Suppose that the blood cholesterol levels of all men aged 20 to 34 years follows

Question

Suppose that the blood cholesterol levels of all men aged 20 to 34 years follows a Normal distribution with a mean of 188 milligrams per deciliter (mg/dl) and standard deviation 41 mg/dl.

(a) If x represents the blood cholesterol level of a randomly selected man aged 20 to 34, complete the following notation to describe the distribution of x by filling in each of the three blanks:                  x ~ ___ ( ___, ___ ). NOTE: This instruction could have also asked you to describe the shape, center, and spread of the distribution of x.

(b) What is the probability that a randomly selected man aged 20 to 34 years has a blood cholesterol level between 185 and 191 mg/dl?

(c) Suppose a simple random sample of 100 men aged 20 to 34 years is chosen. Describe the sampling distribution of the sample mean, .

(d) Find the probability the average (mean) blood cholesterol level for the sample of 100 men aged 20 to 34 years is between 185 and 191 mg/dl.

(e) Suppose a simple random sample of 1000 men aged 20 to 34 years is chosen. Describe the sampling distribution of the sample mean, .

(f) Find the probability the average (mean) blood cholesterol level for the sample of 1000 men aged 20 to 34 years is between 185 and 191 mg/dl.  

Explanation / Answer

(a) If x represents the blood cholesterol level of a randomly selected man aged 20 to 34, the

x ~ _Normal__ ( _mu__,sigma ___ )

or x ~ _Normal__ ( _188 mg/dl__, 41 mg/dl___ )

(b)For X=185, z=(x-mu)/sigma

=(185-188)/41

=-0.07

For X=191, z=(191-188)/41

=0.07

Thus, P(185<X<191)

=P(X<191)-P(X<185)

=P(z<0.07)-P(z<-0.07)

=0.5279-0.4721=0.0558

(c)Using central limit theorem the sampling distribution of sample mean is also normal with mean 188 and standard deviation 41/root over 100=4.1

(d) For X=185, z=(x-mu)/sigma

=(185-188)/4.1

=-0.73

For X=191, z=(191-188)/4.1

=0.73

Thus, P(185<X<191)

=P(X<191)-P(X<185)

=P(z<0.73)-P(z<-0.73)

=0.7673-0.2327=0.5346

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