Question 22 2 pts A random sample of 17 residents of the Cook County in IL has a mean annual income of $\$ 35,800$ with a standard deviation of $\$ 7800$. A random sample of 20 residents of the DuPage County in IL has a mean annual income of $\$ 45,100$ with a standard deviation of $\$ 7300$. Test the claim that the mean annual incomes in the Cook County and the DuPage County are not the same, using the significance level of 0.05 . The following hypotheses were set: $H_{0} \quad \mu_{\text {Cook }}=\mu_{\text {DuPage }}$ $H_{a} \quad \mu_{C o o k} \neq \mu_{D u P a g e}$ Which of the following would describe the Type II error? Sample data provided evidence to support that the mean annual incomes in the Cook County and the DuPage County are different, but actually mean annual incomes are different. Sample data did not provide evidence to support that the mean annual incomes in the Cook County and the DuPage County are different, but actually mean annual incomes are the same. None of these Sample data provided evidence to support that the mean annual incomes in the Cook County and the DuPage County are different, but actually mean annual incomes are the same. Sample data did not provide evidence to support that the mean annual incomes in the Cook County and the DuPage County are different, but actually mean annual incomes are different.
Solution
Step 1 ：The question is asking for the definition of a Type II error in the context of the given problem. A Type II error occurs when we fail to reject a null hypothesis that is actually false. In this case, the null hypothesis is that the mean annual incomes in the Cook County and the DuPage County are the same.
Step 2 ：Therefore, a Type II error would occur if the sample data did not provide evidence to support that the mean annual incomes in the Cook County and the DuPage County are different, but in reality, the mean annual incomes are different.
Step 3 ：Final Answer: \(\boxed{\text{'Sample data did not provide evidence to support that the mean annual incomes in the Cook County and the DuPage County are different, but actually mean annual incomes are different.'}}\)
