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 10 residents has a mean annual income of $ 40 comma 300 and a standard deviation of $ 8500. In County​ B, a random sample of 2 residents has a mean annual income of $ 39 comma 900 and a standard deviation of $ 5500. At alphaequals0.10​, answer parts​ (a) through​ (e). Assume the population variances are not equal. If​ convenient, use technology to solve the problem.
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Part 1
​(a) Identify the claim and state Upper H 0 and Upper H Subscript a.
Which is the correct claim​ below?
A.
​"The mean annual income in county A is less than in county​ B."
B.
​"The mean annual incomes in counties A and B are not​ equal."
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."

Let's first identify the claim made by the personnel director. The claim is that the mean annual income is greater in County A than in County B.

Given this claim, we need to formulate the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is typically a statement of no effect or no difference, while the alternative hypothesis is what we are trying to provide evidence for.

**Claim:** The mean annual income in County A is greater than in County B.

### (a) Identify the claim and state $H_0$ and $H_a$.

**Correct claim:**

D. "The mean annual income in county A is greater than in county B."

**Hypotheses:**

- $ H_0 $: $\mu_A \leq \mu_B$ (The mean annual income in County A is less than or equal to the mean annual income in County B.)
- $ H_a $: $\mu_A > \mu_B$ (The mean annual income in County A is greater than the mean annual income in County B.)

### Do you want details or have any questions?

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Here are 8 relative questions to expand on this information:

1. How do you calculate the test statistic for a two-sample t-test when population variances are not equal?
2. What are the degrees of freedom for the two-sample t-test when population variances are unequal?
3. How do you determine the critical value for a t-test at $\alpha = 0.10$?
4. What steps are involved in performing a hypothesis test using technology (e.g., a statistical calculator or software)?
5. How does the sample size affect the results of the hypothesis test?
6. What assumptions must be met for the two-sample t-test to be valid?
7. How can we interpret the p-value obtained from the t-test?
8. What are the potential implications of Type I and Type II errors in this context?

**Tip:** Always check the assumptions of normality and equality of variances before performing a t-test, especially when dealing with small sample sizes.

Explore hypothesis testing for mean annual income between County A and County B using a two-sample t-test. Understand how to formulate hypotheses, calculate test statistics, interpret results, and determine significance levels at alpha equals 0.10.

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