Question

According the National Center for Health Statistics, the distribution of serum cholesterol levels for 20- to 74-year-old males living in the United States has mean 211 mg/dl, and a standard deviation of 46 mg/dl.

a. We are planning to collect a sample of 25 individuals and measure their cholesterol levels. What is the probability that our sample average will be above 230?

b. Which numbers are 1.96 standard errors away rom the expected value of the sampling distributions(i.e. 95% of sample averages will fall between what two numbers?) Show all work.

c. Let's say we had only collected samples of 10. What is the standard error now? What is the probability that our sample average will be above 230?

d. What tow numbers would contain 95% of our sample averages now? How does this compare with #2? Show all work.

e. Now consider a sample of 50. What two numbers would contain 95% of our sample averages? How does this compare with #2 and #4? Show all work.

f. How large would the sample size need to be in order to ensure a 95% confidence interval will have a length of 10 mg/dl? Show all work.

g. How large would the sample size need to be in order to ensure that the bounds of a 90% confidence interval will be within 5 mg/dl of the sample mean? Show all work.

Answer 1

Let X is a random variable shows the serum cholesterol levels for 20- to 74-year-old males living in the United States. Here X has normal distribution with following parameters:

(a)

Sample size: n=25

The- z-score for is

The probability that our sample average will be above 230 is

Excel function used: "=1-NORMSDIST(2.07)"

(b)

Here we need to find the sampel averages for which z-scores are -1.96 and 1.96.

Lower limit:

Upper limit:

Hence, required two numbers are 192.968 and 229.032.

(c)

Sample size: n=10

The standard error is:

The- z-score for is

The probability that our sample average will be above 230 is

Excel function used: "=1-NORMSDIST(1.31)"

(d)

Here we need to find the sampel averages for which z-scores are -1.96 and 1.96.

Lower limit:

Upper limit:

Hence, required two numbers are 182.489 and 239.511.

As the sample size decreases, width of interval increases.

(e)

Here we need to find the sampel averages for which z-scores are -1.96 and 1.96.

Lower limit:

Upper limit:

Hence, required two numbers are 198.249 and 223.7505.

As the sample size decreases, width of interval increases.

(f)

For 95% confidence interval, critical value is 1.96.

The required sample size is:

hence, required sample size 326.

(g)

For 90% confidence interval, critical value is 1.645.

The required sample size is:

hence, required sample size 230.