AP Statistics

Type of Errors and Power

First, some basics, using the Error Types Desmos graph to visualize:

For a given significance level, null hypothesis, and specific alternative, how does an increase in the sample size affect the probability of a type II error?

For a given significance level, null hypothesis, and sample size, how does a move of the specific alternative further away from the null hypothesis affect the power of the test?

For a given sample size, null hypothesis, and specific alternative, how does an increase in the significance level affect the probability of a type II error?

The PGA Tour records the percentage of fairways hit by their players. At the $2010$ Master's Tournament, players hit $60\mathrm{.}25\%$ of fairways. David Benne, the tournament director, wants to know if the percentage of fairways hit has significantly changed. He records the number of drives on holes with a fairway, and how many drives landed within the fairway. He will test $H_0$:$p=0\mathrm{.}6025$ versus $H_A$:$p>0\mathrm{.}6025,$ using a sample size of $110,$ a significance level $()$ of $0\mathrm{.}01,$ and a specific alternative of $p=0\mathrm{.}7.$

a$.$ Describe a Type I and Type II error in the context of the problem.

b$.$ What is the probability of Benne making a Type II error for this specific alternative?

c$.$ What is the power of the test?

Sasha is a park ranger at Yellowstone National Park. During her activities in past years she has seen a bear on average twice a week $(7$ days$).$ This year, she wants to know if bear activity around humans has increased. She takes data on how many of her working days she sees a bear. She will test $H_0$:$p=\frac{2}{7}$ versus $H_A$:$p>\frac{2}{7}.$

a$.$ Using a sample size of $25,$ a significance level $()$ of $0\mathrm{.}05,$ and a specific alternative of $\frac{3}{7},$ calculate the probability of a type I error, the probability of a Type II error, and the statistical power of the test.

b$.$ If she wants to increase the power of the test to $90\%$ or more versus that specific alternative, how large of a sample will she require?