Inferences from Two Populations: Hypothesis Testing and Confidence Intervals for Two Proportions

Inferences from Two Populations

Introduction

This study guide covers statistical methods for making inferences about two populations, focusing on hypothesis testing and confidence intervals for the difference between two population proportions. These concepts are essential for comparing groups in experimental and observational studies.

Sampling Methods

Independent vs. Dependent Sampling

Understanding the type of sampling is crucial for selecting the correct statistical test.

• Independent Sampling: Individuals selected for one sample do not influence the selection in the other sample.

• Dependent Sampling (Matched-Pairs): Selection in one sample is used to determine selection in the other, often matching individuals against themselves or similar subjects.

Example: Comparing pass rates between students in traditional and lab-based math courses (independent samples), or comparing prices of identical products at two retailers (matched-pairs).

Comparing Two Population Proportions

Formulating Hypotheses

When comparing proportions, we define:

• p1: Proportion in population 1

• p2: Proportion in population 2

Sample proportions are calculated as:

• $\hat{p}_1 = \frac{x_1}{n_1}$

• $\hat{p}_2 = \frac{x_2}{n_2}$

Hypotheses can be:

Type	Null Hypothesis	Alternative Hypothesis
Two-Tailed	$H_0: p_1 = p_2$	$H_1: p_1 \neq p_2$
Left-Tailed	$H_0: p_1 = p_2$	$H_1: p_1 < p_2$
Right-Tailed	$H_0: p_1 = p_2$	$H_1: p_1 > p_2$

Sampling Distribution of the Difference Between Two Proportions

The difference $\hat{p}_1 - \hat{p}_2$ is approximately normally distributed if:

• $n_1 \hat{p}_1 (1 - \hat{p}_1) \geq 10$

• $n_2 \hat{p}_2 (1 - \hat{p}_2) \geq 10$

• Each sample size is no more than 5% of the population size

Mean: $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$

Standard deviation:

$\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }$

Pooled Estimate

When the null hypothesis is $p_1 = p_2$, the pooled estimate $\hat{p}$ is:

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

Test Statistic for Comparing Two Proportions

The standardized test statistic is:

$z_0 = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{ \hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) }}$

Hypothesis Testing Algorithm for Two Proportions

Steps

1. State Hypotheses: Choose null and alternative hypotheses based on the research question.

2. Select Significance Level ($\alpha$): Common choices are 0.05 or 0.01.

3. Compute Test Statistic: Use the formula above.

4. Classical Approach: Compare the test statistic to the critical value from the standard normal table.

5. P-Value Approach: Calculate the p-value and compare to $\alpha$.

6. State Conclusion: Reject or fail to reject the null hypothesis based on the comparison.

Decision Rules

Test Type	Decision Rule
Two-Tailed	If $z_0 < -z_{\alpha/2}$ or $z_0 > z_{\alpha/2}$, reject $H_0$
Left-Tailed	If $z_0 < -z_{\alpha}$, reject $H_0$
Right-Tailed	If $z_0 > z_{\alpha}$, reject $H_0$

Example: Comparing Proportions of A's on Two Exams

• Midterm 1: $n_1 = 122$, $x_1 = 60$ ($\hat{p}_1 = 0.49$)

• Midterm 2: $n_2 = 113$, $x_2 = 37$ ($\hat{p}_2 = 0.33$)

• Observed difference: $0.49 - 0.33 = 0.16$

• Use two-proportion z-test to determine if the difference is statistically significant.

Example: Clinical Trial of Nasonex

• Group 1 (Nasonex): $n_1 = 2103$, $x_1 = 547$ ($\hat{p}_1 = 0.260$)

• Group 2 (Placebo): $n_2 = 1671$, $x_2 = 368$ ($\hat{p}_2 = 0.220$)

• Pooled estimate: $\hat{p} = 0.242$

• Test statistic: $z_0 = 2.85$

• Critical value for $\alpha = 0.05$ (right-tailed): $z_{0.05} = 1.645$

• Since $2.85 > 1.645$, reject $H_0$; sufficient evidence that Nasonex users have a higher rate of headaches.

• P-value: 0.002 < 0.05, also leads to rejection of $H_0$.

Constructing Confidence Intervals for the Difference Between Two Proportions

Requirements

• Samples are independent and randomly selected.

• $n_1 \hat{p}_1 (1 - \hat{p}_1) \geq 10$ and $n_2 \hat{p}_2 (1 - \hat{p}_2) \geq 10$

• Sample sizes are no more than 5% of the population size.

Confidence Interval Formula

For a $(1-\alpha) \times 100\%$ confidence interval:

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:

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

Example: Gallup Poll on Moral Values

• Sample 1 (2022): $n_1 = 1007$, $x_1 = 784$ ($\hat{p}_1 = 0.779$)

• Sample 2 (2002): $n_2 = 1100$, $x_2 = 737$ ($\hat{p}_2 = 0.67$)

• 90% confidence interval for $\hat{p}_1 - \hat{p}_2$ is (0.077, 0.141)

• Interpretation: We are 90% confident that the difference in proportions is between 0.077 and 0.141. Since the interval does not contain 0, there is evidence of a real difference.

Margin of Error and Sample Size Determination

Margin of Error Formula

$E = z_{\alpha/2} \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} }$

Sample Size for Desired Margin of Error

To achieve a margin of error $E$ for a $(1-\alpha) \times 100\%$ confidence interval:

If prior estimates $\hat{p}_1$ and $\hat{p}_2$ are available:

$n_1 = n_2 = \left[ \hat{p}_1(1-\hat{p}_1) + \hat{p}_2(1-\hat{p}_2) \right] \left( \frac{z_{\alpha/2}}{E} \right)^2$

If no prior estimates:

$n_1 = n_2 = 0.5 \left( \frac{z_{\alpha/2}}{E} \right)^2$

Example: Political Scientist Estimating Trust in State Government

• Desired margin of error: $E = 0.03$

• Confidence level: 95% ($z_{\alpha/2} = 1.96$)

• Prior estimates: $\hat{p}_1 = 0.62$, $\hat{p}_2 = 0.55$

• Required sample size: $n_1 = n_2 = 2063$

• If no prior estimates: $n_1 = n_2 = 2135$

Summary Table: Key Formulas

Concept	Formula
Sample Proportion	$\hat{p} = \frac{x}{n}$
Pooled Proportion	$\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$
Test Statistic	$z_0 = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{ \hat{p}(1-\hat{p}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) }}$
Confidence Interval	$(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} }$
Margin of Error	$E = z_{\alpha/2} \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} }$
Sample Size (with prior estimates)	$n_1 = n_2 = [\hat{p}_1(1-\hat{p}_1) + \hat{p}_2(1-\hat{p}_2)] \left( \frac{z_{\alpha/2}}{E} \right)^2$
Sample Size (no prior estimates)	$n_1 = n_2 = 0.5 \left( \frac{z_{\alpha/2}}{E} \right)^2$

Additional info:

• These methods are covered in Chapter 11: Inference on Two Population Parameters.

• All examples and formulas are applicable for large sample sizes and independent samples.