively. In the PROXI trial, 4 685 patients received oxygen 80%, and 701 patients received oxygen 30%. In the 80% group there were 131 infections, while in the 30% group there were 141 infections. In this study, the groups were oxygen 80% and oxygen 30%, and the outcome measure was the presence of a surgical site infection.

The summary is a table ( Table 1 ), and the hypothesis test compares this table (the ‘observed’ table) with the table that would be expected if the proportion of infections in each group was the same (the ‘expected’ table). The test statistic is χ 2 , from which a p- value is calculated. In this instance the p -value is 0.64, which means that results like this would occur 64% of the time if the null hypothesis were true. We thus have no evidence to reject the null hypothesis; the observed difference probably results from sampling variation rather than from an inherent difference between the two groups.

Summary of the results of the PROXI trial. Figures are numbers of patients.

Continuous data: the t- test

The t- test is a statistical method for comparing means, and is one of the most widely used hypothesis tests. Imagine a study where we try to see if there is a difference in the onset time of a new neuromuscular blocking agent compared with suxamethonium. We could enlist 100 volunteers, give them a general anaesthetic, and randomise 50 of them to receive the new drug and 50 of them to receive suxamethonium. We then time how long it takes (in seconds) to have ideal intubation conditions, as measured by a quantitative nerve stimulator. Our data are therefore a list of times. In this case, the groups are ‘new drug’ and suxamethonium, and the outcome is time, measured in seconds. This can be summarised by using means; the hypothesis test will compare the means of the two groups, using a p- value calculated from a ‘ t statistic’. Hopefully it is becoming obvious at this point that the test statistic is usually identified by a letter, and this letter is often cited in the name of the test.

The t -test comes in a number of guises, depending on the comparison being made. A single sample can be compared with a standard (Is the BMI of school leavers in this town different from the national average?); two samples can be compared with each other, as in the example above; or the same study subjects can be measured at two different times. The latter case is referred to as a paired t- test, because each participant provides a pair of measurements—such as in a pre- or postintervention study.

A large number of methods for testing hypotheses exist; the commonest ones and their uses are described in Table 2 . In each case, the test can be described by detailing the groups being compared ( Table 2 , columns) the outcome measures (rows), the summary, and the test statistic. The decision to use a particular test or method should be made during the planning stages of a trial or experiment. At this stage, an estimate needs to be made of how many test subjects will be needed. Such calculations are described in detail elsewhere. 5

The principle types of hypothesis test. Tests comparing more than two samples can indicate that one group differs from the others, but will not identify which. Subsequent ‘post hoc’ testing is required if a difference is found.

Controversies surrounding hypothesis testing

Although hypothesis tests have been the basis of modern science since the middle of the 20th century, they have been plagued by misconceptions from the outset; this has led to what has been described as a crisis in science in the last few years: some journals have gone so far as to ban p -value s outright. 6 This is not because of any flaw in the concept of a p -value, but because of a lack of understanding of what they mean.

Possibly the most pervasive misunderstanding is the belief that the p- value is the chance that the null hypothesis is true, or that the p- value represents the frequency with which you will be wrong if you reject the null hypothesis (i.e. claim to have found a difference). This interpretation has frequently made it into the literature, and is a very easy trap to fall into when discussing hypothesis tests. To avoid this, it is important to remember that the p- value is telling us something about our sample , not about the null hypothesis. Put in simple terms, we would like to know the probability that the null hypothesis is true, given our data. The p- value tells us the probability of getting these data if the null hypothesis were true, which is not the same thing. This fallacy is referred to as ‘flipping the conditional’; the probability of an outcome under certain conditions is not the same as the probability of those conditions given that the outcome has happened.

A useful example is to imagine a magic trick in which you select a card from a normal deck of 52 cards, and the performer reveals your chosen card in a surprising manner. If the performer were relying purely on chance, this would only happen on average once in every 52 attempts. On the basis of this, we conclude that it is unlikely that the magician is simply relying on chance. Although simple, we have just performed an entire hypothesis test. We have declared a null hypothesis (the performer was relying on chance); we have even calculated a p -value (1 in 52, ≈0.02); and on the basis of this low p- value we have rejected our null hypothesis. We would, however, be wrong to suggest that there is a probability of 0.02 that the performer is relying on chance—that is not what our figure of 0.02 is telling us.

