Question

Test thc claim that the proportion of men who own cats is smaller than the proportion of women who own cats at the .10 significance level. The null and alternative hypothesis would be: Ho:PM = PF Ho: M2 MF HAM = Mp HP SPP HP 2 Pp H,MyMp H: PM * PF HIMx Hr H: 4x4 HPx > Pr HPx < Pr H. Hy > HF The test is left-tailed right-tailed two-tailed Based on a sample of 80 men, 20% owned cats Based on a sample of 60 women, 50% owned cats Critical Value (three decimal accuracy Test Statistic three decimal accuracy Based on this we: Fail to reject the null hypothesis Reject the null hypothesis

Answer 1

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Solution: For sample 1, we have that the sample size is N1 = 80, the number of favorable cases is X1 = 16, so then the sample proportion is i X; = 16 = 0.2 For sample 2, we have that the sample size is N2 = 60, the number of favorable cases is X2 = 30, so then the sample proportion is ô2 = x3 = 30 = 0.5 The value of the pooled proportion is computed as p = x + x = 0.3286 Also, the given significance level is a = 0.10. 16+30 80+60 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: HO: P = P2 Ha: P1 P2 This corresponds to a left-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 = 0.10, and the critical value for a left-tailed test is ze = -1.28. The rejection region for this left-tailed test is R = {z: z < -1.28} (3) Test Statistics The Z-statistic is computed as follows: P1 - P2 Võ(1-7)(1/nı +1/n2) 0.2 - 0.5 70.3286 . (1 -0.3286)(1/80 + 1/60) = -3.74 (4) Decision about the null hypothesis Since it is observed that z = -3.74 < Ze = -1.28, it is then concluded that the null hypothesis is rejected. Using the P-value approach: The p-value is p = 0.0001, and since p = 0.0001 0.10, it is concluded that the null hypothesis is rejected. (5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion pı is less than p2, at the 0.10 significance level