Do You Use a T-Test to Compare Means? A Comprehensive Guide

Do you use a T-test to compare means effectively? Absolutely, the T-test is a statistical hypothesis test used to determine if there’s a significant difference between the means of two groups, making it a powerful tool for data analysis. compare.edu.vn provides in-depth comparisons and analyses to help you understand and apply statistical methods like the T-test correctly, ensuring accurate and reliable results. Learn how to choose the right test, interpret results, and make informed decisions based on your data, ensuring the most effective approach for your research or analysis.

1. Understanding the T-Test: A Statistical Tool for Comparing Means

The T-test is a staple in the world of statistics, serving as a foundational method for comparing the means of two groups. Its versatility allows researchers and analysts to determine whether the observed difference between two group means is statistically significant or simply due to random chance. This significance is crucial in drawing meaningful conclusions from data across various fields.

The T-test falls under the umbrella of hypothesis testing, where the goal is to evaluate the validity of a claim or hypothesis about a population based on a sample of data. In the case of the T-test, the hypothesis typically centers around whether the means of two populations are equal.

1.1 What Is a T-Test?

A T-test is a statistical test that compares the means of two groups to determine if there’s a statistically significant difference between them. It’s widely used in research to analyze data and draw conclusions about populations.

1.2 Why Use a T-Test?

The primary reason to use a T-test is to determine if the difference between the means of two groups is statistically significant. This is essential in various scenarios, such as:

Medical Research: Determining if a new drug is more effective than a placebo.
Marketing: Comparing the effectiveness of two different advertising campaigns.
Education: Assessing whether a new teaching method improves student performance.
Engineering: Evaluating the performance differences between two designs.

1.3 Core Principles of T-Tests

At its core, the T-test operates on several key principles:

Null Hypothesis: This is the assumption that there is no significant difference between the means of the two groups being compared. The T-test aims to either reject or fail to reject this null hypothesis.
Alternative Hypothesis: This is the claim that there is a significant difference between the means of the two groups. It is the hypothesis that the researcher is trying to support.
T-Statistic: This is a calculated value that quantifies the difference between the means of the two groups relative to the variability within the groups. A larger T-statistic suggests a greater difference between the means.
P-Value: This is the probability of observing a T-statistic as extreme as, or more extreme than, the one calculated if the null hypothesis were true. A small P-value (typically less than 0.05) indicates strong evidence against the null hypothesis.
Significance Level (Alpha): This is a pre-determined threshold (usually 0.05) that defines the level of risk the researcher is willing to accept in rejecting the null hypothesis when it is actually true.

1.4 Key Metrics in T-Tests

Several key metrics are essential in understanding and interpreting the results of a T-test:

Mean Difference: This is the difference between the sample means of the two groups being compared. It provides a measure of the magnitude of the difference between the groups.
Standard Deviation: This measures the amount of variability or dispersion in a set of data. It reflects how spread out the data points are around the mean.
Standard Error: This estimates the variability in the sample mean. It indicates how much the sample mean is likely to vary from the true population mean.
Degrees of Freedom: This refers to the number of independent pieces of information available to estimate a parameter. It affects the shape of the T-distribution and the critical value used for hypothesis testing.
Confidence Interval: This is a range of values that is likely to contain the true population mean difference with a certain level of confidence (e.g., 95%). It provides a sense of the precision and uncertainty associated with the estimate.

Alt Text: T-Test statistical analysis explaining the key metrics for understanding the difference between sample means.

2. Types of T-Tests: Choosing the Right Test for Your Data

Selecting the appropriate type of T-test is essential for accurate data analysis. The choice depends on the nature of the data and the research question being addressed. The two primary types of T-tests are the Independent Samples T-test and the Paired Samples T-test.

2.1 Independent Samples T-Test (Unpaired T-Test)

The Independent Samples T-test, also known as the unpaired T-test, is used to compare the means of two independent groups. This means that the individuals or observations in one group are not related to the individuals or observations in the other group.

When to Use:
- When comparing the means of two separate groups.
- When the data from the two groups are not related or matched.
- When you want to determine if there is a significant difference between the means of two independent populations.
Example:
- Comparing the test scores of students in two different schools.
- Analyzing the effectiveness of a drug by comparing outcomes for a treatment group versus a control group.
- Comparing the average sales performance of two different marketing strategies.
Assumptions:
- Independence: The observations within each group are independent of each other.
- Normality: The data in each group are approximately normally distributed.
- Homogeneity of Variance: The variances of the two groups are approximately equal (assessed using Levene’s test).

