A **t-test** is a statistical method used to compare the means of two groups to determine whether they are significantly different from each other. Here's a step-by-step example:

### Scenario:
Suppose a teacher wants to know if a new teaching method (Method A) is more effective than the traditional method (Method B) in improving student performance. The teacher randomly selects two groups of students: Group A, which uses the new method, and Group B, which uses the traditional method. After a test, the teacher collects the following scores:

- **Group A (New Method)**: 85, 87, 90, 92, 88
- **Group B (Traditional Method)**: 78, 80, 82, 85, 79

### Step-by-Step Example for an Independent Samples t-Test:

1. **State the hypotheses**:
   - **Null hypothesis (H₀)**: There is no significant difference in the average test scores between the two groups (Group A and Group B).
   - **Alternative hypothesis (H₁)**: There is a significant difference in the average test scores between the two groups.

2. **Choose the significance level**: Usually, the significance level (α) is set to 0.05.

3. **Calculate the sample means**:
   - Mean of Group A: \( \frac{85 + 87 + 90 + 92 + 88}{5} = 88.4 \)
   - Mean of Group B: \( \frac{78 + 80 + 82 + 85 + 79}{5} = 80.8 \)

4. **Calculate the standard deviations**:
   - Standard deviation of Group A: \( s_A = \sqrt{\frac{\sum (x_i - \bar{x}_A)^2}{n_A - 1}} \)
   - Standard deviation of Group B: \( s_B = \sqrt{\frac{\sum (x_i - \bar{x}_B)^2}{n_B - 1}} \)

5. **Calculate the t-statistic**: The formula for the t-statistic in an independent samples t-test is:

\[
t = \frac{\bar{x}_A - \bar{x}_B}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}}
\]

Where:
- \( \bar{x}_A \) and \( \bar{x}_B \) are the means of Group A and Group B.
- \( s_A^2 \) and \( s_B^2 \) are the variances (squared standard deviations) of the two groups.
- \( n_A \) and \( n_B \) are the sample sizes of Group A and Group B.

6. **Compare the t-statistic to the critical t-value**: Using a t-distribution table, compare the calculated t-value with the critical value for a given degree of freedom (df) and significance level (α = 0.05).

7. **Make a decision**:
   - If the calculated t-value is greater than the critical t-value, reject the null hypothesis.
   - If the calculated t-value is less than the critical t-value, fail to reject the null hypothesis.

In this example, based on the t-test, you can conclude whether there is a statistically significant difference between the performance of the two groups.

Would you like more details on the calculations or a step-by-step breakdown?

### Related Questions:
1. What are the different types of t-tests and their uses?
2. How does the significance level affect the outcome of a t-test?
3. How can the sample size influence the result of a t-test?
4. What are the assumptions of an independent samples t-test?
5. How do we interpret p-values in a t-test?

### Tip:
Always check if your data meets the assumptions of the t-test (normality, equal variances) before applying it to ensure valid results.

This page provides a detailed step-by-step example of conducting an independent samples t-test. Learn how to calculate sample means, standard deviations, and the t-statistic to compare the effectiveness of two teaching methods. Suitable for college-level statistics students.

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