Question

According to a recent​ poll, 26​% of adults in a certain area have high levels of cholesterol. They report that such elevated levels​ "could be financially devastating to the regions healthcare​ system" and are a major concern to health insurance providers. Assume the standard deviation from the recent studies is accurate and known. According to recent​ studies, cholesterol levels in healthy adults from the area average about 207 ​mg/dL, with a standard deviation of about 25 ​mg/dL, and are roughly Normally distributed. If the cholesterol levels of a sample of 42 healthy adults from the region is​ taken, answer parts ​(a) through ​(d). ​(a) What is the probability that the mean cholesterol level of the sample will be no more than 207​? ​P(y overbarless than or equals207​)equals nothing ​(Round to three decimal places as​ needed.)

Answer 1

0.5 = 50% probability that the mean cholesterol level of the sample will be no more than 207

Explanation:

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

Normal probability distribution

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 = (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.

Central Limit Theorem

The Central Limit Theorem estabilishes 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 = (\sigma)/(√(n))`.

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

In this problem, we have that:

`\mu = 207, \sigma = 25, n = 42, s = (25)/(√(42)) = 3.86`

What is the probability that the mean cholesterol level of the sample will be no more than 207​?

This is the pvalue of Z when X = 207. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (207 - 207)/(3.86)`

`Z = 0`

`Z = 0` has a pvalue of 0.5

0.5 = 50% probability that the mean cholesterol level of the sample will be no more than 207