Question

The mean family income for a random sample of 550 suburban households in Nettlesville shows that a 92 percent confidence interval is ($45,700, $59,150). Braxton is conducting a test of the null hypothesis H0: µ = 44,000 against the alternative hypothesis Ha: µ ≠ 44,000 at the α = 0.08 level of significance. Does Braxton have enough information to conduct a test of the null hypothesis against the alternative?

a. Yes, because $44,000 is not contained in the 92% confidence interval, the null hypothesis would be rejected in favor of the alternative, and it could be concluded that the mean family income is significantly different from $44,000 at the α = 0.08 level
b. Yes, because $44,000 is not contained in the 92% confidence interval, the null hypothesis would not be rejected, and it could be concluded that the mean family income is not significantly different from $44,000 at the α = 0.08 level
c. No, because the value of α is not known
d. No, because it is not known whether the data are Normally distributed
e. No, because the entire data set is needed to do this test

Answer 1

The correct answer would be:

a. Yes, because $44,000 is not contained in the 92% confidence interval, the null hypothesis would be rejected in favor of the alternative, and it could be concluded that the mean family income is significantly different from $44,000 at the α = 0.08 level.

This scenario involves conducting a hypothesis test about the mean family income based on a confidence interval and a null hypothesis.

The 92% confidence interval for the mean family income is ($45,700, $59,150), which does not contain the value $44,000.

The null hypothesis (H0: µ = 44,000) being tested against the alternative hypothesis (Ha: µ ≠ 44,000) suggests that we're evaluating whether $44,000 falls within the range of potential mean incomes.

Given that $44,000 is outside the 92% confidence interval and the null hypothesis suggests it should be inside, Braxton does have enough information to conduct the test. In this case, the correct answer would be:

a. Yes, because $44,000 is not contained in the 92% confidence interval, the null hypothesis would be rejected in favor of the alternative, and it could be concluded that the mean family income is significantly different from $44,000 at the α = 0.08 level.