Exam 7: The Sampling Distribution of the Sample Mean

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean $\overline { \mathbf { X } } .$ -The mean and the standard deviation of the sampled population are, respectively, 83.6 and 5.5. $\mathrm { n } = 256$

The mean annual income for adult women in one city is $\$ 28,520$ and the standard deviation of the incomes is $\$ 5,190$ . The distribution of incomes is skewed to the right. For samples of size 30 , which of the following statements best describes the sampling distribution of the mean?

$\text { Use the given table to determine the mean, } \mu _ { \chi ^ { \prime } } ^ { - } \text {of the variable } \bar { x } \text { for the given sample size. }$ - Sample 4,5 4.5 4,6 5 4,7 5.5 4,8 6 5,6 5.5 5,7 6 5,8 6.5 6,7 6.5 6,8 7 7,8 7.5

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean $\overline { \mathbf { X } _ { * } }$ -The National Weather Service keeps records of rainfall in valleys. Records indicate that in a certain valley, the annual rainfall has a mean of 95 inches and a standard deviation of 12 inches. Suppose the rainfalls are sampled during randomly picked years and $\bar { x }$ is the mean amount of rain in these years. For samples of size 36 , determine the mean and standard deviation of $\bar { x }$ .

Find the requested probability. -The test scores of 5 students are under consideration. The following is the dotplot for the sampling distribution of the sample mean for samples of size 2. Find the probability, expressed as a percent, that the sample mean will be within 1 point of the population mean.

Provide an appropriate response. -As a general rule, you cannot expect to exactly determine the sampling distribution of a statistic. Why?

Find the indicated probability or percentage for the sampling error. -The monthly expenditures on food by single adults living in one neighborhood of Los Angeles are normally distributed with a mean of $410 and a standard deviation of $60. Determine the Percentage of samples of size 9 that will have mean monthly expenditures on food within $16 of The population mean expenditure of $410.

For the population of one town, the number of siblings is a random variable whose relative frequency histogram has a reverse J-shape. Let $\bar { x }$ denote the mean number of siblings for a random sample of size 30. For samples of size 30, which of the following statements concerning the sampling distribution of the mean is true?

Find the indicated probability or percentage for the sampling error. -The amount of coffee that a filling machine puts into an 8-ounce jar is normally distributed with a mean of 8.2 ounces and a standard deviation of 0.18 ounce. Determine the percentage of samples of Size 16 that will have mean amounts of coffee within 0.1 ounce of the population mean of 8.2 Ounces.

Draw the specified dotplot. -The heights (in inches)of 5 players on a basketball team are given in the table. Player A B C D E Height (inches) 65 78 72 68 57 Draw a dotplot for the sampling distribution of the sample mean for samples of size 2.

The weights of five players on a football team are shown below. Player A B C D E Weight (lb) 290 310 250 255 220 Consider these players to be a population of interest. The mean weight, $\mu$ , for the population is 265 pounds. Construct a table that shows all of the possible samples of size three. For each of the possible samples, list the players in the sample, their weights, and the sample mean. The first line of the table is shown below. Sample Weights A, B, C 290,310,250 283.3 Use your table to find the probability that, for a random sample of size three, the sample mean will be within 15 lb of the population mean.

Let $x$ represent the number which shows up when a balanced die is rolled. Then $x$ is a random variable with a uniform distribution. Let $\bar { x }$ denote the mean of the numbers obtained when the die is rolled 32 times. For samples of size 32, which of the following statements concerning the sampling distribution of the mean is true?

Draw the specified dotplot. -The heights (in inches)of 5 players on a basketball team are given in the table. Player A B C D E Height (inches) 66 69 72 69 72 Draw a dotplot for the sampling distribution of the sample mean for samples of size 3.

The mean height for a population of people is 65 inches. Suppose that you pick a sample of 50 people and determine the sample mean, $\bar { x }$ . You then repeat this procedure three more times. You learned in class that $\mu _ { - \bar { x } } = \mu$ . Can you conclude that the mean of the four sample means will be equal to the population mean of 65 inches? Why or why not?

The weights of five players on a football team are shown below. Player A B C D E Weight (lb) 290 310 250 255 220 Consider these players to be a population of interest. The mean weight, µ, for the population is 265 pounds. Construct a table that shows all of the possible samples of size two. For each of the possible samples, list the players in the sample, their weights, and the sample mean. The first line of the table is shown below. Sample Weights , 290,310 300 Use your table to find the probability that, for a random sample of size two, the sample mean will equal the population mean.

Provide an appropriate response. -Population data: 4, 5, 8, 9. a. Find the mean, $\mu$ , of the variable. b. Use the population data below to construct a table giving the sample means of sample size $n = 3$ . c. Draw a dotplot for the sampling distribution of the sample mean. d. Find the probability that the sample mean will equal the population mean. e. Find the probability that the sampling error made in estimating the population mean by the sampling mean will be $0.5$ or less (in magnitude), that is, that the absolute value of the difference between the sample mean and the population mean is at most $0.5$ .

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean $\bar { x }$ -The mean and the standard deviation of the sampled population are, respectively, $211.7$ and $21.9$ $\mathrm { n } = 64$

Provide an appropriate response. -What generally happens to the sampling distribution of the sample mean as the sample size is decreased?

Find the indicated probability or percentage for the sampling error. -The distribution of weekly salaries at a large company is right skewed with a mean of $1000 and a standard deviation of $350. What is the probability that the sampling error made in estimating the Mean weekly salary for all employees of the company by the mean of a random sample of weekly Salaries of 50 employees will be at most $50?

Find the requested probability. -The test scores of 5 students are under consideration. The following is the dotplot for the sampling distribution of the sample mean for samples of size 2. Find the probability, expressed as a percent, that the sample mean will be equal to the population Mean.