For a binomial random variable ny with parameter ????, consider the null model.

a. Explain how to invert the Wald, likelihood-ratio, and score tests of H0:

???? = ????0 against H1: ???? ≠ ????0 to obtain 95% confidence intervals for ????.

b. In teaching an introductory statistics class, one year I collected data from the students to use for lecture examples. One question in the survey asked whether the student was a vegetarian. Of 25 students, 0 said “yes.” Treating this as a random sample from some population, find the 95% confidence interval for ???? using each method in (a).

c. Do you trust the Wald interval in (b)? (Your answer may depend on whether you regard the standard error estimate for the interval to be credible.)

Explain why the Wald method may behave poorly when a parameter takes value near the parameter space boundary.