Deck 7: The Sampling Distribution of the Sample Mean

Question 1

Provide an appropriate response.
SAT verbal scores are normally distributed with a mean of 430 and a standard deviation of
120 (based on data from the College Board ATP). Consider the following two problems,
which appear at a glance to be very similar. Which one requires the application of the
Central Limit Theorem, and in what way does the solution process differ between the two
problems?
(a)If a student is randomly selected, what is the probability that his or her score is above
500?
(b)If a sample of 35 students is selected randomly, what is the probability that the sample
mean will be above 500?

Answer 1

Question 2

Provide an appropriate response.

Answer 2

D

Question 3

Provide an appropriate response.
The typical computer random-number generator yields numbers in a uniform distribution
between 0 and 1, with a mean of 0.500 and a standard deviation of 0.289. Consider the
following two problems, which appear at a glance to be very similar. One can be solved
using the Central Limit Theorem. Which one and why?
(a)Suppose a sample of size 50 is randomly generated. Find the probability that the mean
is below 0.300. (b)Suppose a sample of size 15 is randomly generated. Find the probability
that the mean is below 0.300.

Answer 3

The first may be solved using the Central Limit Theorem. The second has too small a
sample size.

Question 4

Provide an appropriate response.

Question 5

Solve the problem.
The weights of five players on a football team are shown below. 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.

Question 6

Provide an appropriate response.
Population data: 4, 5, 8, 9.

Question 7

Provide an appropriate response.

Question 8

Provide an appropriate response.

Question 9

Provide an appropriate response.
Do you agree with Tony's reasoning below? Explain why you do or do not agree. Refer to
the Central Limit Theorem in your explanation.
Tony: "When a balanced die is rolled, each of the numbers 1, 2, 3, 4, 5, and 6 has an equal
chance of showing up. So, if I roll the die 50 times and find the mean of the 50 numbers, the
mean has the same chance of falling between 1 and 2 as it has of falling between 3 and 4."

Question 10

Provide an appropriate response.
The weights of five players on a football team are shown below. 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. Use your table to find the probability that, for a random sample of size two, the sample
mean will equal the population mean.

Question 11

Provide an appropriate response.

Question 12

Provide an appropriate response.

Question 13

Solve the problem.
The ages of six members on a board of directors of a nonprofit organization are shown
below.

Question 14

Provide an appropriate response.

Question 15

Solve the problem.
The ages of six members of a board of directors of a nonprofit organization are shown
below. Use your table to find the probability that, for a random sample of size two, the sample
mean will equal the population mean.

Question 16

Draw the specified dotplot.
The heights (in inches)of 5 players on a basketball team are given in the table.

Question 17

Solve the problem.
The weights of five players on a football team are shown below. Use your table to find the probability that, for a random sample of size four, the sample
mean will be within 10 lb of the population mean.

Question 18

Provide an appropriate response.

Question 19

Draw the specified dotplot.
The heights (in inches)of 5 players on a basketball team are given in the table.

Question 20

Solve the problem.
The ages of six members of a board of directors of a nonprofit organization are shown
below. Consider these board members to be a population of interest. The mean age, µ, for the
population is 47. Construct a table that shows all of the possible samples of size four. For
each of the possible samples, list the people in the sample, their ages, and the sample mean.
The first line of the table is shown below. Use your table to find the probability that, for a random sample of size four, the sample
mean will be within 4 years of the population mean.

Question 21

Draw the specified dotplot.
The heights (in inches)of 5 players on a basketball team are given in the table.

Question 22

Provide an appropriate response.
Consider the following two problems.
(a)A random-number generator yields numbers in a uniform distribution between 0 and 1
with a mean of 0.5 and a standard deviation of 0.289. You wish to find the probability that
the mean of a sample of 50 random numbers is greater than 0.6.
(b)Scores on an aptitude test are normally distributed with a mean of 82 and a standard
deviation of 11. You wish to find the probability that the score for a randomly selected
person is greater than 90.
Which of these two problems requires application of the Central Limit Theorem? Explain
your reasoning.

Question 23

Solve the problem.
The ages of six members on a board of directors of a nonprofit organization are shown
below.

Question 24

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean

Question 25

Solve the problem.
The weights of five players on a football team are shown below.

Question 26

Solve the problem.
The ages of six members of a board of directors of a nonprofit organization are shown
below. Use your table to find the probability that, for a random sample of size five, the sample
mean will be within 2 years of the population mean.

Question 27

Draw the specified dotplot.
The heights (in inches)of 5 players on a basketball team are given in the table.

Question 28

Provide an appropriate response.
What generally happens to the sampling error as the sample size is decreased?

