Review of Hypothesis Testing for Two Samples (Chapter 10)

Hypothesis Testing for Two Samples

Introduction to Two-Sample Hypothesis Testing

Two-sample hypothesis testing is a fundamental statistical method used to compare the means or proportions of two independent groups. This technique helps determine whether observed differences are statistically significant or likely due to random variation.

• Purpose: To assess if there is a significant difference between two population parameters (means or proportions).

• Common Applications: Comparing treatment vs. control groups, male vs. female responses, or before-and-after measurements.

Types of Two-Sample Tests

• Independent Samples t-Test: Used when comparing the means of two independent groups.

• Pooled vs. Unpooled Variance: If population variances are assumed equal, use pooled variance; otherwise, use unpooled (Welch's t-test).

• Two-Proportion z-Test: Used to compare proportions between two groups.

Formulating Hypotheses

Hypotheses are statements about population parameters. The null hypothesis ($H_0$) typically states that there is no difference, while the alternative hypothesis ($H_a$) suggests a difference exists.

• Null Hypothesis ($H_0$): $\mu_1 = \mu_2$ or $p_1 = p_2$

• Alternative Hypothesis ($H_a$): $\mu_1 \neq \mu_2$, $\mu_1 > \mu_2$, $\mu_1 < \mu_2$ (similarly for proportions)

• One-tailed vs. Two-tailed Tests: Choose based on the research question.

Test Statistics and Formulas

• t-Test for Independent Samples:

If variances are assumed equal (pooled):

$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

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: $df = n_1 + n_2 - 2$

If variances are not assumed equal (Welch's t-test):

$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Degrees of Freedom (Welch's): Calculated using the Welch-Satterthwaite equation.

• z-Test for Two Proportions:

$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

Where $\hat{p}$ is the pooled sample proportion:

$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$

Critical Values and Decision Making

• Critical Value: The threshold value from the t or z distribution used to determine statistical significance.

• Significance Level ($\alpha$): Commonly set at 0.05 or 0.01.

• Decision Rule: If the test statistic exceeds the critical value (in absolute value for two-tailed tests), reject $H_0$.

Example: Comparing Two Means

Suppose we want to test if the mean scores of two classes are different. We collect sample means, standard deviations, and sample sizes for each class.

• Calculate the test statistic using the appropriate formula.

• Find the critical value for the chosen $\alpha$ and degrees of freedom.

• Compare the test statistic to the critical value and make a conclusion.

Interpreting Results

• Reject $H_0$: There is sufficient evidence to conclude a difference exists.

• Fail to Reject $H_0$: There is not enough evidence to conclude a difference.

Common Critical Values Table

Critical values for t and z tests are often referenced from statistical tables. Below is a sample table of critical t-values for common significance levels and degrees of freedom.

df	t (0.05, two-tailed)	t (0.01, two-tailed)
10	2.228	3.169
20	2.086	2.845
30	2.042	2.750
∞ (z)	1.96	2.576

Additional info: Table values inferred for illustration; actual critical values may vary by textbook.

Summary Table: Two-Sample Test Types

Test Type	Parameter Compared	Assumptions	Test Statistic
Independent t-Test (pooled)	Means	Normality, equal variances	t
Welch's t-Test	Means	Normality, unequal variances	t
Two-Proportion z-Test	Proportions	Large sample size	z

Key Points to Remember

• Always check assumptions before choosing a test.

• State hypotheses clearly and select the correct test statistic.

• Use the correct degrees of freedom and critical values.

• Interpret results in the context of the research question.