Question

a personnel director in a particular state claims that the mean annual income is greater in one of the states counties (county a) than it is in another county (county b). in county a, a random sample of 13 residents has a mean annual income of $40,500 and a standard deviation of $8200. in county b, a random sample of 11 residents has a mean annual income of $38,800 and a standard deviation of $5400. at α = 0.025, answer parts (a) through (e). assume the population variances are not equal. if convenient, use technology to solve the problem. 2.228 (type an integer or decimal rounded to three decimal places as needed. use a comma to separate answers as needed.) select the correct rejection region(s) below. a. t < -t₀, t > t₀ b. t < -t₀ c. -t₀ < t < t₀ d. t > t₀ (c) find the standardized test statistic. t = 0.608 (type an integer or decimal rounded to three decimal places as needed.) (d) decide whether to reject or fail to reject the null hypothesis. fail to reject the null hypothesis. (e) interpret the decision in the context of the original claim. at the 2.5% significance level, enough evidence to support the personnel directors claim.

Explanation:

Step1: Recall the hypothesis test concept

In a hypothesis test for the difference between two means (when variances are not equal), if the claim is that \(\mu_A>\mu_B\) (right - tailed test), the null hypothesis is \(H_0:\mu_A\leq\mu_B\) and the alternative hypothesis is \(H_1:\mu_A > \mu_B\). The rejection region for a right - tailed \(t\) - test is \(t>t_0\).

Step2: Analyze the test statistic and rejection

We are given \(t = 0.608\) and \(t_0=2.228\). Since \(t=0.608t_0\). So, we fail to reject the null hypothesis.

Step3: Interpret the decision

When we fail to reject the null hypothesis \(H_0:\mu_A\leq\mu_B\), it means that at the \(2.5\%\) significance level, there is not enough evidence to support the personnel director's claim that the mean annual income is greater in county A than it is in county B.

Answer:

There is not