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

The Sampling Distribution of the Sample Mean

Exercises

Basic

1. A population has mean 128 and standard deviation 22.

a. Find the mean and standard deviation of \bar{X} for samples of size 36.
b. Find the probability that the mean of a sample of size 36 will be within 10 units of the population mean, that is, between 118 and 138.

3. A population has mean 73.5 and standard deviation 2.5.

a. Find the mean and standard deviation of \bar{X} for samples of size 30.
b. Find the probability that the mean of a sample of size 30 will be less than 72.

5. A normally distributed population has mean 25.6 and standard deviation 3.3.

a. Find the probability that a single randomly selected element X of the population exceeds 30.
b. Find the mean and standard deviation of \bar{X} for samples of size 9.
c. Find the probability that the mean of a sample of size 9 drawn from this population exceeds 30.

7. A population has mean 557 and standard deviation 35.

a. Find the mean and standard deviation of \bar{X} for samples of size 50.
b. Find the probability that the mean of a sample of size 50 will be more than 570.

9. A normally distributed population has mean 1,214 and standard deviation 122.

a. Find the probability that a single randomly selected element X of the population is between 1,100 and 1,300.
b. Find the mean and standard deviation of \bar{X} for samples of size 25.
c. Find the probability that the mean of a sample of size 25 drawn from this population is between 1,100 and 1,300.

11. A population has mean 72 and standard deviation 6.

a. Find the mean and standard deviation of \bar{X} for samples of size 45.
b. Find the probability that the mean of a sample of size 45 will differ from the population mean 72 by at least 2 units, that is, is either less than 70 or more than 74. (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 22, with standard deviation 2.3 days. Find the probability that the mean germination time of a sample of 160 seeds will be within 0.5 day of the population mean.

15. Suppose the mean amount of cholesterol in eggs labeled "large" is 186 milligrams, with standard deviation 7 milligrams. Find the probability that the mean amount of cholesterol in a sample of 144 eggs will be within 2 milligrams of the population mean.

17. Suppose speeds of vehicles on a particular stretch of roadway are normally distributed with mean 36.6 mph and standard deviation 1.7 mph.

a. Find the probability that the speed X of a randomly selected vehicle is between 35 and 40 mph.
b. Find the probability that the mean speed \bar{X} of 20 randomly selected vehicles is between 35 and 40 mph.

19. Suppose the mean cost across the country of a 30-day supply of a generic drug is \$ 46.58, with standard deviation \$ 4.84. Find the probability that the mean of a sample of 100 prices of 30-day supplies of this drug will be between \$ 45 and \$ 50.

21. Scores on a common final exam in a large enrollment, multiple-section freshman course are normally distributed with mean 72.7 and standard deviation 13.1.

a. Find the probability that the score X on a randomly selected exam paper is between 70 and 80.
b. Find the probability that the mean score \bar{X} of 38 randomly selected exam papers is between 70 and 80.

23. Suppose that in a certain region of the country the mean duration of first marriages that end in divorce is 7.8 years, standard deviation 1.2 years. Find the probability that in a sample of 75 divorces, the mean age of the marriages is at most 8 years.


Additional Exercises

25. A high-speed packing machine can be set to deliver between 11 and 13 ounces of a liquid. For any delivery setting in this range the amount delivered is normally distributed with mean some amount \mu and with standard deviation 0.08 ounce. To calibrate the machine it is set to deliver a particular amount, many containers are filled, and 25 containers are randomly selected and the amount they contain is measured. Find the probability that the sample mean will be within 0.05 ounce of the actual mean amount being delivered to all containers.