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. b. the mean precipitation in seattle was less than in birmingham in a recent year. c. the mean precipitation in seattle was the same as in birmingham in a recent year. d. the mean precipitation in seattle was different than in birmingham in a recent year. let the data from seattle be sample 1 and let the data from birmingham be sample 2. identify h₀ and hₐ. 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 = 1.645. b. the critical values are z₀ = ±

Explanation:

Step1: Determine Test Type

The alternative hypothesis \( H_a: \mu_1 > \mu_2 \) is a right - tailed test? Wait, no, wait. Wait, the original claim is that precipitation in Seattle (\(\mu_1\)) is greater than in Birmingham (\(\mu_2\)). But the critical value option B is two - tailed? Wait, no, maybe I made a mistake. Wait, the null hypothesis is \( H_0:\mu_1\leq\mu_2 \) and alternative is \( H_a:\mu_1 > \mu_2 \), which is a right - tailed test. But the option B is for two - tailed. Wait, maybe there is a mis - selection. Wait, for a right - tailed test with \(\alpha = 0.05\), the critical value for a z - test (since population standard deviations are known) is \( z_{0.05}=1.645\) (from standard normal distribution, the z - score that leaves 0.05 area in the right tail). But if we mistakenly chose option B (two - tailed), for a two - tailed test with \(\alpha = 0.05\), the significance level in each tail is \(\alpha/2=0.025\). The critical values are \( z_{\alpha/2}=\pm1.96\) (since \( P(Z > 1.96)=0.025\) and \( P(Z < - 1.96)=0.025\)). But in our case, the test is right - tailed, so the correct critical value is 1.645. But since the option B is selected, maybe we need to check. Wait, the problem says "can you support the climatologist’s claim" which is a one - tailed (right - tailed) test. But if we consider the option B (two - tailed), the critical values for \(\alpha = 0.05\) two - tailed are \(\pm1.96\).

Step2: Recall Z - critical Values

For a two - tailed hypothesis test with \(\alpha=0.05\), we split the \(\alpha\) into two tails, each with area \(\alpha/2 = 0.025\). From the standard normal distribution table, the z - score that corresponds to an area of \(0.975\) (since \(1 - 0.025=0.975\)) to the left is \(z = 1.96\), and the z - score that corresponds to an area of \(0.025\) to the left is \(z=- 1.96\). So the critical values for a two - tailed test with \(\alpha = 0.05\) are \(z_0=\pm1.96\).

Answer:

The critical values are \( z_0=\pm\boldsymbol{1.96}\)