Section 11.1 Inference about Two Population Proportions: Independent Samples

Section 11.1: Inference about Two Population Proportions: Independent Samples

Distinguishing Between Independent and Dependent Sampling

Understanding the difference between independent and dependent sampling is crucial when comparing two population proportions. The sampling method affects the validity and interpretation of statistical inference.

• Independent Sampling: Samples are drawn such that the selection of one individual does not influence the selection of another. Each sample comes from a distinct population or group.

• Dependent Sampling: Also known as paired or matched sampling, where samples are related or matched in some way (e.g., before-and-after measurements on the same subjects).

• Qualitative vs. Quantitative Variables: When determining sampling methods, also consider whether the response variable is categorical (qualitative) or numerical (quantitative).

Example:

• Comparing pass rates between students in a lab-based course and a lecture-based course (independent samples).

• Comparing prices of the same product purchased at two different stores by the same customers (dependent samples).

Testing Hypotheses Regarding Two Population Proportions from Independent Samples

Statistical inference about the difference between two population proportions involves hypothesis testing using independent samples. The goal is to determine if the observed difference is statistically significant.

• Sampling Distribution: The difference in sample proportions $\hat{p}_1 - \hat{p}_2$ is approximately normal if sample sizes are large enough.

• Mean: $p_1 - p_2$

• Standard Error: $\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$

• Test Statistic: $z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n_1} + \frac{\hat{p}(1-\hat{p})}{n_2}}}$, where $\hat{p}$ is the pooled sample proportion.

Example:

• Clinical trial comparing allergy rates between two groups receiving different treatments.

Steps for Hypothesis Testing (Difference Between Two Proportions)

1. State the null and alternative hypotheses.

2. Check conditions (randomness, independence, sample size).

3. Calculate the test statistic (by hand or using technology).

4. Find the p-value and compare to significance level.

5. Draw a conclusion in context.

Constructing and Interpreting Confidence Intervals for the Difference Between Two Population Proportions

A confidence interval estimates the range in which the true difference between two population proportions lies, with a specified level of confidence.

• Conditions: Random sampling, independence, and sufficiently large sample sizes.

• Confidence Interval Formula:

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

• Lower Bound: $(\hat{p}_1 - \hat{p}_2) - z_{\alpha/2} \times SE$

• Upper Bound: $(\hat{p}_1 - \hat{p}_2) + z_{\alpha/2} \times SE$

Example:

• Survey comparing the proportion of adults who believe moral values are declining in two different years.

Determining the Sample Size Necessary for Estimating the Difference Between Two Population Proportions

Calculating the required sample size ensures that the confidence interval for the difference between two proportions meets a desired margin of error.

• 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 Formula (when estimates of $p_1$ and $p_2$ are available):

$n = \frac{z_{\alpha/2}^2 \cdot [p(1-p)]}{E^2}$

• Sample Size Formula (when no prior estimates are available): Use $p = 0.5$ for maximum variability.

Example:

• Determining the sample size needed to estimate the difference in political trust between Democrats and Republicans within a specified margin of error.

Summary Table: Key Concepts in Inference about Two Population Proportions

Concept	Definition	Formula	Example
Independent Sampling	Samples drawn from separate populations, no pairing	N/A	Comparing pass rates in two different courses
Difference in Proportions	Difference between two sample proportions	$\hat{p}_1 - \hat{p}_2$	Proportion of allergy sufferers in two treatment groups
Standard Error	Measure of variability for difference in proportions	$\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$	Calculating variability in survey results
Confidence Interval	Range of plausible values for difference in proportions	$(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \times SE$	Estimating difference in political trust
Sample Size	Number of observations needed for desired margin of error	$n = \frac{z_{\alpha/2}^2 \cdot [p(1-p)]}{E^2}$	Determining sample size for survey

Additional info: Academic context and formulas have been expanded for clarity and completeness. Examples have been generalized for broader understanding.