Chapter 7 If the expected value of a sample statistic is equal to the parameter

Sampling Distributions 7-1

CHAPTER 7: SAMPLING DISTRIBUTIONS

1. Sampling distributions describe the distribution of

a) parameters.

b) statistics.

c) both parameters and statistics.

d) neither parameters nor statistics.

2. The standard error of the mean

a) is never larger than the standard deviation of the population.

b) decreases as the sample size increases.

c) measures the variability of the mean from sample to sample.

d) All of the above.

3. The Central Limit Theorem is important in statistics because

a) for a large n, it says the population is approximately normal.

b) for any population, it says the sampling distribution of the sample mean is approximately

normal, regardless of the sample size.

c) for a large n, it says the sampling distribution of the sample mean is approximately

normal, regardless of the shape of the population.

d) for any sized sample, it says the sampling distribution of the sample mean is

approximately normal.

4. If the expected value of a sample statistic is equal to the parameter it is estimating, then we call

that sample statistic

a) unbiased.

b) minimum variance.

c) biased.

d) random.

7-2 Sampling Distributions

5. For air travelers, one of the biggest complaints is of the waiting time between when the airplane

taxis away from the terminal until the flight takes off. This waiting time is known to have a right

skewed distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose

100 flights have been randomly sampled. Describe the sampling distribution of the mean waiting

time between when the airplane taxis away from the terminal until the flight takes off for these

100 flights.

a) Distribution is right skewed with mean = 10 minutes and standard error = 0.8 minutes.

b) Distribution is right skewed with mean = 10 minutes and standard error = 8 minutes.

c) Distribution is approximately normal with mean = 10 minutes and standard error = 0.8

minutes.

d) Distribution is approximately normal with mean = 10 minutes and standard error = 8

minutes.

6. Which of the following statements about the sampling distribution of the sample mean is

incorrect?

a) The sampling distribution of the sample mean is approximately normal whenever the

sample size is sufficiently large (n≥30 ).

b) The sampling distribution of the sample mean is generated by repeatedly taking samples

of size n and computing the sample means.

c) The mean of the sampling distribution of the sample mean is equal to

μ

.

d) The standard deviation of the sampling distribution of the sample mean is equal to

σ

.

7. Which of the following is true about the sampling distribution of the sample mean?

a) The mean of the sampling distribution is always

μ

.

b) The standard deviation of the sampling distribution is always

σ

.

c) The shape of the sampling distribution is always approximately normal.

d) All of the above are true.

Sampling Distributions 7-3

8. True or False: The amount of time it takes to complete an examination has a left skewed

distribution with a mean of 65 minutes and a standard deviation of 8 minutes. If 64 students were

randomly sampled, the probability that the sample mean of the sampled students exceeds 71

minutes is approximately 0.

9. Suppose the ages of students in Statistics 101 follow a right skewed distribution with a mean of

23 years and a standard deviation of 3 years. If we randomly sampled 100 students, which of the

following statements about the sampling distribution of the sample mean age is incorrect?

a) The mean of the sampling distribution is equal to 23 years.

b) The standard deviation of the sampling distribution is equal to 3 years.

c) The shape of the sampling distribution is approximately normal.

d) The standard error of the sampling distribution is equal to 0.3 years.

10. Why is the Central Limit Theorem so important to the study of sampling distributions?

a) It allows us to disregard the size of the sample selected when the population is not

normal.

b) It allows us to disregard the shape of the sampling distribution when the size of the

population is large.

c) It allows us to disregard the size of the population we are sampling from.

d) It allows us to disregard the shape of the population when n is large.

11. A sample that does not provide a good representation of the population from which it was

collected is referred to as a(n) sample.

7-4 Sampling Distributions

12. True or False: The Central Limit Theorem is considered powerful in statistics because it works

for any population distribution provided the sample size is sufficiently large and the population

mean and standard deviation are known.

13. Suppose a sample of n = 50 items is selected from a population of manufactured products and the

weight, X, of each item is recorded. Prior experience has shown that the weight has a probability

distribution with

μ

= 6 ounces and

σ

= 2.5 ounces. Which of the following is true about the

sampling distribution of the sample mean if a sample of size 15 is selected?

a) The mean of the sampling distribution is 6 ounces.

b) The standard deviation of the sampling distribution is 2.5 ounces.

c) The shape of the sampling distribution is approximately normal.

d) All of the above are correct.

