Question

you want to buy a washing machine, and a salesperson tells you that the mean repair costs for model a and model b are equal. you research the repair costs. the mean repair cost of 23 model a washing machines is $212. assume the population standard deviation is $17. the mean repair cost of 20 model b washing machines is $219. assume the population standard deviation is $24. at α=0.01, can you reject the salespersons claim? assume the samples are random and independent, and the populations are normally distributed. complete parts (a) through (e).

(c) find the standardized test statistic z for μ₁ - μ₂.

z = -1.09 (round to two decimal places as needed.)

(d) decide whether to reject or fail to reject the null hypothesis. choose the correct answer below.

a. fail to reject h₀. the standardized test statistic is not in a rejection region.

b. reject h₀. the standardized test statistic is not in a rejection region.

c. reject h₀. the standardized test statistic is in a rejection region.

d. fail to reject h₀. the standardized test statistic is in a rejection region.

(e) interpret the decision in the context of the original claim.

at the % significance level, there is evidence to the claim that the mean repair costs for model a are the mean repair costs for model b.

Explanation:

Part (d)

1. Identify the test type and significance level: This is a two - tailed z - test (since we are testing if the means are equal, \(H_0:\mu_1=\mu_2\) or \(H_0:\mu_1 - \mu_2 = 0\)) with \(\alpha=0.01\). For a two - tailed test, the critical values are \(z=\pm z_{\alpha/2}\). We know that \(z_{\alpha/2}=z_{0.005}\approx\pm 2.576\) (from standard normal distribution tables).
2. Compare test statistic with critical values: The standardized test statistic \(z=- 1.09\). We check if \(|z|>z_{\alpha/2}\). Since \(|-1.09| = 1.09<2.576\), the test statistic is not in the rejection region (the rejection regions for a two - tailed test with \(\alpha = 0.01\) are \(z < - 2.576\) or \(z>2.576\)). So we fail to reject the null hypothesis \(H_0\).

Part (e)

1. Significance level: The significance level \(\alpha = 0.01\), so it is a \(1\%\) significance level.
2. Evidence and claim interpretation: Since we failed to reject the null hypothesis \(H_0:\mu_1=\mu_2\) (the salesperson's claim that the mean repair costs are equal), at the \(1\%\) significance level, there is not enough evidence to reject the claim that the mean repair costs for Model A are equal to the mean repair costs for Model B.

Answer:

Part (d)

A. Fail to reject \(H_0\). The standardized test statistic is not in a rejection region.

Part (e)

At the \(1\%\) significance level, there is not enough evidence to reject the claim that the mean repair costs for Model A are equal to the mean repair costs for Model B.

(For the blanks: First blank: \(1\); Second blank: not enough; Third blank: reject; Fourth blank: equal to)