A demographics researcher claims that the mean household income in a recent year is different in Polk County, Iowa, than it is in Woodward County, Oklahoma. In Polk County, a sample of 13 residents has a mean household income of $61,300 and a standard deviation of $1770. In Woodward County, a sample of 15 residents has a mean household income of $59,800 and a standard deviation of $8350. At α = 0.01, can you support the demographics researcher’s claim? Assume the populations are normally distributed and the population variances are not equal.

For each exercise, perform the steps below.

(a) Identify the claim and state H₀ and H_a.

(b) Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.

(c) Find the critical value(s) and identify the rejection region(s).

(d) Find the appropriate standardized test statistic.

(e) Decide whether to reject or fail to reject the null hypothesis.

(f) Interpret the decision in the context of the original claim.