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 $8400. in county b, a random sample of 8 residents has a mean annual income of $38,400 and a standard deviation of $5700. at α = 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.
(a) identify the claim and state h₀ and hₐ.
which is the correct claim below?
○ a. “the mean annual incomes in counties a and b are not equal.”
○ b. “the mean annual income in county a is less than in county b.”
○ c. “the mean annual incomes in counties a and b are equal.”
○ d. “the mean annual income in county a is greater than in county b.”
what are h₀ and hₐ?
the null hypothesis, h₀, is dropdown the alternative hypothesis, hₐ, is dropdown

Explanation:

The personnel director claims the mean annual income is the same in County A and County B. So the correct claim is that their means are equal (option C). For hypotheses, the null hypothesis (\(H_0\)) represents the claim being tested, so \(H_0: \mu_A = \mu_B\) (the means are equal). The alternative hypothesis (\(H_a\)) is the opposite, so \(H_a: \mu_A
eq \mu_B\) (the means are not equal), since we're testing if the claim (equality) is true, and the alternative is a two - tailed test here (as the claim is about equality vs. inequality).

Answer:

The correct claim is C. "The mean annual incomes in Counties A and B are equal."
\(H_0: \mu_A=\mu_B\) (The null hypothesis, \(H_0\), is "The mean annual income in County A is equal to the mean annual income in County B.")
\(H_a: \mu_A
eq\mu_B\) (The alternative hypothesis, \(H_a\), is "The mean annual income in County A is not equal to the mean annual income in County B.")