Question

a climatologist claims that the precipitation in seattle, washington, was greater than in birmingham, alabama, in a recent year. the daily precipitation amounts (in inches) for 30 days in a recent year in seattle and a recent year in birmingham are given in the accompanying table. assume the population standard deviation is 0.247 inch for seattle and 0.523 inch for birmingham. at α = 0.05, can you support the climatologist’s claim? complete parts (a) through (e).
click the icon to view the precipitation data.

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 value is z₀ = 1.64.
b. the critical values are 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 <
b. z >
c. z < or z >

Explanation:

Step1: Identify Test Type

This is a right - tailed z - test (since \(H_a:\mu_1>\mu_2\)) with \(\alpha = 0.05\).

Step2: Find Critical Value

For a right - tailed test with \(\alpha=0.05\), we look for the z - value such that the area to the right of \(z_0\) is \(0.05\). The area to the left of \(z_0\) is \(1 - 0.05=0.95\). Using the standard normal distribution table, the \(z\) - value corresponding to an area of \(0.95\) (or the critical value for a right - tailed test with \(\alpha = 0.05\)) is \(z_0 = 1.64\) (this matches option A for the critical value).

Step3: Determine Rejection Region

For a right - tailed test, we reject the null hypothesis when the test statistic \(z\) is greater than the critical value \(z_0\). So the rejection region is \(z>1.64\) (which corresponds to option B for the rejection region, with the value being \(1.64\)).

Answer:

Critical Value:

A. The critical value is \(z_0 = 1.64\)

Rejection Region:

B. \(z>1.64\)