One personality test avallable on the World Wide Web has a subsection designed to assess the "honesty" of the test-taker. You are interested in the mean score, $\mu$, among the general population on this subsection. The website reports that $\mu$ is 148 , but you have good reason to believe that $\mu$ differs from 148 . You decide to do a statistical test. You choose a random sample of people and have them take the personality test. You find that their mean score on the subsection is 144 and that the standard deviation of their scores is 20.
Based on this information, complete the parts below.
(a) What are the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$ that should be used for the test?
\[
\begin{array}{l}
H_{0}: \square \\
H_{1}: \square
\end{array}
\]
(b) Suppose that you decide to reject the null hypothesis. What sort of error might you be making?
\[
\text { (Choose one) } \mathbf{V}
\]
(c) Suppose the true mean score among the general population on the subsection is 141. Fill in the blanks to describe a Type II error.
A Type II error would be (Choose one) $\mathbf{V}$ the hypothesis that $\mu$ is (Choose one) (Choose one) $\mathbf{V}$ when, in fact, $\mu$ is (Choose one) $\mathbf{V}$.
Answer
(c) A Type II error is when we fail to reject a false null hypothesis, i.e., we say there is no effect or difference when there actually is. Given that the true mean score is 141, a Type II error would be failing to reject the hypothesis that the mean score is 148 when, in fact, it is 141.
Step 1 ：(a) The null hypothesis, denoted by \(H_{0}\), is a statement of no effect or no difference. In this case, the null hypothesis would be that the mean score is 148 (as reported by the website). The alternative hypothesis, denoted by \(H_{1}\), is what we are trying to prove, a statement that indicates the presence of an effect or difference. In this case, the alternative hypothesis would be that the mean score is not 148 (since we have reason to believe it differs). So, \(H_{0}: \mu = 148\) and \(H_{1}: \mu \neq 148\).
Step 2 ：(b) If we reject the null hypothesis, we might be making a Type I error. This is when we incorrectly reject a true null hypothesis, i.e., we say there is an effect or difference when there actually isn't.
Step 3 ：(c) A Type II error is when we fail to reject a false null hypothesis, i.e., we say there is no effect or difference when there actually is. Given that the true mean score is 141, a Type II error would be failing to reject the hypothesis that the mean score is 148 when, in fact, it is 141.