2.2 Paired Samples T-Test (Dependent T-Test)

The Paired Samples T-test, also known as the dependent T-test, is used to compare the means of two related groups. This means that the individuals or observations in one group are paired or matched with the individuals or observations in the other group.

When to Use:
- When comparing the means of two related groups.
- When the data from the two groups are paired or matched.
- When you want to determine if there is a significant difference between the means of two measurements taken on the same subjects.
Example:
- Comparing pre-test and post-test scores of students after an intervention.
- Analyzing the blood pressure of patients before and after taking a medication.
- Comparing the performance of employees before and after a training program.
Assumptions:
- Dependence: The observations within each pair are dependent on each other.
- Normality: The differences between the paired observations are approximately normally distributed.

2.3 Welch’s T-Test

Welch’s T-test is a variation of the independent samples T-test that does not assume equal variances between the two groups. It is more robust than the standard independent samples T-test when the assumption of homogeneity of variance is violated.

When to Use:
- When comparing the means of two independent groups.
- When the variances of the two groups are unequal.
- When you want to determine if there is a significant difference between the means of two independent populations, even when their variances differ.
Example:
- Comparing the income levels of two different cities, where the income distribution may vary significantly.
- Analyzing the performance of two different types of software, where the performance variability may differ.
Assumptions:
- Independence: The observations within each group are independent of each other.
- Normality: The data in each group are approximately normally distributed.

2.4 Choosing the Right T-Test

To select the appropriate T-test, consider the following questions:

Are the two groups independent or related?
- If the groups are independent, use an Independent Samples T-test or Welch’s T-test.
- If the groups are related, use a Paired Samples T-test.
Are the variances of the two groups equal?
- If the variances are approximately equal, use an Independent Samples T-test.
- If the variances are unequal, use Welch’s T-test.

By carefully considering these questions, you can choose the appropriate T-test for your data and ensure accurate and reliable results.

Alt Text: This image presents the various types of T-tests that can be applied in statistical analysis, each suited for different data conditions.

3. Assumptions of the T-Test: Ensuring Validity

Before conducting a T-test, it is crucial to verify that the underlying assumptions are met. Violating these assumptions can lead to inaccurate or unreliable results. The key assumptions of the T-test include:

3.1 Normality

The data in each group should be approximately normally distributed. Normality refers to the assumption that the data follows a bell-shaped curve, with the majority of observations clustering around the mean.

How to Assess:
- Visual Inspection: Use histograms, Q-Q plots, or box plots to visually assess the distribution of the data.
- Statistical Tests: Use formal statistical tests such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test to assess normality.
What to Do If Violated:
- Transform the Data: Apply mathematical transformations such as logarithmic, square root, or inverse transformations to make the data more normally distributed.
- Use Non-Parametric Tests: Consider using non-parametric alternatives to the T-test, such as the Mann-Whitney U test or the Wilcoxon signed-rank test, which do not assume normality.
- Increase Sample Size: If the sample size is large enough (typically > 30), the T-test may still be valid due to the central limit theorem, even if the data are not perfectly normally distributed.

3.2 Independence

The observations within each group should be independent of each other. Independence means that the value of one observation does not influence the value of another observation.

How to Assess:
- Study Design: Ensure that the study design prevents dependence between observations.
- Random Sampling: Use random sampling techniques to select participants or observations.
What to Do If Violated:
- Use Mixed-Effects Models: Consider using mixed-effects models or hierarchical models, which can account for dependence between observations.
- Adjust Standard Errors: Use robust standard errors or clustered standard errors to adjust for dependence.

3.3 Homogeneity of Variance (for Independent Samples T-Test)

The variances of the two groups should be approximately equal. Homogeneity of variance means that the spread or dispersion of data points around the mean is similar for both groups.

How to Assess:
- Visual Inspection: Compare the spread of data in box plots or histograms for both groups.
- Statistical Tests: Use formal statistical tests such as Levene’s test or Bartlett’s test to assess homogeneity of variance.
What to Do If Violated:
- Use Welch’s T-Test: Use Welch’s T-test, which does not assume equal variances.
- Transform the Data: Apply mathematical transformations to make the variances more equal.

3.4 Dependence (for Paired Samples T-Test)

The observations within each pair should be dependent on each other. Dependence means that the two measurements are taken on the same subject or matched in some way.

