Inference for Two Samples: Proportions, Means, and Variances

Chapter 11: Inference for Two Samples

This chapter focuses on statistical inference procedures involving two samples. It covers hypothesis testing and confidence intervals for differences between two population proportions, means, and standard deviations. Both independent and dependent (paired) samples are considered, and methods include both classical (critical value) and p-value approaches.

Section 11.1: Inference About Two Population Proportions

When comparing two population proportions, we use hypothesis testing to determine if there is a significant difference between them. This section covers both the traditional (critical value) and p-value methods.

• Independent Samples: The two samples are independent if the selection of one sample does not influence the other. Example: Comparing mean grades of students in traditional vs. hybrid course formats.

• Dependent Samples: The two samples are dependent (paired) if each observation in one sample is paired with a related observation in the other. Example: Pre- and post-test scores for the same students.

Hypothesis Testing Procedure (Two Proportions)

1. State the null and alternative hypotheses:

   • Null hypothesis: $H_0: p_1 = p_2$

   • Alternative hypothesis: $H_a: p_1 \neq p_2$, $p_1 > p_2$, or $p_1 < p_2$ (depending on the research question)

2. Determine the direction of the test based on the alternative hypothesis (two-tailed or one-tailed).

3. Find the critical value using the standard normal (z) distribution.

4. Compute the test statistic:

   • Test statistic formula: $z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$ where $\hat{p}_1$ and $\hat{p}_2$ are sample proportions, $n_1$ and $n_2$ are sample sizes, and $\hat{p}$ is the pooled proportion: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$

5. Decision: Reject $H_0$ if the test statistic falls in the critical region; otherwise, do not reject $H_0$.

6. Conclusion: State the result in the context of the problem.

Confidence Interval for Difference of Proportions

• Formula: $$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$$

• Interpretation: The interval estimates the difference between two population proportions with a specified level of confidence.

Sample Size Formula for Estimating Difference Between Two Proportions

• Formula (for each group): $$n = \frac{(z^*)^2 [p_1(1-p_1) + p_2(1-p_2)]}{E^2}$$where$E$ is the desired margin of error.

Example Application

• Suppose a researcher wants to compare the effectiveness of two teaching methods by comparing the proportion of students passing in each group. The above procedures are used to test if the difference is statistically significant.

Section 11.2: Inference About Two Means – Dependent Samples (Paired Data)

This section addresses hypothesis testing and confidence intervals for the mean difference in paired (dependent) samples, such as before-and-after studies.

• Paired Samples: Each subject or entity is measured twice, resulting in pairs of observations (e.g., pre- and post-test scores).

Hypothesis Testing Procedure (Paired t-Test)

1. State hypotheses:

   • Null hypothesis: $H_0: \mu_d = 0$ (no mean difference)

   • Alternative hypothesis: $H_a: \mu_d \neq 0$, $\mu_d > 0$, or $\mu_d < 0$

2. Calculate the differences for each pair: $d_i = x_{i,\text{post}} - x_{i,\text{pre}}$

3. Compute the mean and standard deviation of the differences:

   • Mean: $\bar{d}$

   • Standard deviation: $s_d$

4. Compute the test statistic: $$t = \frac{\bar{d} - \mu_{d,0}}{s_d / \sqrt{n}}$$where$\mu_{d,0}$is the hypothesized mean difference (often 0),$n$ is the number of pairs.

5. Find the critical t-value from the t-distribution with $n-1$ degrees of freedom.

6. Decision: Compare the test statistic to the critical value; reject $H_0$ if in the rejection region.

7. Conclusion: Interpret the result in context.

Confidence Interval for Mean Difference

• Formula: $$\bar{d} \pm t^* \frac{s_d}{\sqrt{n}}$$

Example Application

• Given pre-course SAT scores: 920, 1000, 1200, 1100, 950, 860

• Post-course SAT scores: 930, 990, 1250, 1200, 960, 860

• Differences: -10, 10, -50, -100, -10, 0; Mean difference: -18.3; Standard deviation: 40.7; $n=6$

• Test statistic: $$t = \frac{-18.3}{40.7/\sqrt{6}} = -1.101$$

• Critical t-value (for $\alpha = 0.10$, left-tailed, df=5): $-1.476$

• Since $-1.101$ is not less than $-1.476$, do not reject $H_0$.

Section 11.3: Inference About Two Means – Independent Samples

This section covers hypothesis testing and confidence intervals for the difference between two independent population means. The procedures are similar to those in Chapter 10, with some extensions.

• Independent Samples: The two groups are unrelated (e.g., different students in two teaching methods).

Hypothesis Testing Procedure (Two-Sample t-Test)

1. State hypotheses:

   • Null hypothesis: $H_0: \mu_1 = \mu_2$

   • Alternative hypothesis: $H_a: \mu_1 \neq \mu_2$, $\mu_1 > \mu_2$, or $\mu_1 < \mu_2$

2. Check assumptions: Normality, equal or unequal variances (use pooled or unpooled test as appropriate).

3. Compute the test statistic:

   • 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:$$s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$$

   • If variances are not assumed equal (unpooled): $$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$

4. Find the critical t-value (degrees of freedom depend on the test type).

5. Decision: Compare the test statistic to the critical value; reject $H_0$ if in the rejection region.

6. Conclusion: Interpret the result in context.

Confidence Interval for Difference of Means

• Formula (equal variances): $$(\bar{x}_1 - \bar{x}_2) \pm t^* s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$$

• Formula (unequal variances): $$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$$

Pooled Two-Sample t-Test

• Used when population variances are assumed equal.

• Pooled standard deviation combines information from both samples.

Section 11.4: Inference About Two Population Standard Deviations

This section introduces inference procedures for comparing two population standard deviations, using the F-distribution.

• F-Test for Equality of Variances: Used to test if two population variances are equal.

Hypothesis Testing Procedure (F-Test)

1. State hypotheses:

   • Null hypothesis: $H_0: \sigma_1^2 = \sigma_2^2$

   • Alternative hypothesis: $H_a: \sigma_1^2 \neq \sigma_2^2$, $\sigma_1^2 > \sigma_2^2$, or $\sigma_1^2 < \sigma_2^2$

2. Compute the test statistic: $$F = \frac{s_1^2}{s_2^2}$$where$s_1^2$and$s_2^2$ are the sample variances.

3. Find the critical value from the F-distribution with appropriate degrees of freedom.

4. Decision: Compare the test statistic to the critical value; reject $H_0$ if in the rejection region.

5. Conclusion: Interpret the result in context.

Section 11.5: Summary and Practice

This section provides a summary of all procedures in the chapter and emphasizes the importance of selecting the correct test and approach (distribution, formula, or technology).

Comparison Table: Two-Sample Inference Procedures

Parameter	Sample Type	Test Statistic	Distribution
Difference of Proportions	Independent	$z$	Standard Normal
Difference of Means	Independent	$t$ (pooled or unpooled)	t-distribution
Mean Difference	Dependent (Paired)	$t$	t-distribution
Difference of Variances	Independent	$F$	F-distribution

Additional info: The chapter emphasizes the importance of understanding when to use each test, the similarities between procedures for one and two samples, and the use of both classical and p-value approaches. Technology (such as Minitab) can be used to compute p-values and confidence intervals efficiently.