Billy's mother asks him, "Shouldn't you be studying for your spelling test instead of watching television?" Billy replies, "I already studied." "You know all of the words on this list?" his mother asked skeptically. "Well, maybe not all of them," he replies, "but i bet I know over 90% of them." "I am going to pick four words from this list at random. If you spell all four of them correctly, you can watch television. If not, you have to study for an hour." "Okay, it's a deal!"

Think of this situation as a hypothesis test about the proportion of words on the list that Billy can spell. Call this proportion p.

a. Billy claims that he can spell over 90% of the words. Is this the null hypothesis or the alternative? Formulate this hypothesis.

b. Formulate the competing hypothesis.

c. Suppose that Billy's claim is false, and he can really only spell 75% of the words. He could still spell all four words correctly, leading his mother to make an incorrect decision. What type of error would this be? What is the probability that he spells all four words correctly?

d. Suppose that Billy can spell exactly 92% of the words (more than 90%, just as he claimed). He could still miss at least one of the four words and have to study for an hour anyway. What type of error would this be? What is the probability that he misses at least one word?

e. Give a reason why the error probabilities are so large in this situation.