To explore this further we can create two populations, and watch what happens when we use simulation to take repeated samples to compare these populations. Computers allow us to do this repeatedly, and to see what p- value s are generated (see Supplementary online material). 7 Fig 1 illustrates the results of 100,000 simulated t -tests, generated in two set of circumstances. In Fig 1 a , we have a situation in which there is a difference between the two populations. The p- value s cluster below the 0.05 cut-off, although there is a small proportion with p >0.05. Interestingly, the proportion of comparisons where p <0.05 is 0.8 or 80%, which is the power of the study (the sample size was specifically calculated to give a power of 80%).

The p- value s generated when 100,000 t -tests are used to compare two samples taken from defined populations. ( a ) The populations have a difference and the p- value s are mostly significant. ( b ) The samples were taken from the same population (i.e. the null hypothesis is true) and the p- value s are distributed uniformly.

Figure 1 b depicts the situation where repeated samples are taken from the same parent population (i.e. the null hypothesis is true). Somewhat surprisingly, all p- value s occur with equal frequency, with p <0.05 occurring exactly 5% of the time. Thus, when the null hypothesis is true, a type I error will occur with a frequency equal to the alpha significance cut-off.

Figure 1 highlights the underlying problem: when presented with a p -value <0.05, is it possible with no further information, to determine whether you are looking at something from Fig 1 a or Fig 1 b ?

Finally, it cannot be stressed enough that although hypothesis testing identifies whether or not a difference is likely, it is up to us as clinicians to decide whether or not a statistically significant difference is also significant clinically.

Hypothesis testing: what next?

As mentioned above, some have suggested moving away from p -values, but it is not entirely clear what we should use instead. Some sources have advocated focussing more on effect size; however, without a measure of significance we have merely returned to our original problem: how do we know that our difference is not just a result of sampling variation?

One solution is to use Bayesian statistics. Up until very recently, these techniques have been considered both too difficult and not sufficiently rigorous. However, recent advances in computing have led to the development of Bayesian equivalents of a number of standard hypothesis tests. 8 These generate a ‘Bayes Factor’ (BF), which tells us how more (or less) likely the alternative hypothesis is after our experiment. A BF of 1.0 indicates that the likelihood of the alternate hypothesis has not changed. A BF of 10 indicates that the alternate hypothesis is 10 times more likely than we originally thought. A number of classifications for BF exist; greater than 10 can be considered ‘strong evidence’, while BF greater than 100 can be classed as ‘decisive’.

Figures such as the BF can be quoted in conjunction with the traditional p- value, but it remains to be seen whether they will become mainstream.

Declaration of interest

The author declares that they have no conflict of interest.

The associated MCQs (to support CME/CPD activity) will be accessible at www.bjaed.org/cme/home by subscribers to BJA Education .

Jason Walker FRCA FRSS BSc (Hons) Math Stat is a consultant anaesthetist at Ysbyty Gwynedd Hospital, Bangor, Wales, and an honorary senior lecturer at Bangor University. He is vice chair of his local research ethics committee, and an examiner for the Primary FRCA.

Matrix codes: 1A03, 2A04, 3J03

Supplementary data to this article can be found online at https://doi.org/10.1016/j.bjae.2019.03.006 .

