Final answer:
For the hypothesis test about pizza brand preference, where 42 percent is the expected preference rate and 39 percent is observed in a sample of 100, the normal approximation to the binomial distribution is used because the sample size criteria for normal approximation are satisfied.
Step-by-step explanation:
To determine what distribution to use for performing a hypothesis test on a proportion, like in the scenario where 42 percent of respondents are expected to prefer Brand A, and 39 percent from a sample of 100 people actually preferred it, you would typically use the binomial distribution if you were only dealing with that one sample. However, since the question implies we'd want to approximate for a normal distribution due to a large sample size or multiple samples, the correct distribution to use is a normal approximation to the binomial distribution. This is because as the sample size grows, the binomial distribution tends to approximate a normal distribution, which is easier to work with when conducting hypothesis tests.
It is important to check the criteria for normal approximation: the sample size (n) should be large enough, and np and n(1-p) both should be greater than or equal to 5, where n is the sample size and p is the population proportion. In this case, for a sample of 100 people, and the expected proportion being 0.42, both conditions (100*0.42 and 100*(1-0.42)) are satisfied, justifying the use of normal approximation.