To collect data in an introductory statistics course, recently 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 let us use these data to illustrate inference for a proportion. (You may wish to refer to Section 1.4.1 on methods of inference.) 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 try to conduct the “Wald test,” for which z = (p − π0)/
√
[p(1 − p)/n] uses the estimated standard error?

b. Find the 95% “Wald confidence interval” (1.3) for π. Is it believable?
(When the observation falls at the boundary of the sample space, often Wald methods do not provide sensible answers.)

c. Conduct the “score test,” for which z = (p − π0)/
√
[π0(1 − π0)/n] uses the null standard error. Report the P-value.

d. Verify that the 95% score confidence interval (i.e., the set of π0 for which |z| < 1.96 in the score test) equals (0.0, 0.133). (Hint: What do the z test statistic and P-value equal when you test H0: π = 0.133 against Ha: π =

0.133.)