Question

a personnel director in a particular state claims that the mean annual income is greater in one of the state counties (county a) than it is in another county (county b). in county a, a random sample of 17 residents h annual income of $42,400 and a standard deviation of $8900. in county b, a random sample of 8 resident annual income of $38,000 and a standard deviation of $5100. at α = 0.05, answer parts (a) through (e). as population variances are not equal. if convenient, use technology to solve the problem. 1.895 (type an integer or decimal rounded to three decimal places as needed. use a comma to separate answer select the correct rejection region(s) below. a. t > t₀ b. t < -t₀ c. -t₀ < t < t₀ d. t < -t₀, t > t₀ (c) find the standardized test statistic. t = 1.564 (type an integer or decimal rounded to three decimal places as needed.) (d) decide whether to reject or fail to reject the null hypothesis. the null hypothesis.

Explanation:

Step1: Recall Hypothesis Test Setup

The claim is that mean income in County A (\(\mu_1\)) is greater than in County B (\(\mu_2\)), so the alternative hypothesis is \(H_a: \mu_1 > \mu_2\), and null is \(H_0: \mu_1 \leq \mu_2\). This is a right - tailed test.

Step2: Compare Test Statistic and Critical Value

The critical value \(t_0 = 1.895\) (from the given data, likely calculated for the right - tailed test with appropriate degrees of freedom). The standardized test statistic \(t = 1.564\). For a right - tailed test, we reject \(H_0\) if \(t>t_0\). Here, \(1.564<1.895\), so we fail to reject the null hypothesis.

Answer:

Fail to reject