Question

an energy company wants to choose between two regions in a state to install energy - producing wind turbines. a researcher claims that the wind speed in region a is less than the wind speed in region b. to test the regions, the average wind speed is calculated for 60 days in each region. the mean wind speed in region a is 13.8 miles per hour. assume the population standard deviation is 2.9 miles per hour. the mean wind speed in region b is 15.1 miles per hour. assume the population standard deviation is 3.1 miles per hour. at \\(\alpha = 0.05\\), can the company support the researcher’s claim? complete parts (a) through (d) below.

what is the claim?
a. the wind speed in region a is not less than the wind speed in region b.
b. the wind speed in region a is less than the wind speed in region b.
c. the wind speed in region a is the same as the wind speed in region b.
d. the wind speed in region a is not greater than the wind speed in region b.

let region a be sample 1 and let region b be sample 2. identify \\(h_0\\) and \\(h_a\\).
\\(h_0: \mu_1 \\) \\(\square\\) \\(\mu_2\\)
\\(h_a: \mu_1 \\) \\(\square\\) \\(\mu_2\\)

(b) find the critical value(s) and identify the rejection region.
the critical value(s) is/are \\(z_0 = \square\\).
(round to two decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Identify Test Type

This is a left - tailed z - test for two population means (since population standard deviations are known). The significance level $\alpha = 0.05$.

Step2: Find Critical Value

For a left - tailed test with $\alpha=0.05$, we look for the z - score such that $P(Z < z_{\alpha})=\alpha$. From the standard normal distribution table, the z - score corresponding to a left - tailed area of 0.05 is $z_{0.05}=- 1.645$ (we can also use the formula or a calculator to find this value. The standard normal distribution has a mean of 0 and standard deviation of 1. The critical value for a left - tailed test at $\alpha = 0.05$ is the value where the cumulative probability is 0.05. Using the inverse of the standard normal CDF, we get $z=- 1.645$ (rounded to two decimal places, it can also be written as - 1.65, but more accurately - 1.645)).

Answer:

The critical value is $\boldsymbol{-1.645}$ (or - 1.65 when rounded to two decimal places as per some conventions).