A geologist has collected 150 specimens of basaltic rock and

A geologist has collected 150 specimens of basaltic rock and 50 specimens of granite. The geologist instructs a laboratory assistant to randomly select 8 specimens for analysis. Let Y= number of selected specimens that are granite.

A.) What is the distribution of Y? Identify all relevant parameters and write down the pmf.

B.) Calculate (using the exact distribution) P(Y less than or equal to 1).

C.) Suggest a distribution that can be used to approximate the true distribution of Y. Verify any conditions necessary for the approximation to work.

D.) Regardless of the conclusion in (C), use the suggested approximation to calculate P(Y greater than or equal to 4).

Solution

A.) What is the distribution of Y? Identify all relevant parameters and write down the pmf.

Y follows Binomial distribution with n=8 and p=50/(150+50) = 0.25

P(X=x)=8Cx*(0.25^x)*(0.75^(8-x)) for x=0,1,2,...,8

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B.) Calculate (using the exact distribution) P(Y less than or equal to 1).

P(Y<=1)= P(Y=0)+P(Y=1)

=8C0*(0.25^0)*(0.75^(8-0))+8C1*(0.25^1)*(0.75^(8-1))

=0.3670807

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C.) Suggest a distribution that can be used to approximate the true distribution of Y. Verify any conditions necessary for the approximation to work.

Normal distribution can be used to approximate the true distribution because the sample size n=50+150=200 is large.

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D.) Regardless of the conclusion in (C), use the suggested approximation to calculate P(Y greater than or equal to 4).

mean=n*p=8*0.25=2

standard deviation =sqrt(n*p*(1-p))=sqrt(8*0.25*0.75) =1.224745

P(Y>=4) = P((Y-mean)/s >(4-2)/1.224745)

=P(Z>1.63) = 0.0516 (from standard normal table)