9.2 Inferences About Two Proportions Key Concept: This section presents methods for using ___________________________________________________________ ___________________________ for constructing a ________________________________________________________________________ ____________________ between the corresponding population proportions, or _____________________________________ ________________________________________________________________________________________________________________________. Requirements: 1. We have proportions from two ____________________________________________________ that are independent. (Samples are independent if the sample values selected from on population are ______ __________________________ to or somehow paired or matched with the sample values selected from the other population.) 2. For each of the two samples, the number of ____________________________ is at least ______ and the number of _______________________ is at least ________. Notation for Two Proportions: For population 1, we let: p1 = population proportion n1= size of the sample x1 = number of successes in the sample ̂ (the sample proportion) ̂

̂

The corresponding meanings are attached to p2, n2, x2, ̂ , and ̂ , which come from population 2. Pooled Sample Proportion The ____________________ sample proportion is denoted by ̅ and is given by: We denote the complement of ̅ by ̅, so ̅ Test Statistic for Two Proportions (with H0: p1 = p2) where p1 – p2 = 0 (assumed in the null hypothesis) P-Value: Use Table ___________. (Use the computed value of the test statistic z and find the P-Value by finding the area of the tail(s).) Critical Values: Use Table ___________. (Based on the significance level α as the area of the tail(s), find the critical value(s) boundary.)

9.2 Inferences About Two Proportions Confidence Interval Estimate of p1 – p2 The confidence interval estimate of the difference p1 – p2 is: where the margin of error E = Example – Hypothesis Tests where H0: p1 = p2 Carpal tunnel syndrome is a common wrist complaint resulting from a compressed nerve. It is often the result of extended use of repetitive wrist movements, such as those associated with the use of the keyboard. Among the various treatments available two are common: apply a splint or perform surgery. The table below summarizes the results from a clinical trial, comparing patients who received a splint or underwent surgery. Surgery Splint Success one year after treatment 67 60 Total number treated 73 83 Success Rate 92% 72% Use a 0.05 significance level to test the claim that the success rate with surgery is better than the success rate with splinting. P-Value Method:

Traditional Method:

9.2 Inferences About Two Proportions Example – Constructing Confidence Intervals Use the sample data given in the previous example to construct a 90% confidence interval estimate of the difference between the two population proportions. (The confidence level of 90% is comparable to the significance level of α = 0.05 use in the preceding right-tailed test).