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. 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. fail to reject the null hypothesis. (e) interpret the decision in the context of the original claim. at the 5% significance level, enough evidence to support the personnel director’s claim.

Explanation:

Step1: Identify the test type

This is a two - sample t - test for the difference in means with unequal variances. The claim is that the mean annual income in County A is greater than in County B, so the alternative hypothesis is \(H_1:\mu_A-\mu_B > 0\), and the null hypothesis is \(H_0:\mu_A-\mu_B\leqslant0\). This is a right - tailed test.

Step2: Determine the rejection region

For a right - tailed t - test with significance level \(\alpha = 0.05\), the rejection region is \(t>t_0\), where \(t_0\) is the critical value from the t - distribution table (or calculated using technology) corresponding to \(\alpha = 0.05\) and the appropriate degrees of freedom.

Step3: Analyze the test statistic and decision

The standardized test statistic \(t = 1.564\). We compare this test statistic with the critical value \(t_0\). For a right - tailed test, if \(t>t_0\), we reject the null hypothesis; otherwise, we fail to reject it. Since we failed to reject the null hypothesis (as given in part (d)), it means that \(t = 1.564\) is not in the rejection region \(t>t_0\) (i.e., \(1.564\leqslant t_0\)).

Step4: Interpret the decision in context

Since we failed to reject the null hypothesis, at the 5% significance level, there is not enough evidence to support the personnel director's claim that the mean annual income in County A is greater than in County B.

Answer:

(e) At the 5% significance level, there is not enough evidence to support the personnel director's claim. So the answer for the blank in part (e) is "there is not".