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 salesperson’s claim? assume the samples are random and independent, and the populations are normally distributed. complete parts (a) through (e). (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.) a. z < -2.58, z > 2.58 b. z < c. z > (c) find the standardized test statistic z for μ₁ - μ₂. z = (round to two decimal places as needed.)

Explanation:

Step1: Recall the formula for z - test statistic for two means

The formula for the standardized test statistic \( z \) for the difference between two population means (\( \mu_1 - \mu_2 \)) when the population standard deviations (\( \sigma_1 \) and \( \sigma_2 \)) are known 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 salesperson claims that \( \mu_1=\mu_2 \), so \( \mu_1 - \mu_2 = 0 \). We have \( \bar{x}_1 = 212 \), \( \bar{x}_2=219 \), \( \sigma_1 = 17 \), \( \sigma_2 = 24 \), \( n_1 = 23 \), \( n_2=20 \).

Step2: Substitute the values into the formula

First, calculate the numerator: \( (\bar{x}_1 - \bar{x}_2)-(\mu_1 - \mu_2)=(212 - 219)-0=- 7 \)
Then, calculate the denominator:

$$ \sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}=\sqrt{\frac{17^2}{23}+\frac{24^2}{20}}=\sqrt{\frac{289}{23}+\frac{576}{20}} $$

Calculate \( \frac{289}{23}\approx12.565 \) and \( \frac{576}{20} = 28.8 \)
Then \( \frac{289}{23}+\frac{576}{20}\approx12.565 + 28.8=41.365 \)
And \( \sqrt{41.365}\approx6.4316 \)

Step3: Calculate the z - value

Now, find \( z \) by dividing the numerator by the denominator:

$$ z=\frac{-7}{6.4316}\approx - 1.09 $$

Answer:

\( z\approx - 1.09 \)