Chapter 12: Inference for Proportions

12.1: Inference for the Population Proportion

This section deals with tests and confidence intervals for sample proportions. When the sample size is large, the distribution of p(hat) is approximately normal with mean π and standard deviation sqrt(π(1-π)/n). The inference procedures are reasonably accurate if the population is at least ten times larger than the sample, and if the sample size is such that nπ and n(1-π) are both at least 10.

A confidence interval for a sample proportion p(hat) has the form

The expression

is called the standard error of p(hat)

Tests of a null hypothesis: proportion = π are based on the statistic

12.2: Comparing Two Proportions

There are two standard error formulas that are commonly used when comparing proportions from two independent samples.

The two-proportion interval is constructed using the same estimate±critical*standard error format used for means and single proportions:

The logic behind this inferential procedure is the same as the logic behind all of our procedures. The only thing that changes is the formula used to calculate the test statistic and P-value. Do not get hung up in the formula...make sure you can interpret the results!

When comparing two proportions, we often test to determine whether or not two population proportions are equal. Since we are generally observing a difference in our p-hats, we must make an estimate as to what the two proportions may really be equal to. We call this estimate a ‘pooled’ proportion.

A significance test for two proportions uses a test statistic based on the pooled proportion:

Chapter 12: Inference for Proportions

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