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). (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 < -1.96 or z > 1.96 b. z > c. z < (c) find the standardized test statistic z. z = (round to two decimal places as needed.)

Explanation:

Step1: Identify the formula for z - test statistic for two - sample means

The formula for the z - test statistic when comparing two population means (with known population standard deviations) is:

$$z=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}}$$

Here, the null hypothesis \(H_0:\mu_1=\mu_2\), so \(\mu_1-\mu_2 = 0\). Let \(\bar{x}_1 = 349237\) (mean of Casper), \(n_1 = 35\), \(\sigma_1=152867\); \(\bar{x}_2 = 435244\) (mean of Cheyenne), \(n_2 = 41\), \(\sigma_2 = 137829\).

Step2: Substitute the values into the formula

First, calculate the numerator: \(\bar{x}_1-\bar{x}_2-0=349237 - 435244=- 86007\)
Then, calculate the denominator:
\(\sqrt{\frac{\sigma_1^{2}}{n_1}+\frac{\sigma_2^{2}}{n_2}}=\sqrt{\frac{(152867)^{2}}{35}+\frac{(137829)^{2}}{41}}\)
Calculate \(\frac{(152867)^{2}}{35}=\frac{152867\times152867}{35}\approx\frac{2.336\times10^{10}}{35}\approx6.674\times10^{8}\)
Calculate \(\frac{(137829)^{2}}{41}=\frac{137829\times137829}{41}\approx\frac{1.900\times10^{10}}{41}\approx4.634\times10^{8}\)
Sum of the two terms inside the square root: \(6.674\times10^{8}+4.634\times10^{8}=1.1308\times10^{9}\)
Take the square root: \(\sqrt{1.1308\times10^{9}}\approx33627.37\)

Step3: Calculate the z - statistic

Now, \(z=\frac{- 86007}{33627.37}\approx - 2.56\)

Answer:

\(z\approx - 2.56\)