# Question: Supporters claim that a new windmill can generate an average

Supporters claim that a new windmill can generate an average of at least 800 kilowatts of power per day. Daily power generation for the windmill is assumed to be normally distributed with a standard deviation of 120 kilowatts. A random sample of 100 days is taken to test this claim against the alternative hypothesis that the true mean is less than 800 kilowatts. The claim will not be rejected if the sample mean is 776 kilowatts or more and rejected otherwise.
a. What is the probability a of a Type I error using the decision rule if the population mean is, in fact, 800 kilowatts per day?
b. What is the probability b of a Type II error using this decision rule if the population mean is, in fact, 740 kilowatts per day?
c. Suppose that the same decision rule is used, but with a sample of 200 days rather than 100 days.
i. Would the value of a be larger than, smaller than, or the same as that found in part a? ii. Would the value of b be larger than, smaller than, or the same as that found in part b?
d. Suppose that a sample of 100 observations was taken, but that the decision rule was changed so that the claim would not be rejected if the sample mean was at least 765 kilowatts.
i. Would the value of a be larger than, smaller than, or the same as that found in part a?
ii. Would the value of b be larger than, smaller than, or the same as that found in part b?

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