How to Assess:
- Study Design: Ensure that the study design involves paired or matched observations.
- Matching Criteria: Verify that the matching criteria are appropriate and relevant to the research question.
What to Do If Violated:
- Re-evaluate Pairing: Re-evaluate the pairing criteria to ensure that the observations are appropriately matched.
- Use Alternative Tests: Consider using alternative tests that do not assume dependence, such as the Wilcoxon signed-rank test.

3.5 Addressing Violations

When the assumptions of the T-test are violated, it is important to take appropriate corrective measures to ensure the validity of the results. This may involve transforming the data, using non-parametric tests, or adjusting the standard errors.

Alt Text: T-Test analysis and the importance of verifying assumptions for ensuring accurate and reliable statistical results.

4. Conducting a T-Test: A Step-by-Step Guide

Performing a T-test involves a series of steps, from formulating the hypothesis to interpreting the results. Here is a step-by-step guide to conducting a T-test:

4.1 Formulate the Hypotheses

The first step in conducting a T-test is to formulate the null and alternative hypotheses.

Null Hypothesis (H0): This is the assumption that there is no significant difference between the means of the two groups being compared.
Alternative Hypothesis (H1): This is the claim that there is a significant difference between the means of the two groups.

The alternative hypothesis can be one-tailed or two-tailed, depending on the research question:

Two-Tailed Hypothesis: This states that there is a difference between the means of the two groups, without specifying the direction of the difference.
- Example: The mean test score of students in School A is different from the mean test score of students in School B.
One-Tailed Hypothesis: This states that there is a difference between the means of the two groups, specifying the direction of the difference.
- Example: The mean test score of students in School A is higher than the mean test score of students in School B.

4.2 Choose the Appropriate T-Test

Select the appropriate T-test based on the nature of the data and the research question:

Independent Samples T-Test: Use this test when comparing the means of two independent groups.
Paired Samples T-Test: Use this test when comparing the means of two related groups.
Welch’s T-Test: Use this test when comparing the means of two independent groups with unequal variances.

4.3 Check the Assumptions

Before conducting the T-test, verify that the underlying assumptions are met:

Normality: Assess whether the data in each group are approximately normally distributed.
Independence: Ensure that the observations within each group are independent of each other.
Homogeneity of Variance: Check whether the variances of the two groups are approximately equal (for Independent Samples T-test).
Dependence: Verify that the observations within each pair are dependent on each other (for Paired Samples T-test).

4.4 Calculate the T-Statistic and Degrees of Freedom

Calculate the T-statistic and degrees of freedom using the appropriate formulas:

Independent Samples T-Test:
- T-Statistic:
```
t = (x̄1 - x̄2) / √(s1²/n1 + s2²/n2)
```
  Where:
  - x̄1 and x̄2 are the sample means of the two groups.
  - s1² and s2² are the sample variances of the two groups.
  - n1 and n2 are the sample sizes of the two groups.
- Degrees of Freedom:
```
df = n1 + n2 - 2
```
Paired Samples T-Test:
- T-Statistic:
```
t = d̄ / (sd / √n)
```
  Where:
  - d̄ is the mean of the differences between the paired observations.
  - sd is the standard deviation of the differences.
  - n is the number of pairs.
- Degrees of Freedom:
```
df = n - 1
```

Welch’s T-Test:

T-Statistic:

t = (x̄1 - x̄2) / √(s1²/n1 + s2²/n2)

Degrees of Freedom:

df ≈ (s1²/n1 + s2²/n2)² / ((s1²/n1)²/(n1-1) + (s2²/n2)²/(n2-1))

4.5 Determine the P-Value

Determine the P-value associated with the calculated T-statistic and degrees of freedom. The P-value represents the probability of observing a T-statistic as extreme as, or more extreme than, the one calculated if the null hypothesis were true.

Using Statistical Software: Use statistical software such as R, Python, SPSS, or Excel to calculate the P-value.
Using T-Table: Consult a T-table to find the P-value corresponding to the T-statistic and degrees of freedom.

4.6 Make a Decision

Compare the P-value to the significance level (alpha) to make a decision about the null hypothesis:

If P-value ≤ Alpha: Reject the null hypothesis. This indicates that there is a statistically significant difference between the means of the two groups.
If P-value > Alpha: Fail to reject the null hypothesis. This indicates that there is not enough evidence to conclude that there is a significant difference between the means of the two groups.

