Question

A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 83 and standard deviation σ = 22. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

(a) x is more than 60


(b) x is less than 110


(c) x is between 60 and 110


(d) x is greater than 140 (borderline diabetes starts at 140)

Answer 1

Since the random variable x have a distribution that is approximately normal with mean μ = 83 and standard deviation σ = 22.

hence Z statistic is applicable here for probability calculation.

a) P(X>60) is calculated by finding the Z score at X>60 by

Now P(X<60)=P(Z>-1.05)

the probability for Z>-1.05 is calculated by Z score table is shown below as

The P-Value is (1-0.1469)=0.8531

b) Again at X<110

P(X<110)=P(Z<1.23) which is also calculated by Z score table is shown below as

The P-Value is (1-0.1094)

=0.8906

c) P(60<X<110)

=P(-1.05<Z<1.23)

=P(Z<1.23)-P(Z<-1.05)

=0.8906-0.1469

=0.7437

d) Again P(X>140)

P(X>140)=P(Z>2.59)

= The P-Value is .00480

The table is shown below