Comparing Two Population Proportions

Overview

This lecture introduces how to construct and interpret confidence intervals for the difference between two population proportions, building on prior work with a single proportion.

Review of Previous Concepts

• Section 7.4 focused on constructing confidence intervals for a single population proportion.
• The same symbols (p, n, p-hat) are used when analyzing proportions.

Comparing Two Populations

• Section 7.5 addresses comparing two population proportions instead of just one.
• Each group (e.g., men vs. women) has its own proportion, sample size, and sample proportion.
• Use subscripts 1 and 2 to distinguish between the two populations (e.g., p₁, p₂, n₁, n₂, p̂₁, p̂₂).
• The comparison is made by subtracting one proportion from the other to find their difference.

Constructing Confidence Intervals for the Difference

• We estimate the difference between population proportions using the difference between sample proportions (p̂₁ - p̂₂).
• A confidence interval for the difference is constructed to determine if there is a significant difference between groups.
• If the interval includes zero, there may be no significant difference; if not, a significant difference may exist.

Confidence Interval Procedure (Three Steps)

• The process remains:
  1. Check the necessary conditions (Central Limit Theorem conditions).
  2. Calculate the confidence interval.
  3. Interpret the interval.

Checking Necessary Conditions

• For both populations, check:
  • Random sample.
  • Large sample (number of successes and failures).
  • Large population.
• These checks must be performed separately for both groups.
• Independence is a new condition in Section 7.5, ensuring the samples from each population do not influence each other.
• All examples in this section will have independent samples.

Key Terms & Definitions

• Proportion (p) — the fraction of a population with a specific characteristic.
• Sample proportion (p̂) — the fraction of a sample with the characteristic.
• Confidence interval — a range estimating a parameter (e.g., difference between proportions) with a certain confidence level.
• Independence — samples from one population do not affect samples from the other population.

Action Items / Next Steps

• Review the conditions for confidence intervals for both populations.
• Prepare to work through three example problems comparing two population proportions.