Supplementary material

The following is the Supplementary data to this article:

1. McCluskey A., Lalkhen A.G. Statistics II: central tendency and spread of data. CEACCP. 2007;7:127–130. [ Google Scholar ]
2. Altman D.G., Machin D., Bryant T.N., Gardner M.J. 2nd Edn. BMJ Books; London: 2000. Statistics with confidence. [ Google Scholar ]
3. Fisher R.A. The arrangement of field experiments. J Min Agric Gr Br. 1926;33:503–513. [ Google Scholar ]
4. Meyhoff C.S., Wetterslev J., Jorgensen L.N. Effect of high perioperative oxygen fraction on surgical site infection and pulmonary complications after abdominal surgery: the PROXI randomized clinical trial. JAMA. 2009;302:1543–1550. doi: 10.1001/jama.2009.1452. [ DOI ] [ PubMed ] [ Google Scholar ]
5. Columb M.O., Atkinson M.S. Statistical analysis: sample size and power estimations. BJA Educ. 2016;16:159–161. [ Google Scholar ]
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7. Colquhoun D. An investigation of the false discovery rate and the misinterpretation of p-values. R Soc Open Sci. 2014;1:140216. doi: 10.1098/rsos.140216. [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ]
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Lesson 10 of 24 By Avijeet Biswal

In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis and hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life -

A teacher assumes that 60% of his college's students come from lower-middle-class families.
A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

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Importance of Hypothesis Testing in Data Analysis

Here is what makes hypothesis testing so important in data analysis and why it is key to making better decisions:

Avoiding Misleading Conclusions (Type I and Type II Errors)

One of the biggest benefits of hypothesis testing is that it helps you avoid jumping to the wrong conclusions. For instance, a Type I error could occur if a company launches a new product thinking it will be a hit, only to find out later that the data misled them. A Type II error might happen when a company overlooks a potentially successful product because their testing wasn’t thorough enough. By setting up the right significance level and carefully calculating the p-value, hypothesis testing minimizes the chances of these errors, leading to more accurate results.

Making Smarter Choices

Hypothesis testing is key to making smarter, evidence-based decisions. Let’s say a city planner wants to determine if building a new park will increase community engagement. By testing the hypothesis using data from similar projects, they can make an informed choice. Similarly, a teacher might use hypothesis testing to see if a new teaching method actually improves student performance. It’s about taking the guesswork out of decisions and relying on solid evidence instead.

Optimizing Business Tactics

In business, hypothesis testing is invaluable for testing new ideas and strategies before fully committing to them. For example, an e-commerce company might want to test whether offering free shipping increases sales. By using hypothesis testing, they can compare sales data from customers who received free shipping offers and those who didn’t. This allows them to base their business decisions on data, not hunches, reducing the risk of costly mistakes.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

Here, x̅ is the sample mean,
μ0 is the population mean,
σ is the standard deviation,
n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternative Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average.

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps in Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. To infer conclusions accurately, this data should be representative of the population.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

3. Chi-Square

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

ANOVA , or Analysis of Variance, is a statistical method used to compare the means of three or more groups. It’s particularly useful when you want to see if there are significant differences between multiple groups. For instance, in business, a company might use ANOVA to analyze whether three different stores are performing differently in terms of sales. It’s also widely used in fields like medical research and social sciences, where comparing group differences can provide valuable insights.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

The null hypothesis is (H0 <= 90) or less change.
A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true].

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

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Practice Problems on Hypothesis Testing

Here are the practice problems on hypothesis testing that will help you understand how to apply these concepts in real-world scenarios:

A telecom service provider claims that customers spend an average of ₹400 per month, with a standard deviation of ₹25. However, a random sample of 50 customer bills shows a mean of ₹250 and a standard deviation of ₹15. Does this sample data support the service provider’s claim?

Solution: Let’s break this down:

Null Hypothesis (H0): The average amount spent per month is ₹400.
Alternate Hypothesis (H1): The average amount spent per month is not ₹400.
Population Standard Deviation (σ): ₹25
Sample Size (n): 50
Sample Mean (x̄): ₹250

1. Calculate the z-value:

z=250-40025/50 −42.42

2. Compare with critical z-values: For a 5% significance level, critical z-values are -1.96 and +1.96. Since -42.42 is far outside this range, we reject the null hypothesis. The sample data suggests that the average amount spent is significantly different from ₹400.

Out of 850 customers, 400 made online grocery purchases. Can we conclude that more than 50% of customers are moving towards online grocery shopping?

