Inferences from Two Samples: Proportions and Means

Inferences from Two Samples

Overview

This chapter explores statistical methods for comparing two populations using sample data. The focus is on hypothesis testing and confidence interval estimation for differences between two proportions, two means (independent samples), and two means (dependent/matched pairs). These methods are foundational for making inferences about population parameters based on sample evidence.

Inferences About Two Proportions

Key Concepts and Objectives

• Hypothesis Test: Test a claim about two population proportions.

• Confidence Interval: Construct a confidence interval estimate for the difference between two population proportions.

Notation for Two Proportions

• p1: Population proportion for group 1

• \( \hat{p}_1 = \frac{x_1}{n_1} \): Sample proportion for group 1

• n1: Size of the first sample

• \( \hat{q}_1 = 1 - \hat{p}_1 \): Complement of the sample proportion

• x1: Number of successes in the first sample

• Corresponding notations apply for population 2.

Pooled Sample Proportion

The pooled sample proportion combines the two sample proportions into one estimate:

• \( \bar{p} = \frac{x_1 + x_2}{n_1 + n_2} \)

• \( \bar{q} = 1 - \bar{p} \)

Requirements for Inference

• Samples must be simple random samples.

• Samples must be independent (no natural pairing between groups).

• Each sample must have at least 5 successes and 5 failures: \( n\hat{p} \geq 5 \) and \( n\hat{q} \geq 5 \) for both samples.

Test Statistic for Two Proportions

To test the null hypothesis \( H_0: p_1 = p_2 \), use the following z-test statistic:

• $z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\bar{p}\bar{q}\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

• Where \( p_1 - p_2 = 0 \) under the null hypothesis.

Confidence Interval for Difference of Proportions

The confidence interval for \( p_1 - p_2 \) is:

• $(\hat{p}_1 - \hat{p}_2) - E < (p_1 - p_2) < (\hat{p}_1 - \hat{p}_2) + E$

• Where the margin of error \( E \) is:

$E = z_{\alpha/2} \sqrt{\frac{\hat{p}_1 \hat{q}_1}{n_1} + \frac{\hat{p}_2 \hat{q}_2}{n_2}}$

Example: Comparing Proportions of Cars with Rear License Plates Only

Suppose we want to test whether the proportion of cars with rear license plates only is the same in Connecticut and New York.

• Connecticut: \( \hat{p}_1 = \frac{239}{2049} = 0.117 \)

• New York: \( \hat{p}_2 = \frac{9}{550} = 0.016 \)

Step-by-Step Solution

1. State hypotheses: $H_0: p_1 = p_2$, $H_1: p_1 \neq p_2$

2. Significance level: $\alpha = 0.05$

3. Calculate pooled proportion: $\bar{p} = \frac{239 + 9}{2049 + 550} = 0.0954$

4. Calculate test statistic:

$z = \frac{0.117 - 0.016}{\sqrt{0.0954 \times 0.9046 \left(\frac{1}{2049} + \frac{1}{550}\right)}} = 7.11$

Interpretation

• P-value is much less than 0.05, so we reject $H_0$.

• There is sufficient evidence to conclude that the proportions are different.

Confidence Interval Example

Construct a 95% confidence interval for $p_1 - p_2$:

• Margin of error: $E = 1.96 \sqrt{\frac{239}{2049} \frac{1810}{2049} / 2049 + \frac{9}{550} \frac{541}{550} / 550} = 0.017482$

• Confidence interval: $0.100278 \pm 0.017482$ or $0.0828 < (p_1 - p_2) < 0.118$

Theoretical Foundation

• The sampling distribution of $\hat{p}_1$ is approximately normal with mean $p_1$ and variance $\frac{p_1 q_1}{n_1}$.

• The variance of the difference $\hat{p}_1 - \hat{p}_2$ is $\frac{p_1 q_1}{n_1} + \frac{p_2 q_2}{n_2}$.

Inferences About Two Means: Independent Samples

Key Concepts and Objectives

• Hypothesis Test: Test a claim about two independent population means.

• Confidence Interval: Construct a confidence interval estimate for the difference between two independent population means.

Notation for Two Means

• \( \mu_1 \): Population mean for group 1

• \( \bar{x}_1 \): Sample mean for group 1

• \( \sigma_1, s_1 \): Population and sample standard deviation for group 1

• \( n_1 \): Sample size for group 1

• Corresponding notations apply for group 2.

Test Statistic for Two Means (Independent Samples)

When population standard deviations are unknown and not assumed equal, use the following t-test statistic:

• $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Degrees of freedom (df) can be approximated by:

• $df = \frac{(A + B)^2}{\frac{A^2}{n_1 - 1} + \frac{B^2}{n_2 - 1}}$, where $A = \frac{s_1^2}{n_1}$ and $B = \frac{s_2^2}{n_2}$

Confidence Interval for Difference of Means

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

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

Example: Are Male Professors and Female Professors Rated Differently?

Student course evaluation scores for courses taught by female and male professors are compared using a two-sample t-test.

• Hypotheses: $H_0: \mu_1 = \mu_2$, $H_1: \mu_1 \neq \mu_2$

• Significance level: $\alpha = 0.05$

• Test statistic: $t = -0.660$ (calculated from sample data)

• P-value: 0.5172 (greater than 0.05)

• Conclusion: Fail to reject $H_0$; no significant difference in mean scores.

Confidence Interval Example

• 95% confidence interval for $\mu_1 - \mu_2$: $-0.53 < (\mu_1 - \mu_2) < 0.27$

• Since the interval contains 0, there is no significant difference between the means.

Summary Table: Key Formulas

Parameter	Test Statistic	Confidence Interval
Two Proportions	$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\bar{p}\bar{q}(\frac{1}{n_1} + \frac{1}{n_2})}}$	$(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_1 \hat{q}_1}{n_1} + \frac{\hat{p}_2 \hat{q}_2}{n_2}}$
Two Means (independent, unknown variances)	$t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$	$(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Additional info: These methods are essential for comparing groups in experimental and observational studies, especially in health and biological sciences. Technology (such as statistical software) is commonly used to perform these calculations and obtain P-values.