Hypothesis Testing with Two Samples: Two-Sample t-Tests for Means

Chapter 8: Hypothesis Testing with Two Samples

8.2 Testing the Difference Between Means (Independent Samples, Unknown Variances)

This section covers the procedures for comparing the means of two independent populations when the population standard deviations are unknown. The two-sample t-test is a fundamental inferential statistical method used to determine if there is a significant difference between the means of two groups.

• Independent Samples: The samples must be randomly selected and independent of each other.

• Unknown Population Standard Deviations: The population standard deviations (\( \sigma_1, \sigma_2 \)) are not known.

• Normality Assumption: The populations should be normally distributed, or the sample sizes should be large enough for the Central Limit Theorem to apply.

Key Concepts

• Two-Sample t-Test: Used to test the difference between two population means under the above conditions.

• Test Statistic: The standardized value used to determine whether to reject the null hypothesis.

• Equal vs. Unequal Variances: The calculation of the test statistic depends on whether the population variances are assumed to be equal or not.

Test Statistic Formulas

The form of the test statistic depends on the assumption about population variances:

1. Equal Variances (Pooled t-Test)

• When population variances are assumed equal, a pooled estimate of the standard deviation is used.

The test statistic is:

$t = \frac{\overline{X}_1 - \overline{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

where the pooled standard deviation is:

$S_p = \sqrt{\frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}}$

2. Unequal Variances (Welch's t-Test)

• If population variances are not assumed equal, the standard error is calculated separately for each sample.

The test statistic is:

$t = \frac{\overline{X}_1 - \overline{X}_2}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}$

The degrees of freedom are approximated using the Welch-Satterthwaite equation:

$df = \frac{\left( \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} \right)^2}{\frac{\left( \frac{S_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{S_2^2}{n_2} \right)^2}{n_2 - 1}}$

Steps for Conducting a Two-Sample t-Test

1. State the Hypotheses:

   • Null hypothesis (\( H_0 \)): \( \mu_1 = \mu_2 \)

   • Alternative hypothesis (\( H_a \)): \( \mu_1 \neq \mu_2 \), \( \mu_1 > \mu_2 \), or \( \mu_1 < \mu_2 \) depending on the claim.

2. Check Assumptions: Randomness, independence, normality, and unknown variances.

3. Calculate the Test Statistic: Use the appropriate formula based on variance assumptions.

4. Determine the Critical Value(s): Based on the significance level (\( \alpha \)) and degrees of freedom.

5. Make a Decision: Compare the test statistic to the critical value(s) or use the p-value approach.

6. State the Conclusion: Interpret the results in the context of the original claim.

Examples

Example 1: Mathematics Test Scores (Unequal Variances)

• Two random samples of students taught by different teachers are compared.

• Population variances are not equal; use Welch's t-test.

• Null hypothesis: No difference in mean scores.

• Test statistic calculated and compared to critical values for a two-tailed test.

• Conclusion: If the test statistic does not fall in the rejection region, fail to reject the null hypothesis. There is not enough evidence to support a difference in means at the 10% significance level.

Example 2: Automobile Driving Costs (Equal Variances)

• Comparing mean driving costs per mile for two automobile manufacturers.

• Population variances are assumed equal; use pooled t-test.

• Null hypothesis: No difference in mean costs.

• Test statistic calculated and compared to the critical value for a left-tailed test.

• Conclusion: If the test statistic falls in the rejection region, reject the null hypothesis. There is enough evidence at the 5% significance level to support the claim that one manufacturer has a lower mean cost.

Summary Table: Two-Sample t-Test for Means

Assumption	Test Statistic	Degrees of Freedom	When to Use
Equal Variances	$t = \frac{\overline{X}_1 - \overline{X}_2}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$	$n_1 + n_2 - 2$	Pooled t-test
Unequal Variances	$t = \frac{\overline{X}_1 - \overline{X}_2}{\sqrt{\frac{S_1^2}{n_1} + \frac{S_2^2}{n_2}}}$	Welch-Satterthwaite formula	Welch's t-test

Important Notes

• Always check assumptions before applying the test.

• Interpret results in the context of the problem and significance level.

• For small sample sizes, normality is especially important.