Question

A random sample of 100 observations is selected from a binomial population with an unknown probability of success p. The computed value of ˆp is equal to to.70.(You need to write down all the details for problem 9 to get full credits. Hint: Please follow the standard four steps in the handout.)

(a)(10 points)TestH0:p=.63 againstHa:p > .63.Useα=.01

(b)(10 points)TestH0:p=.76 againstHa:p6=.76.Useα=.05

(c)(3 points) Form a 99% confidence interval for p.

Answer 1

a and c)

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p = 0.63 Ha: p > 0.63 .. This corresponds to a right-tailed test, for which a z-test for one population proportion needs to be used. (2) Rejection Region Based on the information provided, the significance level is a = 0.01, and the critical value for a right-tailed test is ze = 2.33. The rejection region for this right-tailed test is R = {z: z > 2.33} (3) Test Statistics The Z-statistic is computed as follows: P - po V po(1 – po)/n 0.7 – 0.63 70.63(1 – 0.63)/100 = 1.45 (4) Decision about the null hypothesis Since it is observed that z = 1.45 < %c = 2.33, it is then concluded that the null hypothesis is not rejected. 0.0735 > Using the P-value approach: The p-value is p = 0.0735, and since p 0.01, 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 p is greater than po, at the a = 0.01 significance level. Confidence Interval The 99% confidence interval for p is: 0.582 < p < 0.818.

b)

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho:P 0.76 Ha: p = 0.76 This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used. (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 zc = 1.96. The rejection region for this two-tailed test is R {z: 121 > 1.96} (3) Test Statistics The Z-statistic is computed as follows: z P – PO V Po(1 – po)/n 0.7 – 0.76 0.76(1 – 0.76)/100 = -1.405 (4) Decision about the null hype esis 1.96, it is then concluded that the null Since it is observed that 21 = 1.405 < ze = hypothesis is not rejected. Using the P-value approach: The p-value is p = 0.1601, and since p= 0.1601 > 0.05, it is concluded that the null hypothesis is not rejected.

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