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)$P(X>120)=.3458$

(b)$P(X>120)=.558$

(c)The central limit theorem cannot be used, because sample size $4$is small and also the original distribution is not known.

Step-by-step solution

Given information (part a) 

Given in the question that, 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.

Explanation(part a)

According to the supplied details, the average blood pressure of women aged between$18$and $24$years is $144.8\mathrm{mmhg}$with a standard deviation of $13.1$.

To estimate the probability of systolic blood pressure $P(X>120)$, utilize the Ti-83 calculator. For this, click on $2^{\text{nd}}$, then DISTR, and then scroll down to the normalcdf option and enter the provided details. After this, click on ENTER button on the calculator to have the desired result. The screenshot is given below:

Therefore, the value probability that$P(X>120)$is:

$=1-.6542$

$=.3458$

Final answer(part a)

Required probability $P(X>120)=.3458$

Given information(part b)

Given in the question that, 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.

Explanation(part b)

According to the provided details, the average blood pressure of women ages between$18$and $24$years is $144.8\mathrm{mmhg}$with a standard deviation $13.1$.

To estimate the probability of$40$women's systolic blood pressure $P(\overline{X}>120)$, use the Ti-83 calculator. For this, click on $2^{\text{nd}}$, then DISTR, and then scroll down to the normalcdf option and enter the provided details. After this, click on ENTER button on the calculator to have the desired result. The screenshot is given below:

Therefore, the value probability that $P(X>120)$is:

$=1-.4420$

$=.558$

Final answer(part b)

Required probability$P(X>120)=.558$

Given information (part c) 

Given in the question that, 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.

Explanation(part c)

According to the provided details, the average blood pressure of women ages between $18$and $24$years is $144.8\mathrm{mmhg}$with a standard deviation $13.1$.

The central limit theorem cannot be used, because sample size$4$ is small and also the original distribution is not known.

Final answer(part c)

The central limit theorem cannot be used, because sample size $4$is small and also the original distribution is not known.