Question

A person’s blood glucose level and diabetes are closely related. Let x be a random variable measuring the glucose in mg per decilitre of blood. After a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean µ = 85 and standard deviation σ = 25. (Source: Diagnostic Tests with Nursing Implications, edited by S. Loeb, Springhouse Press). Note: after 50 years of age, both the mean and standard deviation tend to increase.

(a) What is the probability that, for an adult under 50 years old, after a 12-hour fast, x is less than 60?

(b) What is the probability that, for an adult under 50 years old, after a 12-hour fast, x is between 70 and 100?

(c) What is the probability that, for an adult under 50 years old, after a 12-hour fast, x is more than 125? (borderline diabetes starts at 125)

(d) Find the minimum glucose level (after a 12-hour fast) required to participate in a study reserved for the top 10% of people.

Answer 1

$a:\\P\left( X<60 \right) =P\left( \frac{X-85}{25}<\frac{60-85}{25} \right) =P\left( Z<-1 \right) =\varPhi \left( -1 \right) =0.1587\\b:\\P\left( 70<X<100 \right) =P\left( \frac{70-85}{25}<\frac{X-85}{25}<\frac{100-85}{25} \right) =P\left( -0.6<Z<0.6 \right) =\\=2\varPhi \left( 0.6 \right) -1=2\cdot 0.7257-1=0.4514\\c:\\P\left( X>125 \right) =P\left( \frac{X-85}{25}>\frac{125-85}{25} \right) =P\left( Z>1.6 \right) =\\=1-P\left( Z\leqslant 1.6 \right) =1-\varPhi \left( 1.6 \right) =1-0.9542=0.0458\\d:\\P\left( X>C \right) =0.1\Rightarrow P\left( \frac{X-85}{25}\leqslant \frac{C-85}{25} \right) =0.9\Rightarrow \\\Rightarrow \frac{C-85}{25}=z_{0.9}\Rightarrow C=25z_{0.9}+85=25\cdot 1.28155+85=117.039$