Question

After a college football team once again lost a game to their archrival, the alumni association conducted a survey to see if alumni were in favor of firing the coach. A simple random sample of 100 alumni from the population of all living alumni was taken. Sixty-four of the alumni in the sample were in favor of firing the coach. Let p represent the proportion of all living alumni who favored firing the coach. Suppose the alumni association wished to see if the majority of alumni are in favor of firing the coach. To do this they test the hypotheses H0: p = 0.50 versus Ha: p > 0.50. The alumni association wished to conduct the test at a 5% significance level. What would their decision be? Based on that decision, what type of mistake could they have made?

Reject H₀; Type I error

Do not reject H₀; Type I error

Reject H₀; Type II error

Do not reject H₀; Type II error

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis: p = 0.5
Alternative Hypothesis: p > 0.5

Rejection Region
This is right tailed test, for α = 0.05
Critical value of z is 1.64.
Hence reject H0 if z > 1.64

Test statistic,
z = (pcap - p)/sqrt(p*(1-p)/n)
z = (0.64 - 0.5)/sqrt(0.5*(1-0.5)/100)
z = 2.8

P-value Approach
P-value = 0.0026
As P-value < 0.05, reject the null hypothesis.

Decision made is to fire the coach.

Reject H0; Type I error