Question

Assume that blood pressure readings are normally distributed with a mean of 115 and a standard deviation of 8. If 100 people are randomly selected, find the probability that their mean blood pressure will be less than 117.
A. 0.0062.
B. 0.8615.
C. 0.8819.
D. 0.9938.

Answer 1

Answer:

D. 0.9938.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Mean of 115 and a standard deviation of 8.

This means that $\mu = 115, \sigma = 8$

100 people are randomly selected

This means that $n = 100, s = \frac{8}{\sqrt{100}} = 0.8$

Find the probability that their mean blood pressure will be less than 117.

This is the p-value of Z when X = 117, so:

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

By the Central Limit Theorem

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

$Z = \frac{117 - 115}{0.8}$

$Z = 2.5$

$Z = 2.5$ has a p-value of 0.9938, and thus, the correct answer is given by option D.