Question

Based on data from the National Health Survey, women between the ages of $18$and $24$have an average systolic blood pressures (in mm Hg) of $114.8$with a standard deviation of $13.1.$ Systolic blood pressure for women between the ages of $18$ to $24$follow a normal distribution.
a. If one woman from this population is randomly selected, find the probability that her systolic blood pressure is greater than$120$ .
b. If $40$ women from this population are randomly selected, find the probability that their mean systolic blood pressure is greater than $120$ .
c. If the sample were four women between the ages of $18$to $24$ and we did not know the original distribution, could the central limit theorem be used?

Short answer

(a) The probability that the systolic blood pressure is greater than$120$is$0.3516$
(b) The probability that the mean systolic blood pressure is greater than $120$is$Pr(\overline{X}\ge 120)=0.475$
(c) Can not use the central limit theorem.

Step-by-step solution

Given information Part (a)

Women between the ages of$18$and$24$have an average systolic blood pressures of$114.8$with a standard deviation of$13.1$.

Explanation Part (a)

According to the information, we observed that$\mu =114.8$and $\sigma =13.1$
Let's consider:$Pr(X\ge 120)$.
Now, apply exponential distribution for an individual:

$X~\mathrm{Exp}\left(\frac{1}{114.8}\right)$

$\mathrm{P}(\mathrm{x}>120)=e^{\left(-\left(\frac{1}{114.8}\right)\left(120\right)\right)}$

$=e^{-1.0453}=0.3516$

Given information Part (b)

The probability that their mean systolic blood pressure is greater than $120$. To compute the probability by using$Pr(\overline{X}\ge 120)$.

Explanation Part (b)

From the given information, we observed that$\overline{X}~N(114.8,13.1\times \sqrt{40})$where $n=40$
We have to find the probability for 40 women, that their mean systolic blood pressure is greater than 120

So, let's compute$Pr(\overline{X}\ge 120)$
$\mathrm{Pr}(\overline{X}\ge 120)=\mathrm{Pr}\left(Z\ge \frac{120-114.8}{82.85}\right)=\mathrm{Pr}(Z\ge 0.0628)$

$\mathrm{Pr}(\overline{X}\ge 120)=0.475$

Given information Part (c)

Women between the ages of $18$and $24$have an average systolic blood pressures of $114.8$with a standard deviation of$13.1$

Explanation Part (c)

If the sample were four women between the ages of 18 to 24, did not understand the original distribution and could not utilize the central limit theorem because the sample size was too short and did not notice the distribution.