Chapter 11: Inference on Two Population Parameters: Hypothesis Testing and Confidence Intervals

Inference on Two Population Parameters

Overview

This chapter covers statistical inference procedures for comparing two population parameters, including proportions, means, and standard deviations. The main focus is on hypothesis testing and confidence interval construction for independent and dependent samples.

Inference about Two Population Proportions

Hypothesis Testing for Two Proportions

To compare two population proportions, we use independent samples and test hypotheses about their difference. The hypotheses can be structured as:

• Two-Tailed: $H_0: p_1 = p_2$ vs. $H_a: p_1 \neq p_2$

• Left-Tailed: $H_0: p_1 = p_2$ vs. $H_a: p_1 < p_2$

• Right-Tailed: $H_0: p_1 = p_2$ vs. $H_a: p_1 > p_2$

1. Step 1: State the null and alternative hypotheses.

2. Step 2: Select the significance level $\alpha$.

3. Step 3: Compute the test statistic: $z_0 = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$ where $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$ is the pooled sample proportion.

4. Step 4: Compare the test statistic to the critical value or use the P-value approach.

   • If $|z_0| > z_{\alpha/2}$ (two-tailed), reject $H_0$.

   • If $z_0 < -z_{\alpha}$ (left-tailed), reject $H_0$.

   • If $z_0 > z_{\alpha}$ (right-tailed), reject $H_0$.

5. Step 5: State the conclusion.

Confidence Intervals for the Difference of Two Proportions

A $(1-\alpha)\cdot100\%$ confidence interval for $p_1 - p_2$ is:

• Lower bound: $(\hat{p}_1 - \hat{p}_2) - z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

• Upper bound: $(\hat{p}_1 - \hat{p}_2) + z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

Inference about Two Means: Dependent Samples (Matched Pairs)

Hypothesis Testing for Matched-Pairs Data

When data are paired (dependent samples), we analyze the differences between pairs. The hypotheses are:

• Two-Tailed: $H_0: \mu_d = 0$ vs. $H_a: \mu_d \neq 0$

• Left-Tailed: $H_0: \mu_d = 0$ vs. $H_a: \mu_d < 0$

• Right-Tailed: $H_0: \mu_d = 0$ vs. $H_a: \mu_d > 0$

1. Step 1: State the hypotheses.

2. Step 2: Select $\alpha$.

3. Step 3: Compute the test statistic: $t_0 = \frac{\bar{d} - \mu_d}{s_d/\sqrt{n}}$ where $\bar{d}$ is the mean of the differences, $s_d$ is the standard deviation of the differences, and $n$ is the number of pairs.

4. Step 4: Compare $t_0$ to the critical value from the $t$-distribution with $n-1$ degrees of freedom, or use the P-value approach.

5. Step 5: State the conclusion.

Confidence Interval for the Mean Difference

The confidence interval for the mean difference $\mu_d$ is:

• Lower bound: $\bar{d} - t_{\alpha/2} \frac{s_d}{\sqrt{n}}$

• Upper bound: $\bar{d} + t_{\alpha/2} \frac{s_d}{\sqrt{n}}$

Inference about Two Means: Independent Samples

Hypothesis Testing for Two Independent Means

For independent samples, the hypotheses are:

• Two-Tailed: $H_0: \mu_1 = \mu_2$ vs. $H_a: \mu_1 \neq \mu_2$

• Left-Tailed: $H_0: \mu_1 = \mu_2$ vs. $H_a: \mu_1 < \mu_2$

• Right-Tailed: $H_0: \mu_1 = \mu_2$ vs. $H_a: \mu_1 > \mu_2$

1. Step 1: State the hypotheses.

2. Step 2: Select $\alpha$.

3. Step 3: Compute the test statistic (Welch's t): $t_0 = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

4. Step 4: Compare $t_0$ to the critical value from the $t$-distribution with the smaller of $n_1-1$ or $n_2-1$ degrees of freedom, or use the P-value approach.

5. Step 5: State the conclusion.

Confidence Interval for the Difference of Two Means

The confidence interval for $\mu_1 - \mu_2$ is:

• Lower bound: $(\bar{x}_1 - \bar{x}_2) - t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

• Upper bound: $(\bar{x}_1 - \bar{x}_2) + t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Choosing the Appropriate Inference Procedure

To determine the correct test, consider:

• What parameter is addressed (proportion, mean, or standard deviation)?

• Are the samples independent or dependent?

• Are the population variances assumed equal or unequal?

Inference about Two Population Standard Deviations

F-Distribution and Hypothesis Testing

To compare two population standard deviations, use Fisher's F-distribution. The test statistic is:

• $F_0 = \frac{s_1^2}{s_2^2}$

The F-distribution is right-skewed, and its shape depends on the degrees of freedom in the numerator and denominator.

Critical Values and Table Lookup

Critical values for the F-distribution are found using statistical tables, based on the area in the right tail and the degrees of freedom.

Steps for Hypothesis Testing with Two Standard Deviations

1. Step 1: State the hypotheses:

   • Two-Tailed: $H_0: \sigma_1^2 = \sigma_2^2$ vs. $H_a: \sigma_1^2 \neq \sigma_2^2$

   • Left-Tailed: $H_0: \sigma_1^2 = \sigma_2^2$ vs. $H_a: \sigma_1^2 < \sigma_2^2$

   • Right-Tailed: $H_0: \sigma_1^2 = \sigma_2^2$ vs. $H_a: \sigma_1^2 > \sigma_2^2$

2. Step 2: Select $\alpha$.

3. Step 3: Compute $F_0$ and determine the critical value from the F-table.

4. Step 4: Compare $F_0$ to the critical value or use the P-value approach.

5. Step 5: State the conclusion.

Summary Table: Choosing the Correct Test

Use the following guidelines:

• Two Proportions: Use the z-test for independent samples.

• Two Means (Independent): Use Welch's t-test if variances are unequal.

• Two Means (Dependent): Use the paired t-test.

• Two Standard Deviations: Use the F-test.

Additional info: The images included above are directly relevant to the step-by-step procedures for hypothesis testing and confidence intervals, as well as the visual understanding of the F-distribution and the decision process for selecting the appropriate statistical test.