A candy maker produces mints that have a label weight of 204

A candy maker produces mints that have a label weight of 20.4 grams. Assume that the distribution of the weights of these mints is N(21.37, 0.16). Suppose that 15 mints are independently selected and weighed.

Find the probability that at most two of these mints weigh less than 20.857grams.

What two different distributions are used to solve this problem?

Solution

P(X<20.857) = P((X-mean)/s <(20.857-21.37)/sqrt(0.16))

=P(Z<-1.28) = 0.1003 (from standard normal table)

X follows Binomial distriubtion with n=15 and p=0.1003

P(X=x)=15Cx*(0.1003^x)*((1-0.1003)^(15-x)) for x=0,1,2...,15

So the probability that at most two of these mints weigh less than 20.857grams is

P(X<=2) = P(X=0)+P(X=1)+P(X=2)

=15C0*(0.1003^0)*((1-0.1003)^(15-0))+...+15C2*(0.1003^2)*((1-0.1003)^(15-2))

=0.8147812

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What two different distributions are used to solve this problem?

Binomial distribution and normal distribution