A simple random sample of size n 1460 is obtained from a po

A simple random sample of size n = 1460 is obtained from a population whose size is N = 1,500,000 and whose population proportion with a specified characteristic is p = 0.42

a) describe the sampling distribution of p?.

b) What is the probability of obtaining x = 657 or more individuals with the characteristic?

c) What is the probability of obtaining x = 584 or fewer individuals with the characteristic?

Solution

a) describe the sampling distribution of p?.

It follows Normal distribution with mean=n*p=1460*0.42=613.2 and

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

=sqrt(1460*0.42*(1-0.42))

=18.85884

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b) What is the probability of obtaining x = 657 or more individuals with the characteristic?

P(X>657) = P((X-mean)/s >(657-613.2)/18.85884)

=P(Z>2.32) = 0.0102 (from standard normal table)

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c) What is the probability of obtaining x = 584 or fewer individuals with the characteristic?

P(X<584) = P(Z<(584-613.2)/18.85884))

=P(Z<-1.55) =0.0606 (from standard normal table)