A)It gets more predictable.
B)It gets larger.
C)It gets smaller.
D)It gets less predictable.

Question 29

Draw the specified dotplot.
The heights (in inches)of 5 players on a basketball team are given in the table.

Question 30

Solve the problem.
The weights of five players on a football team are shown below.

Question 31

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 $370 and a standard deviation of $80. Determine the
Percentage of samples of size 4 that will have mean monthly expenditures on food within $72 of
The population mean expenditure of $370.

A)46.41%
B)96.41%
C)63.18%
D)92.82%

Question 32

Provide an appropriate response.
Population data: 3, 4, 5, 6, 7.

Question 33

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean

Question 34

Provide an appropriate response.
Suppose that µ represents the mean height for a population of people. Suppose that you
use a sample mean, to estimate µ. Explain what is meant by sampling error in this
situation. Why does vary from one sample to the next? How can you reduce the likely
sampling error?

Question 35

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 5 points of the
Population mean.

A)90%
B)80%
C)95%
D)100%

Question 36

Provide an appropriate response.
A population of people has a mean height of 65 inches. Andrew picks a person at random
from the population and records his or her height. He repeats this procedure 49 times
more. Bob picks a sample of 30 people at random from the population and records the
mean height of the sample. He repeats this procedure 49 times more. Which set of numbers
(those recorded by Andrew or those recorded by Bob)do you think will have more
variability? Explain your reasoning.

Question 37

Solve the problem.
The ages of six members on a board of directors of a nonprofit organization are shown
below.

Question 38

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.

Question 39

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 2 points of the
Population mean.

A)20%
B)40%
C)30%
D)50%

Question 40

Solve the problem.
The weights of five players on a football team are shown below.

Question 41

Provide an appropriate response.

Question 42

Provide an appropriate response.
The mean height for a population is 65 inches and the standard deviation is 3 inches. Let A and B
denote the events described below.
Event A: The height of a randomly selected person is 5 inches or more from the population mean.
Event B: The mean height in a random sample of 16 people is 5 inches or more from the population
mean.
True or false, the probability of event A is greater than the probability of event B?

Question 43

Provide an appropriate response.
What is the sampling distribution of a statistic?

A)The distribution of observations of a variable in a sample for a given value of the statistic
B)The distribution of observations of the statistic for all possible sizes of samples from a population
C)The distribution of all possible sizes of samples from a population that can be used to make observations of the statistic
D)The distribution of all possible observations of the statistic for samples of a given size from a population

Question 44

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. What is the probability that the
Sampling error made in estimating the mean amount of coffee for all 8-ounce jars by the mean of a
Random sample of 100 jars will be at most 0.02 ounce?

A)0.7330
B)0.0938
C)0.0876
D)0.8665

Question 45

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

A)Many populations are not normal.
B)Many populations are too large.
C)Many populations are too small.
D)Many populations are not uniform.

Question 46

The mean annual income for adult women in one city is $28,520 and the standard deviation of the incomes is $5700. The distribution of incomes is skewed to the right. Determine the sampling
Distribution of the mean for samples of size 132.

A)Approximately normal, mean = $28,520, standard deviation = $5700
B)Normal, mean = $28,520, standard deviation = $43
C)Approximately normal, mean = $28,520, standard deviation = $496
D)Normal, mean = $28,520, standard deviation = $496

Question 47

Question 48

Provide an appropriate response.

Question 49

Find the indicated probability or percentage for the sampling error.
Scores on a chemistry final exam are normally distributed with a mean of 280 and a standard deviation of 50. Determine the percentage of samples of size 4 that will have mean scores within 35
Points of the population mean score of 280.

A)91.92%
B)51.60%
C)99.48%
D)83.84%

Question 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 within 1 point of the
Population mean.

A)10%
B)30%
C)20%
D)25%

Question 51

Provide an appropriate response.
How many different samples of size 2 can be obtained from a population of size 5?

A)5
B)1
C)2
D)10

Question 52

The National Weather Service keeps records of snowfall in mountain ranges. Records indicate that in a certain range, the annual snowfall has a mean of 106 inches and a standard deviation of 10
Inches. Suppose the snowfalls are sampled during randomly picked years. For samples of size 25,
Determine the mean and standard deviation of

Question 53

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.

A)5%
B)20%
C)30%
D)10%

Question 54

Question 55

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?

A)0.3124
B)0.6876
C)0.3438
D)Cannot be determined, because the distribution of the population is not normal.

Question 56

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

A)It is unaffected.
B)It becomes less tightly concentrated around the population mean.
C)It becomes more tightly concentrated around the population mean.
D)None of the above

Question 57

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.

A)21.28%
B)98.36%
C)57.62%
D)91.92%

Question 58

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.

