Question

1. A researcher selected a random sample of 268 former students, 110 of whom were student-athletes, from a list of graduates of a large university. A total of 17% of the sample of student-athletes had earned graduate degrees, compared to 14% of the non-student-athletes. We want to know: Is this difference statistically significant at α = .01 for a two-tailed test?

a) what is the t statistic

b) what is our critical value of our test statistic?
c) What is your decision about your null hypothesis?

Answer 1

Solution:

The following information is provided: The sample size is N = 268, the number of favorable cases is X = 110, and the sample proportion is p = = 110 = 0.4104, and the significance level is a = 0.01 (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: p=0.14 Ha:p +0.14 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 = 0.01, and the critical value for a two-tailed test is ze = 2.58. The rejection region for this two-tailed test is R = {::->2.58} (3) Test Statistics The Z-statistic is computed as follows: P-Po 0.4104 -0.14 po (1 - po)/n 0.14(1 -0.14)/268 = 12.76 (4) Decision about the null hypothesis = 12.76 > = 2.58, it is then concluded that the Since it is observed that null hypothesis is rejected. Using the P-value approach: The p-value is p=0, and since p=0 <0.01, 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 the population proportion p is different than po. at the a= 0.01 significance level.