Question

a personnel director in a particular state claims that the mean annual income is greater in one of the state’s counties (county a) than it is in another county (county b). in county a, a random sample of 17 residents has a annual income of $42,400 and a standard deviation of $8900. in county b, a random sample of 8 residents ha annual income of $38,000 and a standard deviation of $5100. at α = 0.05, answer parts (a) through (e). assum population variances are not equal. if convenient, use technology to solve the problem.

(b) find the critical value(s) and identify the rejection region(s).
enter the critical value(s) below.
1.895
(type an integer or decimal rounded to three decimal places as needed. use a comma to separate answers as
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 = (type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Identify the test type

This is a two - sample t - test for independent samples with unequal variances. The formula for the standardized test statistic (t - statistic) for two - sample t - test with unequal variances is:

$$t=\frac{(\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$$

Here, \(\mu_1-\mu_2 = 0\) (since we are testing if \(\mu_1>\mu_2\), the null hypothesis is \(\mu_1=\mu_2\)), \(\bar{x}_1 = 42400\), \(\bar{x}_2=38000\), \(s_1 = 8900\), \(s_2 = 5100\), \(n_1 = 17\), \(n_2=8\).

Step2: Calculate the numerator

The numerator is \((\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)=(42400 - 38000)-0=4400\)

Step3: Calculate the denominator

First, calculate \(\frac{s_1^2}{n_1}=\frac{8900^2}{17}=\frac{79210000}{17}\approx4659411.7647\)
Second, calculate \(\frac{s_2^2}{n_2}=\frac{5100^2}{8}=\frac{26010000}{8} = 3251250\)
Then, the sum inside the square root is \(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\approx4659411.7647+3251250 = 7910661.7647\)
The square root of this sum is \(\sqrt{7910661.7647}\approx2812.59\)

Step4: Calculate the t - statistic

$$t=\frac{4400}{2812.59}\approx1.564$$

Answer:

\(t\approx1.564\)