Related questions with answers

The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the following school year. She wants to know if each type of food will be equally popular so she can start ordering supplies and making other plans. To find out, she selects a random sample of 100 students and asks them, "Which type of food do you prefer: Asian food, Mexican food, pizza, or hamburgers?" Here are her data:

\scriptstyle\begin{array}{lcccc} \hline \text { Type of Food: } & \text { Asian } & \text { Mexican } & \text { Pizza } & \text { Hamburgers } \\ \text { Count: } & 18 & 22 & 39 & 21 \\ \hline \end{array}

An appropriate null hypothesis to test whether the food choices are equally popular is (a) $H_{0}: \mu=25,$ where $\mu$ = the mean number of students that prefer each type of food. (b) $H_{0}: p=0.25,$ where p = the proportion of all students who prefer Asian food. (c) $H_{0}: n_{A}=n_{M}=n_{P}=n_{H}=25,$ where $n_{A}$ is the number of students in the school who would choose Asian food, and so on. (d) $H_{0}: p_{A}=p_{M}=p_{P}=p_{H}=0.25,$ where $p_{A}$ is the proportion of students in the school who would choose Asian food, and so on. (e) $H_{0}: \hat{p}_{A}=\hat{p}_{M}=\hat{p}_{P}=\hat{p}_{H}=0.25,$ where $\hat{p}_{A}$ is the proportion of students in the sample who chose Asian food, and so on.

Researchers studied the behavior of birds that were searching for seeds and insects in an Oregon forest. In this forest, 54% of the trees were Douglas firs, 40% were ponderosa pines, and 6% were other types of trees. At a randomly selected time during the day, the researchers observed 156 red-breasted nuthatches: 70 were seen in Douglas firs, 79 in ponderosa pines, and 7 in other types of trees. Do these data provide convincing evidence that nuthatches prefer particular types of trees when they're searching for seeds and insects?

A company claims that each batch of its deluxe mixed nuts contains 52% cashews, 27% almonds, 13% macadamia nuts, and 8% brazil nuts. To test this claim, a quality-control inspector takes a random sample of 150 nuts from the latest batch. State appropriate hypotheses for performing a test of the company's claim.

Question

Select the best answer. The manager of a high school cafeteria is planning to offer several new types of food for student lunches in the following school year. She wants to know if each type of food will be equally popular so she can start ordering supplies and making other plans. To find out, she selects a random sample of 100 students and asks them, "Which type of food do you prefer: Asian food, Mexican food, pizza, or hamburgers?" An appropriate null hypothesis to test whether the food choices are equally popular is (a)
$H_0: \mu = 25,$
where
$\mu =$
the mean number of students that prefer each type of food. (b)
$H_0: p = 0.25,$
where p = the proportion of all students who prefer Asian food. (c)
$H_0: n_A = n_M = n_P = n_H = 25,$
where
$n_A$
is the number of students in the school who would choose Asian food, and so on. (d)
$H_0: p_A = p_M = p_P = p_H = 0.25,$
where
$p_A$
is the proportion of students in the school who would choose Asian food, and so on. (e)
$H_0: \hat{p}_A = \hat{p}_M = \hat{p}_P = \hat{p}_H = 0.25,$
where
$\hat{p}_A$
is the proportion of students in the sample who chose Asian food, and so on.

Solution

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We expect that each type of food is equally popular.

Since there are 4 types of food, we would then expect a proportion of 1 out of 4 for each type of food.

p=\dfrac{1}{4}=0.25

The null hypothesis states that the specified distribution of the categorical variable is correct. Moreover, the statement needs to be about the proportions in the population: