The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels more than 200 mg/dl is Normally distributed, with mean (in milligrams) μ and standard deviation σ= 3. The brand advertises that eating its oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels more than 200 mg/dl, but you believe that the mean decrease in cholesterol is actually less than advertised. To explore this question, you test the following hypotheses at the 0.05 level:
H₀: μ= 10, H_a: μ< 10
You carry out the level 0.05 test by randomly selecting 100 people with cholesterol levels more than 200 mg/dl and, after they have eaten this brand of oatmeal for one month, computing the mean decrease x̄ in the subjects' cholesterol levels. What is the probability of a Type II error for your test at μ= 8.5?

Question

Quizplus · Accepted Answer

The probability of a Type II error is the probability of failing to reject the null hypothesis when it is false. In this case, the null hypothesis is that the mean decrease in cholesterol is 10, and the alternative hypothesis is that the mean decrease in cholesterol is less than 10. The probability of a Type II error is the area under the normal distribution curve to the right of the critical value, which is the value of the test statistic that would lead to rejection of the null hypothesis. In this case, the critical value is -1.645. The probability of a Type II error is the area under the normal distribution curve to the right of -1.645, which is 0.0004.

The Decrease in Cholesterol Level After Eating a Certain Brand