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.