4.7 Interpret the Results

Interpret the results of the T-test in the context of the research question. Provide a clear and concise explanation of the findings, including the T-statistic, degrees of freedom, P-value, and the decision regarding the null hypothesis.

Alt Text: Visual representation of the steps involved in conducting a T-test, including formulating hypotheses, checking assumptions, and interpreting results.

5. Interpreting T-Test Results: Understanding P-Values and Significance

Interpreting the results of a T-test involves understanding the P-value and its implications for the null hypothesis. The P-value is a critical component of hypothesis testing and provides valuable information about the strength of evidence against the null hypothesis.

5.1 What Is a P-Value?

The P-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated if the null hypothesis were true. In simpler terms, it measures the likelihood of obtaining the observed results (or more extreme results) if there were truly no difference between the means of the two groups being compared.

Range: The P-value ranges from 0 to 1.
Interpretation:
- Small P-Value (e.g., ≤ 0.05): Indicates strong evidence against the null hypothesis. This suggests that the observed results are unlikely to have occurred by chance alone, and there is a statistically significant difference between the means of the two groups.
- Large P-Value (e.g., > 0.05): Indicates weak evidence against the null hypothesis. This suggests that the observed results are likely to have occurred by chance alone, and there is not enough evidence to conclude that there is a significant difference between the means of the two groups.

5.2 Significance Level (Alpha)

The significance level, denoted by alpha (α), is a pre-determined threshold that defines the level of risk the researcher is willing to accept in rejecting the null hypothesis when it is actually true. Commonly used significance levels include 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Interpretation:
- α = 0.05: There is a 5% risk of rejecting the null hypothesis when it is actually true (Type I error).
- α = 0.01: There is a 1% risk of rejecting the null hypothesis when it is actually true (Type I error).

5.3 Making a Decision

To make a decision about the null hypothesis, compare the P-value to the significance level:

If P-value ≤ Alpha: Reject the null hypothesis. This indicates that there is a statistically significant difference between the means of the two groups.
If P-value > Alpha: Fail to reject the null hypothesis. This indicates that there is not enough evidence to conclude that there is a significant difference between the means of the two groups.

5.4 Example Interpretation

Suppose a researcher conducts an independent samples T-test to compare the test scores of students in two different schools. The results of the T-test are:

T-Statistic: 2.50
Degrees of Freedom: 50
P-Value: 0.015
Significance Level: 0.05

Interpretation:

The P-value (0.015) is less than the significance level (0.05).
Therefore, the researcher rejects the null hypothesis.
Conclusion: There is a statistically significant difference between the mean test scores of students in the two schools.

5.5 Common Misinterpretations

It is important to avoid common misinterpretations of P-values:

P-Value Is Not the Probability That the Null Hypothesis Is True: The P-value is the probability of observing the data (or more extreme data) if the null hypothesis were true, not the probability that the null hypothesis is true.
P-Value Does Not Measure the Size or Importance of an Effect: The P-value indicates the strength of evidence against the null hypothesis, but it does not provide information about the magnitude or practical significance of the effect.
Non-Significant Result Does Not Prove the Null Hypothesis Is True: Failing to reject the null hypothesis does not mean that the null hypothesis is true; it simply means that there is not enough evidence to reject it.

5.6 Reporting T-Test Results

When reporting the results of a T-test, include the following information:

The type of T-test used (e.g., independent samples T-test, paired samples T-test).
The T-statistic.
The degrees of freedom.
The P-value.
The sample means and standard deviations for each group.
A clear and concise interpretation of the findings.

Example: “An independent samples T-test was conducted to compare the test scores of students in School A and School B. The results indicated a statistically significant difference between the means (t(50) = 2.50, p = 0.015), with students in School A scoring significantly higher (M = 85, SD = 5) than students in School B (M = 80, SD = 6).”

Alt Text: A comprehensive guide on interpreting T-test results, including understanding P-values, significance levels, and avoiding common misinterpretations.

6. Practical Examples of T-Tests: Real-World Applications

The T-test is a versatile statistical tool with numerous applications across various fields. Here are some practical examples of how T-tests can be used in real-world scenarios:

6.1 Medical Research

Scenario: A pharmaceutical company wants to test the effectiveness of a new drug in reducing blood pressure. They conduct a clinical trial with two groups: a treatment group receiving the new drug and a control group receiving a placebo.
T-Test Application: An independent samples T-test is used to compare the mean reduction in blood pressure between the treatment group and the control group.
Interpretation: If the P-value is less than 0.05, the company can conclude that the new drug is significantly more effective in reducing blood pressure compared to the placebo.