Solution: Here’s how to approach it:

Proportion of customers who shopped online (p): 400 / 850 = 0.47
Null Hypothesis (H0): The proportion of online shoppers is 50% or more.
Alternate Hypothesis (H1): The proportion of online shoppers is less than 50%.
Sample Size (n): 850
Significance Level (α): 5%

z=p-PP(1-P)/n

z=0.47-0.500.50.5/850 −1.74

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is -1.645. Since -1.74 is less than -1.645, we reject the null hypothesis. This means the data does not support the idea that most customers are moving towards online grocery shopping.

In a study of code quality, Team A has 250 errors in 1000 lines of code, and Team B has 300 errors in 800 lines of code. Can we say Team B performs worse than Team A?

Solution: Let’s analyze it:

Proportion of errors for Team A (pA): 250 / 1000 = 0.25
Proportion of errors for Team B (pB): 300 / 800 = 0.375
Null Hypothesis (H0): Team B’s error rate is less than or equal to Team A’s.
Alternate Hypothesis (H1): Team B’s error rate is greater than Team A’s.
Sample Size for Team A (nA): 1000
Sample Size for Team B (nB): 800

p=nApA+nBpBnA+nB

p=10000.25+8000.3751000+800 ≈ 0.305

z=pA−pBp(1-p)(1nA+1nB)

z=0.25−0.3750.305(1-0.305) (11000+1800) ≈ −5.72

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is +1.645. Since -5.72 is far less than +1.645, we reject the null hypothesis. The data indicates that Team B’s performance is significantly worse than Team A’s.

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Applications of Hypothesis Testing

Apart from the practical problems, let's look at the real-world applications of hypothesis testing across various fields:

Medicine and Healthcare

In medicine, hypothesis testing plays a pivotal role in assessing the success of new treatments. For example, researchers may want to find out if a new exercise regimen improves heart health. By comparing data from patients who followed the program to those who didn’t, they can determine if the exercise significantly improves health outcomes. Such rigorous testing allows medical professionals to rely on proven methods rather than assumptions.

Quality Control and Manufacturing

In manufacturing, ensuring product quality is vital, and hypothesis testing helps maintain those standards. Suppose a beverage company introduces a new bottling process and wants to verify if it reduces contamination. By analyzing samples from the new and old processes, hypothesis testing can reveal whether the new method reduces the risk of contamination. This allows manufacturers to implement improvements that enhance product safety and quality confidently.

Education and Learning

In education and learning, hypothesis testing is a tool to evaluate the impact of innovative teaching techniques. Imagine a situation where teachers introduce project-based learning to boost critical thinking skills. By comparing the performance of students who engaged in project-based learning with those in traditional settings, educators can test their hypothesis. The results can help educators make informed choices about adopting new teaching strategies.

Environmental Science

Hypothesis testing is essential in environmental science for evaluating the effectiveness of conservation measures. For example, scientists might explore whether a new water management strategy improves river health. By collecting and comparing data on water quality before and after the implementation of the strategy, they can determine whether the intervention leads to positive changes. Such findings are crucial for guiding environmental decisions that have long-term impacts.

Marketing and Advertising

In marketing, businesses use hypothesis testing to refine their approaches. For instance, a clothing brand might test if offering limited-time discounts increases customer loyalty. By running campaigns with and without the discount and analyzing the outcomes, they can assess if the strategy boosts customer retention. Data-driven insights from hypothesis testing enable companies to design marketing strategies that resonate with their audience and drive growth.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

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After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science.

1. What is hypothesis testing in statistics with example?

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. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0 and H1 represent the null and alternative hypotheses. The null hypothesis, H0, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

5. What software tools can assist with hypothesis testing?

Several software tools offering distinct features can help with hypothesis testing. R and RStudio are popular for their advanced statistical capabilities. The Python ecosystem, including libraries like SciPy and Statsmodels, also supports hypothesis testing. SAS and SPSS are well-established tools for comprehensive statistical analysis. For basic testing, Excel offers simple built-in functions.

6. How do I interpret the results of a hypothesis test?

Interpreting hypothesis test results involves comparing the p-value to the significance level (alpha). If the p-value is less than or equal to alpha, you can reject the null hypothesis, indicating statistical significance. This suggests that the observed effect is unlikely to have occurred by chance, validating your analysis findings.

