Question

a personnel director in a particular state claims that the mean annual income is the same in one of the states counties (county a) as it is in another county (county b). in county a, a random sample of 18 residents has a m annual income of $41,100 and a standard deviation of $8400. in county b, a random sample of 8 residents has annual income of $38,400 and a standard deviation of $5700. at α = 0.10, answer parts (a) through (e). assume population variances are not equal. assume the samples are random and independent, and the populations are distributed.

(b) find the critical value(s) and identify the rejection region(s).
enter the critical value(s) below.
1.895, -1.895
(type an integer or decimal rounded to three decimal places as needed. use a comma to separate answers as ne
select the correct rejection region(s) below.
a. t < -t₀, t > t₀
b. t > t₀
c. t < -t₀
d. -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 formula for the standardized test statistic (t - test for independent samples with unequal variances)

The formula for the t - statistic in a two - sample t - test with unequal variances is:

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

Since the null hypothesis \(H_0:\mu_1=\mu_2\), so \(\mu_1-\mu_2 = 0\).
We are given:
\(\bar{x}_1 = 41100\), \(s_1=8400\), \(n_1 = 18\)
\(\bar{x}_2=38400\), \(s_2 = 5700\), \(n_2=8\)

Step2: Calculate the numerator

The numerator is \((\bar{x}_1-\bar{x}_2)-(\mu_1 - \mu_2)\). Substituting the values, we get:
\((41100 - 38400)-0=2700\)

Step3: Calculate the denominator

First, calculate \(\frac{s_1^2}{n_1}\) and \(\frac{s_2^2}{n_2}\)
\(\frac{s_1^2}{n_1}=\frac{8400^2}{18}=\frac{70560000}{18}=3920000\)
\(\frac{s_2^2}{n_2}=\frac{5700^2}{8}=\frac{32490000}{8} = 4061250\)
Then, the denominator is \(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=\sqrt{3920000 + 4061250}=\sqrt{7981250}\approx2825.1106\)

Step4: Calculate the t - statistic

\(t=\frac{2700}{2825.1106}\approx0.956\)

Answer:

\(0.956\)