Question

The distribution of results from a cholesterol test has a mean of 180 and a standard deviation of 20. A sample size of 40 is drawn randomly. Find the probability that the sum of the 40 values is less than 7,100. (Round your answer to four decimal places.)

Answer 1

Using the normal distribution, there is a 0.2148 = 21.48% probability that the sum of the 40 values is less than 7,100.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

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

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

• By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For this problem, these parameters are given as follows:

$\mu = 180, \sigma = 20, n = 40, s = \frac{20}{\sqrt{40}} = 3.1623$

A sum of 7100 is equivalent to a sample mean of 7100/40 = 177.5, which means that the probability is the p-value of Z when X = 177.5, hence:

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

By the Central Limit Theorem:

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

$Z = \frac{177.5 - 180}{3.1623}$

Z = -0.79

Z = -0.79 has a p-value of 0.2148.

There is a 0.2148 = 21.48% probability that the sum of the 40 values is less than 7,100.

More can be learned about the normal distribution at brainly.com/question/28135235

#SPJ1