Question

a personnel director in a particular state claims that the mean annual income is the same in one of the state’s counties (county a) as it is in another county (county b). in county a, a random sample of 18 residents has a mean annual income of $41,100 and a standard deviation of $9400. in county b, a random sample of 8 residents has a mean annual income of $38,600 and a standard deviation of $5700. at \\(\alpha = 0.10\\), answer parts (a) through (e). assume the population variances are not equal. assume the samples are random and independent, and the populations are normally distributed.

1.895, -1.895
(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 < -t_0, t > t_0 \\)
b. \\( t > t_0 \\)
c. \\( t < -t_0 \\)
d. \\( -t_0 < t < t_0 \\)
(c) find the standardized test statistic.
\\( t = \\) 0.956 (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.

Explanation:

Step1: Identify the test type

This is a two - sample t - test for the difference in means with unequal variances. The null hypothesis \(H_0:\mu_1=\mu_2\) and the alternative hypothesis \(H_a:\mu_1
eq\mu_2\) (since the claim is that the means are the same, and we are testing against a two - tailed alternative).

Step2: Recall the rejection region for two - tailed t - test

For a two - tailed t - test with significance level \(\alpha\), the rejection region is \(t < - t_{\alpha/2}\) or \(t>t_{\alpha/2}\), where \(t_{\alpha/2}\) is the critical value from the t - distribution table. In this case, since \(\alpha = 0.10\) and it's a two - tailed test, we look for the region where the test statistic is less than the negative critical value or greater than the positive critical value.

Step3: Analyze the test statistic and critical values

The critical values are given as \(1.895,- 1.895\) (which are \(t_{\alpha/2}\) and \(-t_{\alpha/2}\) for a two - tailed test with \(\alpha = 0.10\)). The test statistic \(t = 0.956\). We check if \(0.956\) is in the rejection region. Since \(- 1.895<0.956 < 1.895\), the test statistic does not lie in the rejection region \(t < - 1.895\) or \(t>1.895\).

Step4: Make a decision

If the test statistic lies in the rejection region, we reject the null hypothesis; otherwise, we fail to reject it. Since \(0.956\) is not in the rejection region (\(t < - 1.895\) or \(t>1.895\)), we fail to reject the null hypothesis.

Answer:

(b) Rejection Region

The correct rejection region for a two - tailed t - test (since the alternative hypothesis is \(\mu_1
eq\mu_2\)) is \(t < - t_0\) or \(t>t_0\), so the answer is A. \(t < - t_0,t > t_0\)

(c) Standardized Test Statistic

The standardized test statistic \(t = 0.956\)

(d) Decision

Since the test statistic \(t=0.956\) does not lie in the rejection region (\(t < - 1.895\) or \(t>1.895\)), we fail to reject the null hypothesis. So the answer is "Fail to reject" the null hypothesis.