Question

a real estate agency says that the mean home sales price in casper, wyoming is the same as in cheyenne, wyoming. the mean home sales price for 35 homes in casper, wyoming is $349,237. assume the population standard deviation is $152,867. the mean home sales price for 41 homes in cheyenne, wyoming is $435,244. assume the population standard deviation is $137,829. at α=0.05, is there enough evidence to reject the agency’s claim? complete parts (a) through (e).

h₀: μ₁ = μ₂
hₐ: μ₁ ≠ μ₂

(b) find the critical value(s) and identify the rejection region. select the correct choice below and fill in the answer box within your choice.
(round to two decimal places as needed.)
a. the critical values are z₀ = ± 1.96.
b. the critical value is z₀ =

what is the rejection region? select the correct choice below and fill in the answer box(es) within your choice.
(round to two decimal places as needed.)
a. z < or z >
b. z >
c. z <

Explanation:

Step1: Identify Test Type

This is a two - tailed z - test (since population standard deviations are known, and we are testing a claim about two means). For a two - tailed test with \(\alpha = 0.05\), we split the significance level \(\alpha\) into two tails, so \(\alpha/2=0.025\) in each tail.

Step2: Find Critical Values

We look up the z - value in the standard normal distribution table (or use the property of the standard normal distribution) such that the area to the right of \(z_{\alpha/2}\) is \(0.025\) and the area to the left of \(-z_{\alpha/2}\) is \(0.025\). The z - value corresponding to an area of \(0.975\) (since \(1 - 0.025=0.975\)) to the left of it in the standard normal distribution is \(z = 1.96\). So the critical values are \(z_0=\pm1.96\).

Step3: Determine Rejection Region

For a two - tailed test, we reject the null hypothesis \(H_0\) when the test statistic \(z\) is less than the lower critical value or greater than the upper critical value. Since the critical values are \(\pm1.96\), the rejection region is \(z < - 1.96\) or \(z>1.96\).

Answer:

For the critical value part, the critical values are \(z_0=\pm1.96\) (Option A). For the rejection region, the correct choice is Option A with \(z < - 1.96\) or \(z>1.96\).