The level of cholesterol in the blood for all men aged $20$to $34$follows a Normal distribution with mean $188$milligrams per deciliter (mg/dl) and a standard deviation $41$mg/dl. For $14$-year-old boys, blood cholesterol levels follow a Normal distribution with a mean $170$mg/dl and a standard deviation of $30$mg/dl.

(a) Let M $=$the cholesterol level of a randomly selected $20$to $34$-year-old man and B =the cholesterol level of a randomly selected $14$-year-old boy. Describe the shape, center, and spread of the distribution of$M-B$

(b) Find the probability that a randomly selected $14$-year-old boy has higher cholesterol than a randomly selected man aged $20$to$34$. Show your work.

Men have a normal distribution with mean $=188(mg/dl)$

Standard deviation $=41mg/dl$

$14$year-old boy has a mean $=170mg/dl$

Standard deviation$=30mg/dl$

Distribution M: Normal with ${\mu }_M=188and{\sigma }_M=41$

Distribution B: Normal with ${\mu }_B=170and{\sigma }_B=30$

If M and B are normally distributed, then their difference $M-B$is also normally distributed

Mean and standard deviation

${\mu }_{aX+bY}=a{\mu }_X+b{\mu }_Y$

${\sigma }_{aX+bY}=\sqrt{a^2{\sigma }_X^2+b^2{\sigma }_Y^2}$

Therefore we get,

localid="1650362260757" $$\begin{gathered} {\mu }_{M-B}={\mu }_M-{\mu }_B \\ =188-170 \\ =18 \end{gathered}$$

localid="1650362473508" $$\begin{gathered} {\sigma }_{M-B}=\sqrt{{\sigma }_M^2+{\sigma }_B^2} \\ =\sqrt{{41}^2+{30}^2} \\ =\sqrt{2581} \\ \approx 50.8035 \end{gathered}$$

So the distribution is normal with mean${\mu }_{M-B}=18$and standard deviation${\sigma }_{M-B}\approx 50.8035.$

Men have a normal distribution with mean $=188mg/dl$

Standard deviation$=41mg/dl$

$14$year-old boy has mean$=170mg/dl$

Standard deviation$=30mg/dl$

From part (a) answer,

The $z-$value is the difference between the population mean and the standard deviation divided by the population mean:

$$\begin{gathered} z=\frac{x-\mu }{\sigma } \\ =\frac{0-18}{50.80} \\ =-0.35 \end{gathered}$$

Find the probability using table A

$$\begin{gathered} P(M-B<0)=P(Z<-0.35) \\ =0.3632 \end{gathered}$$.