Question

For women aged 18-24, systolic blood pressure (in mm Hg) are normally distributed with a mean of 114.8 and a standard deviation of 13.1.

a. What is the probability that a randomly selected woman in that age bracket has a blood pressure greater than 140? Round your answer to 4 decimal places. b. If 4 woman in that age bracket are randomly selected, what is the probability that their mean systolic blood pressure is greater than 140? Round to 7 decimal places, not scientific notation.

Answer 1

Let X be the systolic blood pressure (in mm Hg) for women aged 18-24. X is normally distributed with mean and standard deviation

a) the probability that a randomly selected woman in that age bracket has a blood pressure greater than 140 is

ans: the probability that a randomly selected woman in that age bracket has a blood pressure greater than 140 is 0.0274

b) Let be the mean systolic blood pressure for any given sample of size n=4 women in that age bracket.

We know the standard deviation of systolic blood pressure for the population of women in that age bracket to be .

Hence using the central limit theorem we can say that is normally distributed with mean and standard deviation (also called the standard error of mean)

the probability that their mean systolic blood pressure is greater than 140 is

To find this either we can use an approximation from the standard normal table or use a software to estimate the probability.

The standard normal table would list the probability for z=4 as P(Z<4) = 0.99997 and for z=3.5 as P(Z<3.5) = 0.9998. We need to find P(Z<3.85)

Using the linear interpolation we get

Hence P(Z>3.85) = 1- P(Z<3.85) = 1- 0.999919 = 0.000081

ans: The probability that their mean systolic blood pressure is greater than 140 is 0.000081 (using the approximation from the standard normal tables)

Using the Excel function =1-NORM.DIST(3.8473,0,1,TRUE) we get P(Z>3.8473) =  0.0000597

ans: The probability that their mean systolic blood pressure is greater than 140 is 0.0000597 (using the software)