Confidence intervals for the difference between two proportions Section 10.2

Objectives Construct confidence intervals for the difference between two proportions

Objective 1 Construct confidence intervals for the difference between two proportions

Notation Consider the following example. In a study of the effect of air pollution on lung function, a random sample of 50 children living in a community with a high level of ozone pollution had their lung capacities measured, and 14 of them had capacities that were below normal for their size. A second random sample of 80 children was drawn from a community with a low level of ozone pollution, and 12 of them had lung capacities that were below normal for their size. We are interested in studying the difference between the proportions of individuals in two different categories (communities). We begin by associating some notation for the population proportions, the numbers of individuals in each category, and the sample sizes. 𝑝 1 and 𝑝 2 are the population proportions of the category of interest in the two populations. 𝑥 1 and 𝑥 2 are the numbers of individuals in the category of interest in the two samples. 𝑛 1 and 𝑛 2 are the two sample sizes.

Point Estimate and Standard Error The point estimates for the population proportions are the sample proportions 𝑝 1 = 𝑥 1 𝑛 1 𝑝 2 = 𝑥 2 𝑛 2 It is relatively straightforward to see that the point estimate of 𝑝 1 − 𝑝 2 is 𝑝 1 − 𝑝 2 . The sample proportions 𝑝 1 and 𝑝 2 have variances 𝑝 1 1− 𝑝 1 𝑛 1 and 𝑝 2 1− 𝑝 2 𝑛 2 respectively. When the samples are independent, the variance of the difference 𝑝 1 − 𝑝 2 can be shown to be the sum of the variances of the sample proportions. In other words, the variance of 𝑝 1 − 𝑝 2 is 𝑝 1 1− 𝑝 1 𝑛 1 + 𝑝 2 1− 𝑝 2 𝑛 2 , therefore, the standard error is 𝑝 1 1− 𝑝 1 𝑛 1 + 𝑝 2 1− 𝑝 2 𝑛 2 .

Critical Value and Margin of Error When the sample sizes are large enough 𝑝 1 and 𝑝 2 are approximately normally distributed, so the critical value is 𝑧 𝛼 2 . We multiply the standard error by the critical value to obtain the margin of error: Margin of error = 𝑧 𝛼 2 𝑝 1 1 − 𝑝 1 𝑛 1 + 𝑝 2 1− 𝑝 2 𝑛 2 The level 100(1 − 𝛼)% confidence interval is Point estimate ± Margin of Error 𝑝 1 − 𝑝 2 ± 𝑧 𝛼 2 𝑝 1 1 − 𝑝 1 𝑛 1 + 𝑝 2 1− 𝑝 2 𝑛 2

Assumptions The method for constructing a confidence interval for the difference of population proportions requires that the sampling distribution be approximately normal. The following assumptions ensure this: Assumptions: We have two independent simple random samples. Each population is at least 20 times as large as the sample drawn from it. The individuals in each sample are divided into two categories. Both samples contain at least 10 individuals in each category.

Example A random sample of 50 children living in a community with a high level of ozone pollution had their lung capacities measured, and 14 of them had capacities that were below normal. A second random sample of 80 children was drawn from a community with a low level of ozone pollution, and 12 of them had lung capacities that were below normal. Construct a 95% confidence interval for the difference between the proportions of children with diminished lung capacity differ between the two communities. Solution: We have two independent random samples. The populations of children are more than 20 times as large as the samples. The individuals are divided into two categories with at least 10 individuals in each category. The assumptions are satisfied. We summarize the information: High Pollution Low Pollution Sample size 𝑛 1 =50 𝑛 2 =80 Number with below-normal lung capacity 𝑥 1 =14 𝑥 2 =12 Population proportion 𝑝 1 (unknown) 𝑝 2 (unknown)

Solution The point estimate is: 𝑝 1 − 𝑝 2 = 𝑥 1 𝑛 1 − 𝑥 2 𝑛 2 = 14 50 − 12 80 =0.130 The critical value for a 95% confidence interval is 𝑧 𝛼 2 =1.96. The standard error is 𝑝 1 1 − 𝑝 1 𝑛 1 + 𝑝 2 1− 𝑝 2 𝑛 2 = 0.280(1 −0.280) 50 + 0.150(1 −0.150) 80 = 0.075005. The margin of error is 𝑧 𝛼 2 𝑝 1 1 − 𝑝 1 𝑛 1 + 𝑝 2 1− 𝑝 2 𝑛 2 =1.96∙ 0.280(1 −0.280) 50 + 0.150(1 −0.150) 80 = 0.14701. The 95% confidence interval is Point estimate ± Margin of Error 0.130 ± 0.14701 0.130 – 0.14701< 𝑝 1 – 𝑝 2 < 0.130 + 0.14701 – 0.017 < 𝑝 1 – 𝑝 2 < 0.277 We are 95% confident that the difference between the proportions is between −0.017 and 0.277. This confidence interval contains 0. Therefore, we cannot be sure that the proportions of children with diminished lung capacity differ between the two communities. Remember: 𝑥 1 =14 𝑛 1 =50 𝑥 2 =12 𝑛 2 =80

Confidence Intervals on the TI-84 PLUS The 2-PropZInt command constructs confidence intervals for the difference between two proportions. This command is accessed by pressing STAT and highlighting the TESTS menu. Input the values of 𝑥 1 , 𝑛 1 , 𝑥 2 , and 𝑛 2 and the confidence level.

Example (TI-84 PLUS) A random sample of 50 children living in a community with a high level of ozone pollution had their lung capacities measured, and 14 of them had capacities that were below normal. A second random sample of 80 children was drawn from a community with a low level of ozone pollution, and 12 of them had lung capacities that were below normal. Construct a 95% confidence interval for the difference between the proportions of children with diminished lung capacity differ between the two communities. Solution: We have two independent random samples. The populations of children are more than 20 times as large as the samples. The individuals are divided into two categories with at least 10 individuals in each category. The assumptions are satisfied. We summarize the information: High Pollution Low Pollution Sample size 𝑛 1 =50 𝑛 2 =80 Number with below-normal lung capacity 𝑥 1 =14 𝑥 2 =12 Population proportion 𝑝 1 (unknown) 𝑝 2 (unknown)

Example (TI-84 PLUS) Solution: We press STAT and highlight the TESTS menu and select 2-PropZInt. Select Stats as the input method and enter the following values: Enter 0.95 as the confidence level and select Calculate. The confidence interval is (–0.17, 0.277). We are 95% confident that the difference between the proportions is between −0.017 and 0.277. This confidence interval contains 0. Therefore, we cannot be sure that the proportions of children with diminished lung capacity differ between the two communities. 𝑥 1 =14 𝑛 1 =50 𝑥 2 =12 𝑛 2 =80

You Should Know… The assumptions for constructing a confidence interval for the difference between proportions How to construct confidence intervals for the difference between two proportions