An estimated 23 million people are poisoned each year by dan

An estimated 2.3 million people are poisoned each year by dangerous chemicals and products found in the home. 64% involve children under the age of 6.

a) What is the probability that of the next 1000 calls to the poison control center, less than 650 involve children?

An estimated 2.3 million people are poisoned each year by dangerous chemicals and products found in the home. 64% involve children under the age of 6. a) What is the probability that of the next 1000 calls to the poison control center, less than 650 involve children? b) It is widely accepted that a Normal Approximation can be used to approximate any binomial distribution where both np and n(1-p) are greater that ten. Using the sigma) and approximate the probability that of the next 1000 calls to the poison control center, less than 650 involve children. mu/ sigma from the binomial distribution set up the formula for a standard Normal (Z = x - mu and

Solution

(a) Given X follows Binomial distribution with n=1000 and p=0.64

P(X=x)=1000Cx*(0.64^x)*((1-0.64)^(1000-x)) for x=0,1,2,...,1000

So the probablity is

P(X<650) = P(X=0)+P(X=1)+...+P(X=649)

=1000C0*(0.64^0)*((1-0.64)^(1000-0))+...+1000C649*(0.64^649)*((1-0.64)^(1000-649))

=0.7336743

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(b) mean=n*p= 1000*0.64 =640

standard deviation =sqrt(n*p*(1-p))

=sqrt(1000*0.64*(1-0.64))

=15.17893

So the probability is

P(X<650) = P((X-mean)/s <(650-640)/15.17893)

=P(Z<0.66)

=0.7453 (from standard normal table)