NBC News reported that 1 in 20 children in the US have a foo

NBC News reported that 1 in 20 children in the US have a food allergy of some sort. Consider selecting a random sample of 25 children and let X be the number in the sample who have a food allergy.

d) What are expected value and standard deviation of X?

e) In a sample of 50 children, what is the probability that none has a food allergy.

Please show work.

Solution

Prob for any random child having food allergy =1/20 =0.05

Each child is independent and there are only two outcome

X- no of children who have food allergy is binomial with p = 0.05 and n = 25

a) P(X<=3) = P(0,1,2,3) = 0.9659

P(X<3) = P(0,1,2)=0.8729

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b) P(X>=3) = 1-P(X<3) =0.1271

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c) P(1<=x<=3) = P(1,2,3)

P(X<=3)-P(0)

= 0.9659-0.2774

=0.6885

d) E(X) = np = 1.25 and var(X) = npq = 1.1875

Std dev = square root of variance = 1.0897

e) n =50

P(X=0) = 0.05⁵⁰

=0.0769