. In an article exploring blood serum levels of vitamins and lung cancer risks (

Question

. In an article exploring blood serum levels of vitamins and lung cancer risks (The New England Journal of Medicine), the mean serum level of vitamin E in the control group was 11.9 mg/liter with the standard deviation of 4.30 mg/liter. There were 196 patients in the control group. (These patients were free of all cancer, except possible skin cancer, in the subsequent 8 years). mg/liter. Assume that serum levels of vitamin E are normally distributed. Find the number of patients with the serum level of vitamin E (a) (5 points) between 3.3 mg/liter and 16.2 mg/liter. (b) (4 points) more than 20.5 mg/liter.

Explanation / Answer

(a)Using Central limit theorem the sampling distribution of sample mean is also normal with mean mu= 11.9 mg/liter and standard deviation= 4.30 mg/liter/root over 196

=0.307

For X bar=3.3, z=(x-mu)/sigma

=(3.3-11.9)/0.307

=-28.11

For X bar=16.2, z=(16.2-11.9)/0.307=14.006

Thus, P(3.3<X bar<16.2)

=P(X bar<16.2)-P(X bar<3.3)

=P(z<14.00)-P(z<-28.11)

=0.9999-0.00001

=0.9998

(b)For X bar=20.5, z=(20.5-11.9)/0.307=28.11

Thus P(X bar>20.5)=1-P(z<28.11)=1-0.9999=0.0001

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