Question

Consider the following hypothesis statement using a = 0.05 and the following data from two independent samples: HO: p1 - p2 = 0 Hl: pl.p20 x1 = 18 x2 = 24 nl -85 n2 = 105 a. Calculate the appropriate 2-test statistic and interpret the result. b. Calculate the p-value and interpret the result.

Answer 1

a.

For sample 1, we have that the sample size is Ni = 85, the number of favorable cases is X1 = 18, so then the sample proportion is p1 = x = 3% = 0.2118 For sample 2, we have that the sample size is N2 = 105, the number of favorable cases is X2 = 24, so then the sample proportion is p2 = * = P = 0.2286 The value of the pooled proportion is computed as p = 0.2211 18-24 85-105 Also, the given significance level is a = 0.05. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: P1 = P2 Ha: p = P2 This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted

(2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a two- tailed test is ze = 1.96. The rejection region for this two-tailed test is R= {z: 7 > 1.96} (3) Test Statistics The Z-statistic is computed as follows: P1 - 2 (1-7)(1/n1 +1/n2) 0.2118 – 0.2286 0.2211 (1 – 0.2211)(1/85 +1/105) -0.278 (4) Decision about the null hypothesis Since it is observed that |z| = 0.278 <ze = 1.96, it is then concluded that the null hypothesis is not rejected.

b.

Using the P-value approach: The p-value is p=0.7813, and since p=0.7813 > 0.05, it is concluded that the null hypothesis is not rejected. (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pi is different than p2, at the 0.05 significance level.

So is the answer.