7. Why is sample size important in hypothesis testing?

Sample size is crucial in hypothesis testing as it affects the test’s power. A larger sample size increases the likelihood of detecting a true effect, reducing the risk of Type II errors. Conversely, a small sample may lack the statistical power needed to identify differences, potentially leading to inaccurate conclusions.

8. Can hypothesis testing be used for non-numerical data?

Yes, hypothesis testing can be applied to non-numerical data through non-parametric tests. These tests are ideal when data doesn't meet parametric assumptions or when dealing with categorical data. Non-parametric tests, like the Chi-square or Mann-Whitney U test, provide robust methods for analyzing non-numerical data and drawing meaningful conclusions.

9. How do I choose the proper hypothesis test?

Selecting the right hypothesis test depends on several factors: the objective of your analysis, the type of data (numerical or categorical), and the sample size. Consider whether you're comparing means, proportions, or associations, and whether your data follows a normal distribution. The correct choice ensures accurate results tailored to your research question.

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Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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Understanding Hypothesis Testing

Hypothesis testing is a fundamental statistical method employed in various fields, including data science , machine learning , and statistics , to make informed decisions based on empirical evidence. It involves formulating assumptions about population parameters using sample statistics and rigorously evaluating these assumptions against collected data. At its core, hypothesis testing is a systematic approach that allows researchers to assess the validity of a statistical claim about an unknown population parameter. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

Table of Content

What is Hypothesis Testing?

Why do we use hypothesis testing, one-tailed and two-tailed test, what are type 1 and type 2 errors in hypothesis testing, how does hypothesis testing work, real life examples of hypothesis testing, limitations of hypothesis testing.

A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data.

To test the validity of the claim or assumption about the population parameter:

A sample is drawn from the population and analyzed.
The results of the analysis are used to decide whether the claim is true or not.

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

This structured approach to hypothesis testing in data science , hypothesis testing in machine learning , and hypothesis testing in statistics is crucial for making informed decisions based on data.

By employing hypothesis testing in data analytics and other fields, practitioners can rigorously evaluate their assumptions and derive meaningful insights from their analyses.
Understanding hypothesis generation and testing is also essential for effectively implementing statistical hypothesis testing in various applications.

Defining Hypotheses

Null hypothesis (H 0 ): In statistics, the null hypothesis is a general statement or default position that there is no relationship between two measured cases or no relationship among groups. In other words, it is a basic assumption or made based on the problem knowledge. Example : A company’s mean production is 50 units/per da H 0 : [Tex]\mu [/Tex] = 50.
Alternative hypothesis (H 1 ): The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis. Example: A company’s production is not equal to 50 units/per day i.e. H 1 : [Tex]\mu [/Tex] [Tex]\ne [/Tex] 50.

Key Terms of Hypothesis Testing

Level of significance : It refers to the degree of significance in which we accept or reject the null hypothesis. 100% accuracy is not possible for accepting a hypothesis, so we, therefore, select a level of significance that is usually 5%. This is normally denoted with [Tex]\alpha[/Tex] and generally, it is 0.05 or 5%, which means your output should be 95% confident to give a similar kind of result in each sample.
P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

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.

Understanding hypothesis testing in statistics is essential for data scientists and machine learning practitioners, as it provides a structured framework for statistical hypothesis generation and testing. This methodology can also be applied in hypothesis testing in Python , enabling data analysts to perform robust statistical analyses efficiently. By employing techniques such as multiple hypothesis testing in machine learning , researchers can ensure more reliable results and avoid potential pitfalls associated with drawing conclusions from statistical tests.

