Name Mean of Sampling Distribution Standard Deviation/Error of Sampling Distribution 1 sample z-Interval for Proportions 1 sample z-interval for Means ( σ known) 1 sample z-test for proportions 1 sample z-test for Means ( σ known)

p-value Decision about Ho Conclusion about Ha p-value >  p-value < 

AP STATISTICS: CHAPTER 22 Comparing Two Proportions - Notes

We will compare 2 population proportions by looking at independent random samples from the 2 populations and comparing the difference in the sample proportions.

Confidence Interval for the difference in sample proportions ( ): Conditions that must be satisfied when constructing a confidence interval for the difference in 2 population proportions: 1. Random selection for both samples 2. (i.e., the counts of successes and failures in each sample are at least 10—some books use 5) 3. Both populations must be at least 10 times their corresponding sample sizes 4. Independent samples

Confidence Interval for : Standard Error

2 Sample z test for the difference of population proportions: H o :p 1 = p 2 H a: p 1 (> < ≠) p 2 Because we think p 1 = p 2, pool the estimates together to get a better estimate of the true p. Pooled sample proportion : (the pooled estimate of the common value of p 1 and p 2 )

Conditions that must be satisfied when testing hypotheses for the difference in 2 population proportions: 1. Random selection for both samples 2. where ê is the pooled estimate of the common value of p 1 and p 2 (some books use 5) 3. Both populations must be at least 10 times their corresponding sample sizes 4. Independent samples

Test statistic: Standard error = (Notice that you are using the pooled in the standard error formula)

On the formula sheet: use the “unequal variances” formula for standard error with confidence intervals and the “equal variances” formula for standard deviation with significance tests.

Hypothesis testing: Difference of Proportions In the past decade intensive antismoking campaigns have been sponsored by both federal and private agencies. Suppose the American Cancer Society randomly sampled 1500 U.S. adults in 1985 and then sampled 2000 U.S. adults in 1995 to determine whether there was evidence that the percentage of smokers had decreased. The results of the two sample surveys are shown in the table below, where x 1 and x 2 represent the number of smokers in the 1985 and 1995 samples, respectively.

n 1 = 1500n 2 = 2000 x 1 = 576x 2 = Construct a 95% confidence interval for the decrease in the proportion of smokers in the U.S. from 1985 to 1995.

2.Do these data indicate that the proportion of U.S. smokers decreased over this 10-year period? Use  =.05.