6.2 Marketing

Scenario: A marketing team wants to compare the effectiveness of two different advertising campaigns. They run both campaigns and measure the sales generated by each.
T-Test Application: An independent samples T-test is used to compare the mean sales generated by Campaign A and Campaign B.
Interpretation: If the P-value is less than 0.05, the team can conclude that one campaign is significantly more effective than the other in generating sales.

6.3 Education

Scenario: A teacher wants to assess the impact of a new teaching method on student performance. The teacher administers a pre-test to students, implements the new teaching method, and then administers a post-test.
T-Test Application: A paired samples T-test is used to compare the mean pre-test scores and the mean post-test scores for the same students.
Interpretation: If the P-value is less than 0.05, the teacher can conclude that the new teaching method has a significant impact on student performance.

6.4 Engineering

Scenario: An engineer wants to compare the performance of two different designs for a bridge. The engineer builds both designs and measures the maximum load each can withstand before failing.
T-Test Application: An independent samples T-test is used to compare the mean maximum load for Design A and Design B.
Interpretation: If the P-value is less than 0.05, the engineer can conclude that one design is significantly stronger than the other.

6.5 Psychology

Scenario: A psychologist wants to study the effect of a new therapy on reducing anxiety levels. The psychologist measures the anxiety levels of patients before and after undergoing the therapy.
T-Test Application: A paired samples T-test is used to compare the mean anxiety levels before and after the therapy.
Interpretation: If the P-value is less than 0.05, the psychologist can conclude that the therapy has a significant impact on reducing anxiety levels.

6.6 Business

Scenario: A company wants to compare the job satisfaction levels of employees in two different departments. They administer a job satisfaction survey to employees in both departments.
T-Test Application: An independent samples T-test is used to compare the mean job satisfaction scores for Department A and Department B.
Interpretation: If the P-value is less than 0.05, the company can conclude that there is a significant difference in job satisfaction levels between the two departments.

6.7 Environmental Science

Scenario: A researcher wants to compare the levels of pollution in two different rivers. The researcher collects water samples from both rivers and measures the concentration of pollutants.
T-Test Application: An independent samples T-test is used to compare the mean pollution levels in River A and River B.
Interpretation: If the P-value is less than 0.05, the researcher can conclude that there is a significant difference in pollution levels between the two rivers.

These practical examples demonstrate the wide range of applications for T-tests across various fields. By understanding how to apply and interpret T-tests, researchers and analysts can gain valuable insights and make informed decisions based on their data.

Alt Text: The image displays various practical examples of T-test applications in medical research, marketing, education, engineering, and psychology.

7. Limitations of the T-Test: When Not to Use It

While the T-test is a powerful and versatile statistical tool, it is not appropriate for all situations. It is important to be aware of the limitations of the T-test and to consider alternative methods when necessary. Here are some key limitations of the T-test:

7.1 Limited to Two Groups

The T-test is designed to compare the means of two groups. It cannot be used to compare the means of three or more groups simultaneously.

Alternative: If you need to compare the means of three or more groups, consider using analysis of variance (ANOVA), which is specifically designed for this purpose.

7.2 Assumes Normality

The T-test assumes that the data in each group are approximately normally distributed. If the data are not normally distributed, the results of the T-test may be inaccurate.

Alternative: If the data are not normally distributed, consider using non-parametric alternatives to the T-test, such as the Mann-Whitney U test or the Wilcoxon signed-rank test, which do not assume normality.

7.3 Sensitive to Outliers

The T-test is sensitive to outliers, which are extreme values that can disproportionately influence the mean and standard deviation.

Alternative: If the data contain outliers, consider using robust statistical methods that are less sensitive to outliers, such as the trimmed mean or the Winsorized mean.

7.4 Assumes Independence

The T-test assumes that the observations within each group are independent of each other. If the observations are not independent, the results of the T-test may be inaccurate.

Alternative: If the observations are not independent, consider using mixed-effects models or hierarchical models, which can account for dependence between observations.

7.5 Requires Interval or Ratio Data

The T-test requires that the data be measured on an interval or ratio scale. It cannot be used with nominal or ordinal data.

Alternative: If the data are nominal or ordinal, consider using non-parametric tests such as the chi-square test or the Kruskal-Wallis test, which are designed for categorical data.