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

Left-Tailed (Left-Sided) Test: The alternative hypothesis asserts that the true parameter value is less than the null hypothesis. Example: H 0 : [Tex]\mu \geq 50 [/Tex] and H 1 : [Tex]\mu < 50 [/Tex]
Right-Tailed (Right-Sided) Test : The alternative hypothesis asserts that the true parameter value is greater than the null hypothesis. Example: H 0 : [Tex]\mu \leq50 [/Tex] and H 1 : [Tex]\mu > 50 [/Tex]

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]

To delve deeper into differences into both types of test: Refer to link

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

Type I error: When we reject the null hypothesis, although that hypothesis was true. Type I error is denoted by alpha( [Tex]\alpha [/Tex] ).
Type II errors : When we accept the null hypothesis, but it is false. Type II errors are denoted by beta( [Tex]\beta [/Tex] ).

Step 1: Define Null and Alternative Hypothesis

State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ), suggesting an effect or difference.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

If Test Statistic>Critical Value: Reject the null hypothesis.
If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

If the p-value is less than or equal to the significance level i.e. ( [Tex]p\leq\alpha [/Tex] ), you reject the null hypothesis. This indicates that the observed results are unlikely to have occurred by chance alone, providing evidence in favor of the alternative hypothesis.
If the p-value is greater than the significance level i.e. ( [Tex]p\geq \alpha[/Tex] ), you fail to reject the null hypothesis. This suggests that the observed results are consistent with what would be expected under the null hypothesis.

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]

[Tex]\bar{x} [/Tex] is the sample mean,
μ represents the population mean,
σ is the standard deviation
and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]

t = t-score,
x̄ = sample mean
μ = population mean,
s = standard deviation of the sample,
n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]

[Tex]O_{ij}[/Tex] is the observed frequency in cell [Tex]{ij} [/Tex]
i,j are the rows and columns index respectively.
[Tex]E_{ij}[/Tex] is the expected frequency in cell [Tex]{ij}[/Tex] , calculated as : [Tex]\frac{{\text{{Row total}} \times \text{{Column total}}}}{{\text{{Total observations}}}}[/Tex]

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

m = mean of the difference i.e X after, X before
s = standard deviation of the difference (d) i.e d i = X after, i − X before,
n = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Case A

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05.

The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] and we get accordingly , Z =2.039999999999992.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Python Implementation of Case B

Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.

Although hypothesis testing is a useful technique in data science , it does not offer a comprehensive grasp of the topic being studied.

Lack of Comprehensive Insight : Hypothesis testing in data science often focuses on specific hypotheses, which may not fully capture the complexity of the phenomena being studied.
Dependence on Data Quality : The accuracy of hypothesis testing results relies heavily on the quality of available data. Inaccurate data can lead to incorrect conclusions, particularly in hypothesis testing in machine learning .
Overlooking Patterns : Sole reliance on hypothesis testing can result in the omission of significant patterns or relationships in the data that are not captured by the tested hypotheses.
Contextual Limitations : Hypothesis testing in statistics may not reflect the broader context, leading to oversimplification of results.
Complementary Methods Needed : To gain a more holistic understanding, it’s essential to complement hypothesis testing with other analytical approaches, especially in data analytics and data mining .
Misinterpretation Risks : Poorly formulated hypotheses or inappropriate statistical methods can lead to misinterpretation, emphasizing the need for careful consideration in hypothesis testing in Python and related analyses.
Multiple Hypothesis Testing Challenges : Multiple hypothesis testing in machine learning poses additional challenges, as it can increase the likelihood of Type I errors, requiring adjustments to maintain validity.

Hypothesis testing is a cornerstone of statistical analysis , allowing data scientists to navigate uncertainties and draw credible inferences from sample data. By defining null and alternative hypotheses, selecting significance levels, and employing statistical tests, researchers can validate their assumptions effectively.

This article emphasizes the distinction between Type I and Type II errors, highlighting their relevance in hypothesis testing in data science and machine learning . A practical example involving a paired T-test to assess a new drug’s effect on blood pressure underscores the importance of statistical rigor in data-driven decision-making .

Ultimately, understanding hypothesis testing in statistics , alongside its applications in data mining , data analytics , and hypothesis testing in Python , enhances analytical frameworks and supports informed decision-making.

Understanding Hypothesis Testing- FAQs

What is hypothesis testing in data science.

