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 210 mg/dL, with a standard deviation of about 30 mg/dL, and are roughly Normally distributed. If the cholesterol levels of a sample of 41 healthy adults from the region is taken.

a. What shape will the sampling distribution of the mean have?

i. skewed left
ii. skewerd right
iii. mean is normal.
iv. not enough information

b. What is the mean of the sampling distribution?
c. What is the standard deviation?

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Then, the mean of the distribution of sample means is given by,

[tex]\mu_{\bar x}=\mu[/tex]

And the standard deviation of the distribution of sample means is given by,

[tex]\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}[/tex]

The sample of healthy adults selected from the region is n = 41 > 30.

Thus, the shape will the sampling distribution of the mean is normal.

(b)

Compute the mean of the sampling distribution as follows:

[tex]\mu_{\bar x}=\mu=210[/tex]

Thus, the mean of the sampling distribution is 210 mg/dL.

(c)

Compute the standard deviation of the sampling distribution as follows:

[tex]\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}=\frac{30}\sqrt{41}}=4.69[/tex]

Thus, the standard deviation of the sampling distribution is 4.69 mg/dL.