## 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.

### 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.