Question

To qualify for a medical study, an applicant must have systolic blood pressure in the 50% of the middle range. If the systolic blood pressure is normally distributed with a mean of 120 and a standard deviation of 4, find the upper and lower limits of the blood pressure a person must have to qualify for the study.

Answer 1

Answer:

Lower limit: 117.3

Upper limit: 122.7

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 120, \sigma = 4$

Middle 50%

So it goes from X when Z has a pvalue of 0.5 - 0.5/2 = 0.25 to X when Z has a pvalue of 0.5 + 0.5/2 = 0.75

Lower limit

X when Z has a pvalue of 0.25. So X when $Z = -0.675$

$Z = \frac{X - \mu}{\sigma}$

$-0.675 = \frac{X - 120}{4}$

$X - 120 = -0.675*4$

$X = 117.3$

Upper limit

X when Z has a pvalue of 0.75. So X when $Z = 0.675$

$Z = \frac{X - \mu}{\sigma}$

$-0.675 = \frac{X - 120}{4}$

$X - 120 = 0.675*4$

$X = 122.7$