To collect data in an introductory statistics course, I gave the students a questionnaire.

One question asked whether the student was a vegetarian. Of 25 students, 0 answered yes. They were not a random sample, but use these data to illustrate inference for a proportion. Let π denote the population proportion who would say yes. Consider H0: π = 0.50 and Ha: π = 0.50.

a. What happens when you conduct the Wald test, which uses the estimated standard error in the z test statistic?

b. Find the 95% Wald confidence interval (1.3) for π. Is it believable?

c. Conduct the score test, which uses the null standard error in the z test statistic.

Report and interpret the P-value.

d. Verify that the 95% score confidence interval equals (0.0, 0.133). (This is similar to the interval (0.0, 0.137) obtained with a small-sample method of Section 1.4.3, inverting the binomial test with the mid P-value.)