In data science, hypothesis testing is used to validate assumptions or claims about data. It helps data scientists determine whether observed patterns are statistically significant or could have occurred by chance.

How does hypothesis testing work in machine learning?

In machine learning, hypothesis testing helps assess the effectiveness of models. For example, it can be used to compare the performance of different algorithms or to evaluate whether a new feature significantly improves a model’s accuracy.

What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

What is the difference between hypothesis testing and data mining?

Hypothesis testing focuses on evaluating specific claims or hypotheses about a dataset, while data mining involves exploring large datasets to discover patterns, relationships, or insights without predefined hypotheses.

How is hypothesis generation used in business analytics?

In business analytics , hypothesis generation involves formulating assumptions or predictions based on available data. These hypotheses can then be tested using statistical methods to inform decision-making and strategy.

What is the significance level in hypothesis testing?

The significance level, often denoted as alpha (α), is the threshold for deciding whether to reject the null hypothesis. Common significance levels are 0.05, 0.01, and 0.10, indicating the probability of making a Type I error in statistical hypothesis testing .

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Hypothesis Testing
There are 5 main steps in hypothesis testing: State your research hypothesis as a null hypothesis and alternate hypothesis (H o) and (H a or H 1). Collect data in a way designed to test the hypothesis. Perform an appropriate statistical test. Decide whether to reject or fail to reject your null hypothesis. Present the findings in your results ...
6.4: Hypothesis Testing
1. State the hypothesis. This is a necessary first step. Before a study can be designed, a researcher needs to specify exactly what the hypothesis is what they intend to test. Then the process for collecting data (which is the research method) can be developed and carried out, accordingly.
Statistical Hypothesis Testing Overview
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.
Hypothesis Testing, P Values, Confidence Intervals, and Significance
Medical providers often rely on evidence-based medicine to guide decision-making in practice. Often a research hypothesis is tested with results provided, typically with p values, confidence intervals, or both. Additionally, statistical or research significance is estimated or determined by the investigators. Unfortunately, healthcare providers may have different comfort levels in interpreting ...
Hypothesis Testing: Uses, Steps & Example
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.; Conduct A Test: Use a suitable statistical test to analyze your data.; Make a Decision: Based on the statistical test results, decide whether to reject the null hypothesis or fail to reject it.
Hypothesis Testing: A Complete Guide for Beginners
By understanding the basics of null and alternative hypotheses, test statistics, p-values, and the steps in hypothesis testing, you can analyze experiments and surveys effectively. Hypothesis testing is a powerful tool for everything from scientific research to everyday decisions, and mastering it can lead to better data analysis and decision ...
Choosing the Right Statistical Test
Statistical tests are used in hypothesis testing. They can be used to: determine whether a predictor variable has a statistically significant relationship with an outcome variable. estimate the difference between two or more groups. Statistical tests assume a null hypothesis of no relationship or no difference between groups. Then they ...
PDF Introduction to Hypothesis Testing
8.1 Inferential Statistics and Hypothesis Testing 8.2 Four Steps to Hypothesis Testing 8.3 Hypothesis Testing and Sampling Distributions 8.4 Making a Decision: 8.5 Testing a Research Using the z Test 8.6 Research in Focus: Directional Versus Nondirectional Tests 8.7 Measuring the Size of an Effect: Cohen's d 8.8 Effect Size, Power, and
Hypothesis Testing Made Easy
Hypothesis testing is a statistical procedure used to test assumptions or hypotheses about a population parameter. It involves formulating a null hypothesis (H0) and an alternative hypothesis (Ha), collecting data, and determining whether the evidence is strong enough to reject the null hypothesis. The primary purpose of hypothesis testing is ...
Null & Alternative Hypotheses
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.
Hypothesis Testing
The first step in hypothesis testing is to set a research hypothesis. In Sarah and Mike's study, the aim is to examine the effect that two different teaching methods - providing both lectures and seminar classes (Sarah), and providing lectures by themselves (Mike) - had on the performance of Sarah's 50 students and Mike's 50 students.
Introduction to Hypothesis Testing
Hypothesis Tests. A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
1.2: The 7-Step Process of Statistical Hypothesis Testing
Step 7: Based on steps 5 and 6, draw a conclusion about H0. If the F\calculated F \calculated from the data is larger than the Fα F α, then you are in the rejection region and you can reject the null hypothesis with (1 − α) (1 − α) level of confidence. Note that modern statistical software condenses steps 6 and 7 by providing a p p -value.
Hypothesis Testing
Hypothesis testing is a scientific method used for making a decision and drawing conclusions by using a statistical approach. It is used to suggest new ideas by testing theories to know whether or not the sample data supports research. A research hypothesis is a predictive statement that has to be tested using scientific methods that join an ...
Hypothesis Testing
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.
What is Hypothesis Testing? Types and Methods
Hypothesis Testing is a statistical concept to verify the plausibility of a hypothesis that is based on data samples derived from a given population, using two competing hypotheses. ... Alternative Hypothesis (H1) or the research hypothesis states that there is a relationship between two variables (where one variable affects the other). ...
Inferential Statistics
Hypothesis testing is a formal process of statistical analysis using inferential statistics. The goal of hypothesis testing is to compare populations or assess relationships between variables using samples. Hypotheses, or predictions, are tested using statistical tests. Statistical tests also estimate sampling errors so that valid inferences ...
Hypothesis tests
Hypothesis tests are used to assess whether a difference between two samples represents a real difference between the populations from which the samples were taken. ... All of the routine statistical 'tests' used in research—t-tests, ... the t-test. The t-test is a statistical method for comparing means, and is one of the most widely used ...
Hypothesis Testing in Statistics
Let's understand this with an example. A sanitizer manufacturer claims that its product kills 95 percent of germs on average. To put this company's claim to the test, create a null and alternate hypothesis. H0 (Null Hypothesis): Average = 95%. Alternative Hypothesis (H1): The average is less than 95%.
Understanding Hypothesis Testing
Understanding Hypothesis Testing. Hypothesis testing is a fundamental statistical method employed in various fields, including data science, machine learning, and statistics, to make informed decisions based on empirical evidence. It involves formulating assumptions about population parameters using sample statistics and rigorously evaluating ...
Full article: Monitoring and audit quality: Does quality standards
Research constructs statistics. ... Hypothesis testing. Download CSV Display Table. Compliance with quality standards partially mediates the relationship between SMP monitoring and audit quality, according to the empirical findings (H4). H2 and H3 provide an explanation for this. The measurements are derived from elements of ISAs and ISQM 1 ...

