Of the items manufactured at a plant 11 are underweight If t

Of the items manufactured at a plant, 11% are underweight. If the plant produces 15000 items in a day, find the probability that (a) exactly 1600 of them are underweight; and (b) at least 1675 of them are underweight.

Show the steps so i can understand how to do this. Thank you.

Solution

(a) exactly 1600 of them are underweight; and

mean=n*p=15000*0.11 =1650

standard deviation =sqrt(n*p*(1-p)) =sqrt(15000*0.11*(1-0.11)) =38.32101

So P(X=1600) = P(1599.5<X<1600.5)

=P((1599.5-1650)/38.32101 <(X-mean)/s <(1600.5-1650)/38.32101)

=P(-1.32<Z<-1.29) = 0.0051 (from standard normal table)

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(b) at least 1675 of them are underweight.

P(X>1675) = P(Z>(1675-1650)/38.32101)

=P(Z>0.65) =0.2578(from standard normal table)