Comparing Two Proportions

AP Statistics Chap 13-2 Two Population Proportions The point estimate for the difference is p 1 – p 2 Population proportions Assumptions: n 1 p 1  10, n 1 (1-p 1 )  10 n 2 p 2  10, n 2 (1-p 2 )  10

AP Statistics Chap 13-3 Confidence Interval for Two Population Proportions Population proportions The confidence interval for p 1 – p 2 is:

AP Statistics Chap 13-4 Example: Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A? In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes Construct a 95% Confidence Interval

AP Statistics Chap 13-5 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : p 1  p 2 H A : p 1 < p 2 i.e., H 0 : p 1 – p 2  0 H A : p 1 – p 2 < 0 Upper tail test: H 0 : p 1 ≤ p 2 H A : p 1 > p 2 i.e., H 0 : p 1 – p 2 ≤ 0 H A : p 1 – p 2 > 0 Two-tailed test: H 0 : p 1 = p 2 H A : p 1 ≠ p 2 i.e., H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 ≠ 0

AP Statistics Chap 13-6 Two Proportions Hyp. Test Population proportions The test statistic for p 1 – p 2 is:

AP Statistics Chap 13-7 Hypothesis Tests for Two Population Proportions Population proportions Lower tail test: H 0 : p 1 – p 2  0 H A : p 1 – p 2 < 0 Upper tail test: H 0 : p 1 – p 2 ≤ 0 H A : p 1 – p 2 > 0 Two-tailed test: H 0 : p 1 – p 2 = 0 H A : p 1 – p 2 ≠ 0  /2  -z  -z  /2 zz z  /2 Reject H 0 if z < -z  Reject H 0 if z > z  Reject H 0 if z < -z   or z > z 

AP Statistics Chap 13-8 Example: Two population Proportions Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A? In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes Test at the.05 level of significance

AP Statistics Chap 13-9 The hypothesis test is: H 0 : p 1 – p 2 = 0 (the two proportions are equal) H A : p 1 – p 2 ≠ 0 (there is a significant difference between proportions) The sample proportions are: Men: p 1 = 36/72 =.50 Women: p 2 = 31/50 =.62 Example P-value =.59