Question

a pet association claims that the mean annual costs of food for dogs and cats are the same. the the two types of pets are shown below. at α = 0.05, can you reject the pet association’s claim? a variances are equal. assume that the samples are random and independent, and the population distributed. complete parts (a) through (e).

dogs	cats
$s_1 = \\$31$	$s_2 = \\$28$
$n_1 = 15$	$n_2 = 19$

\bigcirc d. \the mean annual costs of food for dogs and cats are not equal.\

the null hypothesis, $h_0$, is $\mu_1 = \mu_2$ the alternative hypothesis, $h_a$, is $\mu_1 \
eq \mu_2$. the claim.

(b) find the critical value(s) and identify the rejection region(s). select the correct choice below answer box(es) within your choice.
(round to two decimal places as needed.)
\bigcirc a. the rejection region is $t > \square$.
\bigcirc b. the rejection region is $\square < t < \square$.
\bigcirc c. the rejection region is $t < \square$.
\bigcirc d. the rejection regions are $t < \square$ and $t > \square$.

Explanation:

Step1: Determine Degrees of Freedom

For two - sample t - test with equal variances, the degrees of freedom \(df=n_1 + n_2-2\). Given \(n_1 = 15\) and \(n_2=19\), so \(df=15 + 19-2=32\).

Step2: Determine Significance Level and Test Type

The significance level \(\alpha = 0.05\) and the alternative hypothesis is \(H_a:\mu_1
eq\mu_2\), which means it is a two - tailed test. So we will split \(\alpha\) into two tails, \(\alpha/2=0.025\) for each tail.

Step3: Find Critical Values

We look up the t - distribution table for \(df = 32\) and \(\alpha/2=0.025\). The critical values are \(t_{\alpha/2}=\pm 2.0369\) (using t - table or statistical software). For a two - tailed test, the rejection regions are \(t < - 2.04\) (rounded to two decimal places) and \(t>2.04\) (rounded to two decimal places). So the correct option is D, and the critical values are approximately \(- 2.04\) and \(2.04\).

Answer:

D. The rejection regions are \(t < - 2.04\) and \(t>2.04\).