Question: Among adults in the United States, the distribution of albumin levels (albumin is a type of protein) in cerebrospinal fluid is roughly symmetric with mean μ = 29.5 mg/100 ml and standard deviation σ = 9.25 mg/100 ml. Suppose that you select repeated samples of size 20 and calculate the mean for each sample. (Note: the units of albumin are "mg/100ml".)

1. (4 pts) If you were to select a very large number of random samples of size 20, what would (approximately) be the mean of these sample means?
2. (4 pts) What would be the standard deviation of these sample means? What is another name for this quantity?
3. (4 pts) How does the standard deviation of the sample means compare to the standard deviation of the albumin levels themselves?
4. (5 pts) If you were to take all the different sample means and use them to construct a histogram, what would be the shape of the distribution? How do you know?
5. (8 pts) What proportion of the means of samples of size 20 will be greater than 33 mg/100 ml?
6. (8 pts) What proportion of the means of samples of size 20 will be less than 28 mg/100 ml?
7. (8 pts) What proportion of the means of samples of size 20 will be between 29 and 31 mg/100 ml?
8. (10 pts) Barry Biostat takes one sample of size 20 and finds that the sample mean is 32.8. Calculate a 95% confidence interval for Barry. (Assume that Barry knows the true standard deviation.)
9. (5 pts) Does Barry’s confidence interval in (h) contain the true parameter value? Comment on whether or not this result is surprising.

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