Hypothesis Tests for Two Population Means: Pooled, Non-Pooled, and Paired t-Tests

Chapter 10 - Hypothesis Tests for Two Population Means

This chapter introduces statistical methods for comparing the means of two populations. The main approaches are the pooled t-test, non-pooled t-test, and paired t-test. Each method is appropriate under different assumptions about the data and experimental design.

Pooled t-Test

The pooled t-test is used to compare the means of two independent populations when it is reasonable to assume that the population variances are equal.

• Assumptions:

  • Both samples are independent and randomly selected.

  • Both populations are normally distributed.

  • Population variances are equal ($\sigma_1^2 = \sigma_2^2$).

• Test Statistic: $t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$ where $s_p$ is the pooled standard deviation: $s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$

• Degrees of Freedom: $df = n_1 + n_2 - 2$

• Example: Comparing the average test scores of students from two schools, assuming similar variability in scores.

Non-Pooled t-Test (Welch's t-Test)

The non-pooled t-test (also known as Welch's t-test) is used when the two populations are independent and may have unequal variances.

• Assumptions:

  • Samples are independent and randomly selected.

  • Populations are normally distributed.

  • Population variances may be unequal ($\sigma_1^2 \neq \sigma_2^2$).

• Test Statistic: $t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Degrees of Freedom: Calculated using the Welch-Satterthwaite equation: $df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}$

• Example: Comparing the average lifespans of two different brands of batteries, where the variability in lifespans may differ.

Paired t-Test

The paired t-test is used when the data consist of paired observations, such as before-and-after measurements on the same subjects.

• Assumptions:

  • Pairs are randomly selected and differences are normally distributed.

  • Observations within each pair are dependent, but pairs are independent of each other.

• Test Statistic: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$ where $\bar{d}$ is the mean of the differences and $s_d$ is the standard deviation of the differences.

• Degrees of Freedom: $df = n - 1$

• Example: Measuring the effect of a drug by comparing patients' blood pressure before and after treatment.

Summary Table: Comparison of t-Tests for Two Means

Test	Data Structure	Variance Assumption	Formula for Test Statistic	Degrees of Freedom
Pooled t-Test	Independent samples	Equal variances	$t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$	$n_1 + n_2 - 2$
Non-Pooled t-Test	Independent samples	Unequal variances	$t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$	Welch-Satterthwaite formula
Paired t-Test	Paired (dependent) samples	Not applicable	$t = \frac{\bar{d}}{s_d / \sqrt{n}}$	$n - 1$

Additional info: The above content expands on the brief list in the original file, providing definitions, assumptions, formulas, and examples for each test. The summary table offers a side-by-side comparison for clarity.