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

## The Sampling Distribution of the Sample Mean

### 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.

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.