## The Mean, Standard Deviation, and Sampling Distribution of the Sample Mean

This section gives several concrete examples of calculating the exact distributions of the sample mean. The corresponding means and standard deviations are computed for demonstration based on these distributions. Next, it discusses sampling distributions of sample means when the sample size is large. It also considers the case when the population is normal. Finally, it uses the central limit theorem for large sample approximations.

### The Sampling Distribution of the Sample Mean

#### Exercises

### Basic

1. A population has mean and standard deviation .

a. Find the mean and standard deviation of for samples of size .b. Find the probability that the mean of a sample of size will be within units of the population mean, that is, between and .

3. A population has mean and standard deviation .

a. Find the mean and standard deviation of for samples of size .

b. Find the probability that the mean of a sample of size will be less than .

5. A normally distributed population has mean and standard deviation .

a. Find the probability that a single randomly selected element of the population exceeds .

b. Find the mean and standard deviation of for samples of size .

c. Find the probability that the mean of a sample of size drawn from this population exceeds .

7. A population has mean and standard deviation .

a. Find the mean and standard deviation of for samples of size .

b. Find the probability that the mean of a sample of size will be more than .

9. A normally distributed population has mean and standard deviation .

a. Find the probability that a single randomly selected element of the population is between and .

b. Find the mean and standard deviation of for samples of size .

c. Find the probability that the mean of a sample of size drawn from this population is between and .

11. A population has mean and standard deviation .

a. Find the mean and standard deviation of for samples of size .

b.
Find the probability that the mean of a sample of size will
differ from the population mean by at least units, that is,
is either less than or more than . (Hint: One way to solve
the problem is to first find the probability of the complementary
event.)

### Applications

13. Suppose the mean number of days to germination of a variety of seed is , with standard deviation days. Find the probability that the mean germination time of a sample of seeds will be within day of the population mean.15. Suppose the mean amount of
cholesterol in eggs labeled "large" is milligrams, with standard
deviation milligrams. Find the probability that the mean amount
of cholesterol in a sample of eggs will be within
milligrams of the population mean.

17. Suppose speeds of vehicles on a particular stretch of roadway are normally distributed with mean mph and standard deviation mph.

a. Find the probability that the speed of a randomly selected vehicle is between and mph.

b. Find the probability that the mean speed of randomly selected vehicles is between and mph.

19. Suppose the mean cost across
the country of a -day supply of a generic drug is ,
with standard deviation . Find the probability that the mean
of a sample of prices of -day supplies of this drug will
be between and .

21. Scores on a common final exam in a large enrollment, multiple-section freshman course are normally distributed with mean and standard deviation .

a. Find the probability that the score on a randomly selected exam paper is between and .

b. Find the probability that the mean score of randomly selected exam papers is between and .

23. Suppose that in a certain
region of the country the mean duration of first marriages that end in
divorce is years, standard deviation years. Find the
probability that in a sample of divorces, the mean age of the
marriages is at most years.