Question

students at a large university have four places to get lunch: the cafeteria, the hut, the taco wagon, or the pizza place. an article in the school newsletter states that 70% of students prefer to get lunch in the cafeteria and the other three options are preferred equally. to investigate this claim, a random sample of 150 students is selected. what are the appropriate hypotheses?

$h_0$: the distribution of lunch preferences is 70% cafeteria, 10% hut, 10% taco wagon, and 10% pizza place.
$h_a$: the distribution of lunch preferences is not 70% cafeteria, 10% hut, 10% taco wagon, and 10% pizza place.

$h_0$: the distribution of lunch preferences is not 70% cafeteria, 10% hut, 10% taco wagon, and 10% pizza place.
$h_a$: the distribution of lunch preferences is 70% cafeteria, 10% hut, 10% taco wagon, and 10% pizza place.

$h_0$: in the sample of 150 students, 105 will prefer the cafeteria, 15 will prefer the hut, 15 will prefer the taco wagon, and 15 will prefer pizza place.
$h_a$: in the sample of 150 students, the distribution will not be that 105 will prefer the cafeteria, 15 will prefer the hut, 15 will prefer the taco wagon, and 15 will prefer the pizza place.

$h_0$: in the sample of 150 students, the distribution will not be that 105 will prefer the cafeteria, 15 will prefer the hut, 15 will prefer the taco wagon, and 15 will prefer the pizza place.
$h_a$: in the sample of 150 students, 105 will prefer the cafeteria, 15 will prefer the hut, 15 will prefer the taco wagon, and 15 will prefer the pizza place.

Explanation:

• The null hypothesis \(H_0\) is a statement of no - difference or the claim that is initially assumed to be true. Here, the claim from the school newsletter (70% cafeteria, 10% hut, 10% taco wagon, 10% pizza place) is the null hypothesis.
• The alternative hypothesis \(H_a\) is the statement that we are trying to find evidence for. It is the opposite of the null hypothesis, so it is that the distribution of lunch preferences is not 70% cafeteria, 10% hut, 10% taco wagon, and 10% pizza place.
• Hypotheses are about the population distribution, not about a specific sample. Options C and D are about a sample, so they are incorrect. Option B has the null and alternative hypotheses reversed.

Answer:

\(H_0\): The distribution of lunch preferences is 70% cafeteria, 10% hut, 10% taco wagon, and 10% pizza place. \(H_a\): The distribution of lunch preferences is not 70% cafeteria, 10% hut, 10% taco wagon, and 10% pizza place. (First option)