Questions: A pet association claims that the mean annual costs of food for dogs and cats are the same. The results for samples for the two types of pets are shown below. At α=0.10, can you reject the pet association's claim? Assume the population variances are equal. Assume that the samples are random and independent, and the populations are normally distributed. Complete parts (a) through (e). Dogs Cats ------ x̄₁=257 x̄₂=231 s₁=29 s₂=27 n₁=17 n₂=18 The null hypothesis, H₀, is μ₁=μ₂ The alternative hypothesis, Hₐ, is μ₁≠μ₂. The null hypothesis is the claim. (b) Find the critical value(s) and identify the rejection region(s). Select the correct choice below and fill in the answer box(es) within your choice. (Round to two decimal places as needed.) A. The rejection region is t< . B. The rejection region is t> . C. The rejection region is < t < . D. The rejection regions are t<-1.697 and t>-1.697.

A pet association claims that the mean annual costs of food for dogs and cats are the same. The results for samples for the two types of pets are shown below. At α=0.10, can you reject the pet association's claim? Assume the population variances are equal. Assume that the samples are random and independent, and the populations are normally distributed. Complete parts (a) through (e). Dogs Cats ------ x̄₁=257 x̄₂=231 s₁=29 s₂=27 n₁=17 n₂=18 The null hypothesis, H₀, is μ₁=μ₂ The alternative hypothesis, Hₐ, is μ₁≠μ₂. The null hypothesis is the claim. (b) Find the critical value(s) and identify the rejection region(s). Select the correct choice below and fill in the answer box(es) within your choice. (Round to two decimal places as needed.) A. The rejection region is t< . B. The rejection region is t> . C. The rejection region is < t < . D. The rejection regions are t<-1.697 and t>-1.697.
Transcript text: A pet association claims that the mean annual costs of food for dogs and cats are the same. The results for samples for the two types of pets are shown below. At $\alpha=0.10$, can you reject the pet association's claim? Assume the population variances are equal. Assume that the samples are random and independent, and the populations are normally distributed. Complete parts (a) through (e). \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|}{ Dogs } & \multicolumn{1}{c|}{ Cats } \\ \hline $\bar{x}_{1}=\$ 257$ & $\bar{x}_{2}=\$ 231$ \\ $\mathrm{~s}_{1}=\$ 29$ & $\mathrm{~s}_{2}=\$ 27$ \\ $\mathrm{n}_{1}=17$ & $\mathrm{n}_{2}=18$ \\ \hline \end{tabular} The null hypothesis, $H_{0}$, is $\mu_{1}=\mu_{2}$ The alternative hypothesis, $H_{a}$, is $\mu_{1} \neq \mu_{2}$. The null hypothesis is the claim. (b) Find the critical value(s) and identify the rejection region(s). Select the correct choice below and fill in the answer box(es) within your choice. (Round to two decimal places as needed.) A. The rejection region is $t<$ $\square$ . B. The rejection region is $t>$ $\square$ . C. The rejection region is $\square$ $<$ t $<$ $\square$ . D. The rejection regions are $\mathrm{t}<-1.697$ and $\mathrm{t}>-1.697$.
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Solution

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Solution Steps

Step 1: Hypotheses

We are testing the following hypotheses:

  • Null hypothesis: \( H_0: \mu_1 = \mu_2 \) (the mean annual costs of food for dogs and cats are the same)
  • Alternative hypothesis: \( H_a: \mu_1 \neq \mu_2 \) (the mean annual costs of food for dogs and cats are different)
Step 2: Test Statistic Calculation

The test statistic is calculated using the formula: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} \] where \( SE \) is the standard error. In this case, the standard error was calculated as: \[ SE = \sqrt{s_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = 0.0 \] This results in an undefined test statistic: \[ t = \frac{257 - 231}{0} = \text{inf} \]

Step 3: Degrees of Freedom

The degrees of freedom for this test is calculated as: \[ df = n_1 + n_2 - 2 = 17 + 18 - 2 = 33 \]

Step 4: P-value Calculation

The p-value is calculated using the formula: \[ P = 2(1 - T(|t|)) = 2(1 - T(\text{inf})) = 0.0 \]

Step 5: Critical Value and Rejection Region

At a significance level of \( \alpha = 0.10 \), the critical value for a two-tailed test with \( df = 33 \) is approximately: \[ \text{Critical value} = 1.6924 \] The rejection regions are: \[ t < -1.69 \quad \text{or} \quad t > 1.69 \]

Step 6: Conclusion

Since the calculated \( t \)-statistic is \( \text{inf} \) and the p-value is \( 0.0 \), we reject the null hypothesis \( H_0 \). This indicates that there is sufficient evidence to conclude that the mean annual costs of food for dogs and cats are different.

Final Answer

The rejection regions are \( t < -1.69 \) and \( t > 1.69 \). Thus, we reject the null hypothesis.

\(\boxed{\text{Reject } H_0}\)

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