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 mean annual income of $42,400 and a standard deviation of $8900. in county b, a random sample of 8 residents has a mean annual income of $38,000 and a standard deviation of $5100. at α = 0.05, answer parts (a) through (e). assume the population variances are not equal. if convenient, use technology to solve the problem.

(a) identify the claim and state h₀ and hₐ.
which is the correct claim below?
a. \the mean annual income in county a is greater than in county b.\
b. \the mean annual incomes in counties a and b are equal.\
c. \the mean annual income in county a is less than in county b.\
d. \the mean annual incomes in counties a and b are not equal.\
what are h₀ and hₐ?
the null hypothesis, h₀, is μ₁ ≤ μ₂. the alternative hypothesis, hₐ, is μ₁ > μ₂.
which hypothesis is the claim?
the null hypothesis, h₀
the alternative hypothesis, hₐ

Explanation:

The claim is that the mean annual income in county A is greater than in county B, which matches the alternative hypothesis \( H_a: \mu_1 > \mu_2 \). The null hypothesis \( H_0 \) is the opposite or a statement of equality (or less than/equal in this case, \( \mu_1 \leq \mu_2 \)). The claim aligns with the alternative hypothesis because it's the statement we are trying to find evidence for, while the null is the default we test against.

Answer:

The alternative hypothesis, \( H_a \)