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Hypothesis Testing: A Complete Guide for Beginners

What is a Hypothesis?

For example:

What is Hypothesis Testing?

Null and Alternative Hypothesis

Key Terms in Hypothesis Testing

1. Test Statistic

3. Significance Level (α)

4. Critical Value

5. Type I and Type II Errors

Types of Hypothesis Testing

2. Two-Tailed Test

Steps in Hypothesis Testing

Step 2: Set the Significance Level (α)

Step 3: Collect and Analyze Data

Step 4: Calculate the P-value or Critical Value

Step 5: Make a Decision

Step 6: Interpret the Results

P-Value and Significance

Common Hypothesis Tests

Example of Hypothesis Testing

What is the difference between the null hypothesis and the alternative hypothesis?

What is the difference between a one-tailed test and a two-tailed test?

Can we always reject the null hypothesis if the p-value is less than 0.05?

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Hypothesis Testing – A Complete Guide with Examples

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What is Hypothesis Testing?

Characteristics of the Hypothesis to be Tested

What is a Null Hypothesis and Alternative Hypothesis?

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How to Conduct Hypothesis Testing?

Step 1: State the Null and Alternative Hypothesis

Step 2: Data Collection

Step 3: Select Appropriate Statistical Test

One-sided Test

Two-sided Test

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Step 4: Select the Level of Significance

Step 5: Find out Whether the Null Hypothesis is Rejected or Supported

Step 6: Present the Outcomes of your Study

Frequently Asked Questions

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What are null hypothesis?

What is the probability value?

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