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U6D9 Ch.Y.U. - Power of a Test

Last updated 7 months ago
6 questions


A high school student is planning a very large graduation party. She needs to know how many people are planning on attending in order to estimate the cost of the party. She plans to select a random sample of 25 guests that were invited to the party and record whether or not each person plans to attend the party. She will then test H0: p = 0.75 versus Ha: p ≠ 0.75 where p = the true proportion of all invited guests that will attend the party. She will use alpha = 0.05.

Notes

  1. A larger sample size gives more information about the true parameter, making it more likely to detect the alternative hypothesis if it is true. If Mr. Y took 1,000 free throws instead of 50, it would become much easier to decide against his 80% claim if he wasn’t really that good.
  2. Using a larger significance level makes it easier to reject H0​ (remember that α tells us how likely it is we would reject H0​), which is needed for more Power.
  3. It is easier to detect a bigger difference between the null and alternative parameter value. If Mr. Y was actually a 22% free throw shooter, a sample of 50 free throws would undoubtedly lead us to reject the claim that he is an 80% free throw shooter.
One final note about the alpha level α. The person doing the significance test can set the alpha level and may do so depending on the potential consequences of the Type I versus the Type II error.
  • Increasing α will increase the probability of a Type I error. But it will also increase the Power, meaning that it will decrease the probability of a Type II error. Do this if the consequences of the Type II error are more serious.
  • Decreasing α will decrease the probability of a Type I error. But it will also decrease the Power, meaning that it will increase the probability of a Type II error. Do this if the consequences of the Type I error are more serious.
1

Describe a Type I error in this setting.

1
What is the probability of making a Type I error?
P(Type I Error) = _______
1

Suppose the true proportion of invited guests that will attend the party is 0.70. Then the power of the given significance test is 0.10. Interpret the power.

1
Find the probability of a Type II error for the test in #3.

P(Type II Error) = _______
1

Determine whether the following change would increase or decrease the power of the test. Explain your answers.

Use alpha = 0.01 instead of alpha = 0.05.
Don't Forget to EXPLAIN (in show your work)!

1

Determine whether the following change would increase or decrease the power of the test. Explain your answers.

Use n = 100 instead of n = 25.
Don't Forget to EXPLAIN (in show your work)!