A)42.46%
B)97.36%
C)71.23%
D)98.68%

Question 59

Provide an appropriate response.
The mean height for a population is 65 inches and the standard deviation is 3 inches. Let A and B
denote the events described below.
Event A: The height of a randomly selected person is within 3 inches of the population mean.
Event B: The mean height in a random sample of 16 people is within 3 inches of the population
mean.
True or false, the probability of event A is greater than the probability of event B?

Question 60

Identify the distribution of the sample mean. In particular, state whether the distribution of is normal or approximatelynormal and give its mean and standard deviation.

Question 61

Find the indicated probability or percentage for the sampling error.
The monthly expenditures on food by single adults in one city are normally distributed with a mean of $410 and a standard deviation of $70. What is the probability that the sampling error made
In estimating the mean monthly expenditure of all single adults in that city by the mean of a
Random sample of 90 such adults will be at most $10?

A)0.9131
B)0.9990
C)0.1114
D)0.8262

Question 62

Identify the distribution of the sample mean. In particular, state whether the distribution of is normal or approximatelynormal and give its mean and standard deviation.
The heights of people in a certain population are normally distributed with a mean of 65 inches and a standard deviation of 3.9 inches. Determine the sampling distribution of the mean for samples of
Size 44.

A)Normal, mean = 65 inches, standard deviation = 0.59 inches
B)Approximately normal, mean = 65 inches, standard deviation = 0.09 inches
C)Normal, mean = 65 inches, standard deviation = 3.9 inches
D)Normal, mean = 65 inches, standard deviation = 0.09 inches

Question 63

Find the indicated probability or percentage for the sampling error.
Scores on an aptitude test are distributed with a mean of 220 and a standard deviation of 30. The shape of the distribution is unspecified. What is the probability that the sampling error made in
Estimating the population mean by the mean of a random sample of 50 test scores will be at most
5 points?

A)0.881
B)0.135
C)0.762
D)Cannot be determined, because the distribution of the population is not known to be normal.

Question 64

Find the indicated probability or percentage for the sampling error.
The distribution of weekly salaries at a large company is reverse J-shaped with a mean of $1000 and a standard deviation of $370. 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 80 employees will be at most $75?

A)0.4649
B)0.0702
C)0.9298
D)Cannot be determined, because the distribution of the population is not normal.

Question 65

Identify the distribution of the sample mean. In particular, state whether the distribution of is normal or approximatelynormal and give its mean and standard deviation.
The weights of people in a certain population are normally distributed with a mean of 152 lb and a standard deviation of 22 lb. Determine the sampling distribution of the mean for samples of size 2.

A)Approximately normal, mean = 152 lb, standard deviation = 11 lb
B)Approximately normal, mean = 152 lb, standard deviation = 15.56 lb
C)Normal, mean = 152 lb, standard deviation = 22 lb
D)Normal, mean = 152 lb, standard deviation = 15.56 lb

Question 66

Provide an appropriate response.
The mean height for a population is 65 inches and the standard deviation is 3 inches. Let A and B
denote the events described below.
Event A: The mean height in a random sample of 16 people is within 1 inch of the population
mean.
Event B: The mean height in a random sample of 50 people is within 1 inch of the population mean.
True or false, the probability of event A is greater than the probability of event B?

Question 67

Find the requested probability.
The table reports the GPA for each of five students in a statistics class. For a random sample of size two, find the probability, expressed as a percent, that the sample mean will be within 0.1 of the population mean.

A)70%
B)80%
C)20%
D)30%

Question 68

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 3 points of the
Population mean.

A)50%
B)60%
C)70%
D)40%

Question 69

Find the indicated probability or percentage for the sampling error.
Scores on a biology final exam are normally distributed with a mean of 220 and a standard deviation of 24. Determine the percentage of samples of size 9 that will have mean scores within 12
Points of the population mean score of 220.

A)86.64%
B)13.36%
C)93.32%
D)38.30%

Question 70

Question 71

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean
The mean and the standard deviation of the sampled population are, respectively, 83.6 and 5.5.

Question 72

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 2 points of the
Population mean.

A)50%
B)30%
C)60%
D)40%

Question 73

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 3 points of the
Population mean.

A)72%
B)50%
C)60%
D)40%

Question 74

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean

Question 75

Find the requested probability.
The table reports the distribution of pocket money, in bills, of the 6 students in a statistics seminar. For a random sample of size two, find the probability, expressed as a percent rounded to the nearest tenth, that the sample mean will be within $1 of the population mean.

A)66.7%
B)73.3%
C)80.0%
D)78.6%

Question 76

For samples of the specified size from the population described, find the mean and standard deviation of the samplemean