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Chapter 19

Chapter 19. Comparing Two Proportions. Outline. Two-sample problems: proportions The sampling distribution of a difference between proportions Large-sample confidence intervals for comparing proportions Significance tests for comparing proportions. 1. Two-sample problems: proportions.

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Chapter 19

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  1. Chapter 19 Comparing Two Proportions

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

  3. 1. Two-sample problems: proportions • In a two sample problem, we compare two populations or the responses to two treatments based on two independentsamples. • Notation:

  4. 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.

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

  6. 2. The sampling distributions of • When the samples are large, the distribution of is approximate normal. • The mean of is (p1-p2). (unbiased) • The standard deviation of is

  7. 3. Large-sample confidence intervals for comparing proportions

  8. 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. • One response variable of interest is the need for social services as adults. Let p1 be the proportion from population 1 who need such services, and similarly for p2. • Compute a 95% confidence interval for (p1-p2).

  9. 4. Significance tests for comparing proportions • H0: p1=p2 vs Ha: one-sided or two-sided • If H0 is true, then the two proportions are equal to some common value p. • Instead of estimating p1 and p2 separately, we will combine the two samples to estimate p. (why is it better?)

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

  11. Then the Standard Error of becomes:

  12. Example: • Example 19.4 and 19.5: Choosing a mate • (page 500)

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