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 \\(\alpha = 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).

\\(\bigcirc\\) a. \\(h_0: \mu_1 < \mu_2\\)\\(h_a: \mu_1 \geq \mu_2\\) \\(\bigcirc\\) b. \\(h_0: \mu_1 \geq \mu_2\\)\\(h_a: \mu_1 < \mu_2\\) \\(\bigcirc\\) c. \\(h_0: \mu_1 = \mu_2\\)\\(h_a: \mu_1 \
eq \mu_2\\)\\(\bigcirc\\) d. \\(h_0: \mu_1 \
eq \mu_2\\)\\(h_a: \mu_1 = \mu_2\\) \\(\bigcirc\\) e. \\(h_0: \mu_1 > \mu_2\\)\\(h_a: \mu_1 \leq \mu_2\\) \\(\bigcirc\\) f. \\(h_0: \mu_1 \leq \mu_2\\)\\(h_a: \mu_1 > \mu_2\\)

(b) find the critical value(s) and identify the rejection region(s).

the critical value(s) is/are \\(-2.58, 2.58\\).
(round to two decimal places as needed. use a comma to separate answers as needed.)
what is/are 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.)
\\(\bigcirc\\) a. \\(z < \square, z > \square\\)\\(\bigcirc\\) b. \\(z < \square\\)\\(\bigcirc\\) c. \\(z > \square\\)

Explanation:

Step1: Identify Test Type

This is a two - tailed z - test for the difference between two means. The alternative hypothesis \(H_{a}:\mu_{1}
eq\mu_{2}\) indicates a two - tailed test.

Step2: Determine Rejection Region

For a two - tailed test with \(\alpha = 0.01\), we split the significance level \(\alpha\) equally between the two tails. So \(\alpha/2=0.005\) in each tail.
The critical values for a two - tailed test with \(\alpha = 0.01\) are \(z=- 2.58\) and \(z = 2.58\) (from standard normal distribution tables, where \(P(Z < - 2.58)=0.005\) and \(P(Z>2.58) = 0.005\)).
In a two - tailed test, the rejection regions are \(z < - 2.58\) and \(z>2.58\). So we choose option A, and fill in the boxes with \(-2.58\) and \(2.58\) respectively.

Answer:

A. \(z < - 2.58\), \(z>2.58\)