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. For this situation, what is the probability of committing a Type II error? a) 0.05 b) 0.10 c) 0.20 d) 0.30 e) 0.40 f) 0.50 g) 0.60 h) 0.70

Explanation / Answer

Here, P(wrapping paper is Red | gift comes from Child A) = 0.60

P(wrapping paper is Red | gift comes from Child B) = 0.30

Null Hypothesis is H0: The gift comes from Child A

When grandma gets her gift, her decision rule is: Reject H0 if the wrapping paper is Green

So, The probability of committing a Type II error

=Probability of accepting the hypothesis when it is actually false

=Probability that grandma concludes the gift is from child A(ie, she receives the gift wrapped in red paper), when actually it is from child B

=P(wrapping paper is Red | gift comes from Child B)

= 0.30 [Option d ]

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