If the null hypothesis is true, each customer is equally likely to select the re

Question

If the null hypothesis is true, each customer is equally likely to select the red box or the blue box of cereal. If the alternative hypothesis is true, customers prefer the blue box 3 to 1 over the red box. Assume each customer selects one box of cereal, independently of the other customers. The null hypothesis will be rejected if 15 or more of the first 20 customers select blue boxes. Find the level of significance. Find the power. If 17 of the first 20 customers actually observed selected blue boxes, find the p-value.

Explanation / Answer

Let Xi denote the random variable which takes the value 1 if ith customer picks blue box and 0 otherwise.

mean (Xi) = 0.5, var(Xi) = 0.25

Let X = X1+X2+X3+...+X20

Significance Level = P(Rejecting H0|H0) = P(X>15 | H0)

Mean(X) = 5, Var(X) = 5 , Std(X) = sqrt(5)

P(X>15) = 1- P(X<15) = 0.01255 = significance level

Calculation of power:

P(X<15|H1)

Mean(X) assuming H1 = 20*(0.75) = 15

Var(X) = 20*(0.1875) = 3.75, std(X) = 1.9365

P(X<15) = P(Z<(15-15)/1.9365) =P(Z<0) = 0.5

Power = 1-P(X<15 | H1) = 0.5

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