Chapter 19 Comparing Two Proportions

Outline Two-sample problems: proportions Two-sample problems: proportions The sampling distribution of a difference between proportions The sampling distribution of a difference between proportions Large-sample confidence intervals for comparing proportions Large-sample confidence intervals for comparing proportions Significance tests for comparing proportions Significance tests for comparing proportions

1. Two-sample problems: proportions In a two sample problem, we compare two populations or the responses to two treatments based on two independent samples. In a two sample problem, we compare two populations or the responses to two treatments based on two independent samples. Notation: Notation: Population Population proportion Sample size Sample proportion 12 n1n1n2n2n1n1n2n2

Case Study - Machine Reliability A study is performed to test of the reliability of products produced by two machines. Machine A produced 8 defective parts in a run of 140, while machine B produced 10 defective parts in a run of 200. This is an example of when to use the two-proportion z procedures. nDefects Machine A 1408 Machine B 20011

Case Study - Summer Jobs u A university financial aid office polled a simple random sample of undergraduate students to study their summer employment. u Not all students were employed the previous summer. Here are the results: u Is there evidence that the proportion of male students who had summer jobs differs from the proportion of female students who had summer jobs? Summer Status MenWomen Employed Not Employed Total797732

2. The sampling distributions of When the samples are large, the distribution of is approximate normal. When the samples are large, the distribution of is approximate normal. The mean of is (p 1 -p 2 ). (unbiased) The mean of is (p 1 -p 2 ). (unbiased) The standard deviation of is The standard deviation of is

3. Large-sample confidence intervals for comparing proportions

Example19.1, 19.2 (page 493, page 495) Does preschool help? To study the long term effects of preschool programs, two groups of poor children were looked at since early childhood. Group 2 attended preschool, but group 1 did not. To study the long term effects of preschool programs, two groups of poor children were looked at since early childhood. Group 2 attended preschool, but group 1 did not. One response variable of interest is the need for social services as adults. Let p 1 be the proportion from population 1 who need such services, and similarly for p 2. One response variable of interest is the need for social services as adults. Let p 1 be the proportion from population 1 who need such services, and similarly for p 2. Compute a 95% confidence interval for (p 1 -p 2 ). Compute a 95% confidence interval for (p 1 -p 2 ).

4. Significance tests for comparing proportions H 0 : p 1 =p 2 vs H a : one-sided or two-sided H 0 : p 1 =p 2 vs H a : one-sided or two-sided If H 0 is true, then the two proportions are equal to some common value p. If H 0 is true, then the two proportions are equal to some common value p. Instead of estimating p 1 and p 2 separately, we will combine the two samples to estimate p. (why is it better?) Instead of estimating p 1 and p 2 separately, we will combine the two samples to estimate p. (why is it better?)

Pooled Sample Proportion This combined or pooled estimate is called the pooled sample proportion, This combined or pooled estimate is called the pooled sample proportion, pooled sample proportion

Then the Standard Error of becomes: Then the Standard Error of becomes:

Example: Example 19.4 and 19.5: Choosing a mate Example 19.4 and 19.5: Choosing a mate –(page 500)