Grandma’s birthday is coming up and my kids are both excited to get her the perf

Question

Grandma’s birthday is coming up and my kids are both excited to get her the perfect gift. We have 2 different colors of wrapping paper at home, Green and Red. Child A loves Green and Red and gravitates toward Red 60% of the time and Child B also loves Green and Red but gravitates toward Red 30% of the time. If on Grandma’s birthday she gets a gift that is wrapped in Red wrapping paper she will conclude that Child A gave her the gift, and if she gets a gift that is wrapped in Green wrapping paper, she will conclude that Child B gave her the gift.

Statistically speaking, the following hypotheses and rejection rule have been stated:

H0: The gift comes from Child A; that is, the P(wrapping paper is Red | gift comes from Child A) = 0.60.
Ha: The gift comes from Child B; that is, the P(wrapping paper is Red | gift comes from Child B) = 0.30.

When grandma gets her gift, her decision rule wll be: Reject H0 if the wrapping paper is Green and thus fail to reject H0 if the wrapping paper is Red.

Question A: For this situation, what is the value of , the significance level (also known as the probability of incorrectly rejecting the null hypothesis)?

Question B: For this situation, what is the power of the test (also known as the probability of correctly rejecting the null hypothesis)?

Explanation / Answer

Question A

An alpha level is the probability of a type I error, or you reject the null hypothesis when it is true. Null hypothesis is gift has come from Child A. GrandMa will reject null hypothesis when gift is green. Hence alpha becomes Green gift from Child A. Grandma in this case will reject null hypothesis based on green color, but it will still be given by A. So we will be rejecting null hypothesis when its true.

Probability of green gift from child A = 40%, so alpha = 0.4

Question B

Power of test - Correctly rejecting the null hypothesis. Null hypothesis should be rejected when it gift has come from B. So correctly rejecting null hypothesis would be - Hence power becomes Green gift from Child B. Power = 1-0.3 = -.7

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