7.6 Limited Information About Effect Size

The T-test provides information about the statistical significance of the difference between the means of two groups, but it does not provide information about the size or importance of the effect.

Alternative: To assess the effect size, consider calculating measures such as Cohen’s d or eta-squared, which provide information about the magnitude of the effect.

7.7 Assumes Homogeneity of Variance

The independent samples T-test assumes that the variances of the two groups are approximately equal. If the variances are unequal, the results of the T-test may be inaccurate.

Alternative: If the variances are unequal, consider using Welch’s T-test, which does not assume equal variances.

7.8 Limited to Linear Relationships

The T-test is designed to assess linear relationships between the means of two groups. It cannot be used to assess non-linear relationships.

Alternative: If you suspect that there is a non-linear relationship between the variables, consider using non-linear regression techniques or other appropriate statistical methods.

By understanding the limitations of the T-test and considering alternative methods when necessary, researchers and analysts can ensure that they are using the most appropriate statistical tools for their data and research questions.

Alt Text: This image presents the limitations of the T-test, including its restriction to two groups, assumptions of normality, and sensitivity to outliers.

8. T-Test vs. Other Statistical Tests: Choosing the Right Tool

Selecting the right statistical test is crucial for accurate data analysis. While the T-test is a versatile tool for comparing means, it is not always the most appropriate choice. Here’s a comparison of the T-test with other statistical tests to help you choose the right tool for your research question.

8.1 T-Test vs. ANOVA

T-Test: Used to compare the means of two groups.
ANOVA (Analysis of Variance): Used to compare the means of three or more groups.

Feature	T-Test	ANOVA
Number of Groups	Two	Three or more
Purpose	Compare means of two groups	Compare means of multiple groups
Type of Data	Interval or ratio	Interval or ratio
Assumptions	Normality, independence, equal variance	Normality, independence, equal variance

When to Use:
- Use a T-test when you want to compare the means of exactly two groups.
- Use ANOVA when you want to compare the means of three or more groups.
Example:
- T-Test: Comparing the test scores of students in two different schools.
- ANOVA: Comparing the test scores of students in three different teaching methods.

8.2 T-Test vs. Chi-Square Test

T-Test: Used to compare the means of two groups with interval or ratio data.
Chi-Square Test: Used to analyze categorical data and determine if there is an association between two categorical variables.

Feature	T-Test	Chi-Square Test
Type of Data	Interval or ratio	Categorical (nominal or ordinal)
Purpose	Compare means of two groups	Test association between two variables
Number of Variables	One independent, one dependent	Two categorical variables
Assumptions	Normality, independence, equal variance	Independence, expected cell counts

When to Use:
- Use a T-test when you want to compare the means of two groups with interval or ratio data.
- Use a Chi-Square Test when you want to analyze categorical data and determine if there is an association between two categorical variables.
Example:
- T-Test: Comparing the blood pressure of patients taking Drug A vs. Drug B.
- Chi-Square Test: Analyzing if there is a relationship between smoking status (smoker vs. non-smoker) and the presence of lung cancer (yes vs. no).

8.3 T-Test vs. Correlation

T-Test: Used to compare the means of two groups.
Correlation: Used to measure the strength and direction of a linear relationship between two continuous variables.

Feature	T-Test	Correlation
Purpose	Compare means of two groups	Measure relationship between two variables
Type of Variables	One independent (categorical), one dependent (continuous)	Two continuous variables
Type of Relationship	Difference in means	Linear relationship
Output	P-value, t-statistic	Correlation coefficient (r)

When to Use:
- Use a T-test when you want to compare the means of two groups.
- Use correlation when you want to measure the strength and direction of a linear relationship between two continuous variables.
Example:
- T-Test: Comparing the exam scores of students who attended a review session vs. those who did not.
- Correlation: Measuring the relationship between hours studied and exam scores.

8.4 T-Test vs. Mann-Whitney U Test

T-Test: Used to compare the means of two groups with interval or ratio data, assuming normality.
Mann-Whitney U Test: A non-parametric test used to compare the medians of two groups when the data are not normally distributed.

Feature	T-Test	Mann-Whitney U Test
Data Type	Interval or ratio	Ordinal or non-normally distributed
Purpose	Compare means of two groups	Compare medians of two groups
Assumptions	Normality, independence, equal variance	Independence
Type	Parametric	Non-parametric

When to Use:
- Use a T-test when you want to compare the means