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Based on data from the National Health Survey, women between the ages of 18and 24have an average systolic blood pressures (in mm Hg) of 114.8with a standard deviation of 13.1. Systolic blood pressure for women between the ages of 18 to 24follow a normal distribution.
a. If one woman from this population is randomly selected, find the probability that her systolic blood pressure is greater than120 .
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 18to 24 and we did not know the original distribution, could the central limit theorem be used?

Short Answer

Expert verified

(a) The probability that the systolic blood pressure is greater than120is0.3516
(b) The probability that the mean systolic blood pressure is greater than 120isPr(X¯120)=0.475
(c) Can not use the central limit theorem.

Step by step solution

01

Given information Part (a)

Women between the ages of18and24have an average systolic blood pressures of114.8with a standard deviation of13.1.

02

Explanation Part (a)

According to the information, we observed thatμ=114.8and σ=13.1
Let's consider:Pr(X120).
Now, apply exponential distribution for an individual:

X~Exp(1114.8)

P(x>120)=e1114.8(120)

=e1.0453=0.3516

03

Given information Part (b)

The probability that their mean systolic blood pressure is greater than 120. To compute the probability by usingPr(X¯120).

04

Explanation Part (b)

From the given information, we observed thatX¯~N(114.8,13.1×40)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 computePr(X¯120)
Pr(X¯120)=PrZ120114.882.85=Pr(Z0.0628)

Pr(X¯120)=0.475

05

Given information Part (c)

Women between the ages of 18and 24have an average systolic blood pressures of 114.8with a standard deviation of13.1

06

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.

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