Inference on Two Population Parameters: Proportions and Means

11.1 Inference About Two Population Proportions

Determining Between Independent and Dependent Sampling

When comparing two populations, it is crucial to determine whether the samples are independent or dependent. This distinction affects the choice of statistical methods for inference.

• Independent Sampling: The selection of individuals in one sample does not influence the selection in the other sample.

• Dependent Sampling (Matched Pairs): The selection of individuals in one sample is related to the selection in the other sample, often through pairing or matching.

• Examples: Comparing two medications on different subjects (independent), or measuring reaction times before and after a stimulus on the same subjects (dependent).

Test Hypotheses Regarding Two Proportions from Independent Samples

Statistical inference about the difference between two population proportions involves hypothesis testing using sample data.

• Sampling Distribution: For large samples, the difference in sample proportions is approximately normal.

• Standard Error: The standard deviation of the sampling distribution is given by: $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$

• Hypothesis Test Steps:

  1. State null and alternative hypotheses: $H_0: p_1 = p_2$ vs. $H_a: p_1 \neq p_2$ (or $p_1 > p_2$, $p_1 < p_2$).

  2. Calculate the test statistic:

  $z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ where $\hat{p}$ is the pooled sample proportion.

• Example: Testing whether the proportion of Democrats differs between two age groups.

Construct and Interpret Confidence Intervals for the Difference Between Two Population Proportions

Confidence intervals estimate the range in which the true difference between two population proportions lies.

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

• Interpretation: The interval provides a range of plausible values for the difference in population proportions.

• Example: Estimating the difference in the proportion of individuals above and below average weight.

Determine the Sample Size Necessary for Estimating the Difference Between Population Proportions

Calculating the required sample size ensures that the confidence interval for the difference in proportions meets a specified margin of error.

• Formula: $n = \left( \frac{z_{\alpha/2}}{E} \right)^2 \left[ p_1(1-p_1) + p_2(1-p_2) \right]$

• Application: Used when planning surveys or experiments to compare proportions.

• Example: Determining sample size for comparing proportions of males and females who exercise regularly.

11.2 Inference About Two Means: Dependent Samples

Test Hypotheses for a Population Mean from Matched-Pairs Data

Matched-pairs designs compare two measurements from the same subjects or closely matched subjects. The analysis focuses on the differences within pairs.

• Difference Calculation: For each pair, compute the difference $d_i$.

• Test Statistic: $t = \frac{\bar{d}}{s_d/\sqrt{n}}$ where $\bar{d}$ is the mean difference, $s_d$ is the standard deviation of differences, and $n$ is the number of pairs.

• Example: Comparing time spent waiting in line before and after a process change for the same individuals.

Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data

Confidence intervals for matched-pairs data estimate the mean difference between paired observations.

• Formula: $\bar{d} \pm t_{\alpha/2, n-1} \frac{s_d}{\sqrt{n}}$

• Interpretation: The interval provides a range of plausible values for the mean difference.

• Example: Estimating the mean difference in distance required to come to a stop before and after a training program.

11.3 Inference About Two Means: Independent Samples

Test Hypotheses Regarding the Difference of Two Independent Means

When samples are independent, the difference in means is analyzed using a two-sample t-test. Assumptions about population variances affect the choice of test.

• Equal Variances: Use pooled variance estimate.

• Unequal Variances: Use Welch's t-test.

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

• Example: Comparing walking speeds of business and leisure travelers at airports.

Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means

Confidence intervals for the difference in means provide a range of plausible values for the true difference between two population means.

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

• Interpretation: The interval estimates the difference in population means.

• Example: Estimating the difference in mean scores between two political groups.

Tables: Summary of Hypothesis Test Steps and Confidence Interval Formulas

The following tables summarize the key formulas and steps for hypothesis testing and confidence interval construction for two population proportions and means.

Test	Statistic	Formula	Assumptions
Difference in Proportions	z	$z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$	Large samples, independent samples
Difference in Means (Matched Pairs)	t	$t = \frac{\bar{d}}{s_d/\sqrt{n}}$	Differences are approximately normal
Difference in Means (Independent Samples)	t	$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$	Samples are independent, normality

Additional info: These notes expand on the original content by providing definitions, formulas, and context for statistical inference regarding two population parameters, including both proportions and means. Examples and tables have been logically grouped and explained for clarity.