14. The mean score of all pro golfers for a particular course has a mean of 70 and a standard

deviation of 3.0. Suppose 36 pro golfers played the course today. Find the probability that the

mean score of the 36 pro golfers exceeded 71.

15. The distribution of the number of loaves of bread sold per week by a large bakery over the past 5

years has a mean of 7,750 and a standard deviation of 145 loaves. Suppose a random sample of n

= 40 weeks has been selected. What is the approximate probability that the mean number of

loaves sold in the sampled weeks exceeds 7,895 loaves?

Sampling Distributions 7-5

16. Sales prices of baseball cards from the 1960s are known to possess a right skewed distribution

with a mean sale price of $5.25 and a standard deviation of $2.80. Suppose a random sample of

100 cards from the 1960s is selected. Describe the sampling distribution for the sample mean sale

price of the selected cards.

a) Right skewed with a mean of $5.25 and a standard error of $2.80

b) Normal with a mean of $5.25 and a standard error of $0.28

c) Right skewed with a mean of $5.25 and a standard error of $0.28

d) Normal with a mean of $5.25 and a standard error of $2.80

17. Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in

a certain year in the past. Suppose a sample of 100 major league players was taken. What was the

standard error for the sample mean salary?

a) $0.012 million

b) $0.12 million

c) $12 million

d) $1,200.0 million

18. Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in

a certain year in the past. Suppose a sample of 100 major league players was taken. Find the

approximate probability that the mean salary of the 100 players exceeded $3.5 million.

a) Approximately 0

b) 0.0228

c) 0.9772

d) Approximately 1

7-6 Sampling Distributions

19. Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in

a certain year in the past. Suppose a sample of 100 major league players was taken. Find the

approximate probability that the mean salary of the 100 players exceeded $4.0 million.

a) Approximately 0

b) 0.0228

c) 0.9772

d) Approximately 1

20. Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in

a certain year in the past. Suppose a sample of 100 major league players was taken. Find the

approximate probability that the mean salary of the 100 players was no more than $3.0 million.

a) Approximately 0

b) 0.0151

c) 0.9849

d) Approximately 1

21. Major league baseball salaries averaged $3.26 million with a standard deviation of $1.2 million in

a certain year in the past. Suppose a sample of 100 major league players was taken. Find the

approximate probability that the mean salary of the 100 players was less than $2.5 million.

a) Approximately 0

b) 0.0151

c) 0.9849

d) Approximately 1

Sampling Distributions 7-7

22. At a computer manufacturing company, the actual size of a particular type of computer chips is

normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A

random sample of 12 computer chips is taken. What is the standard error for the sample mean?

a) 0.029

b) 0.050

c) 0.091

d) 0.120

23. At a computer manufacturing company, the actual size of a particular type of computer chips is

normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A

random sample of 12 computer chips is taken. What is the probability that the sample mean will

be between 0.99 and 1.01 centimeters?

24. At a computer manufacturing company, the actual size of a particular type of computer chips is

normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A

random sample of 12 computer chips is taken. What is the probability that the sample mean will

be below 0.95 centimeters?

25. At a computer manufacturing company, the actual size of a particular type of computer chips is

normally distributed with a mean of 1 centimeter and a standard deviation of 0.1 centimeter. A

random sample of 12 computer chips is taken. Above what value do 2.5% of the sample means

fall?

7-8 Sampling Distributions

26. The owner of a fish market has an assistant who has determined that the weights of catfish are

normally distributed, with mean of 3.2 pounds and standard deviation of 0.8 pound. If a sample of

16 fish is taken, what would the standard error of the mean weight equal?

a) 0.003

b) 0.050

c) 0.200

d) 0.800

27. The owner of a fish market has an assistant who has determined that the weights of catfish are

normally distributed, with mean of 3.2 pounds and standard deviation of 0.8 pound. If a sample of

25 fish yields a mean of 3.6 pounds, what is the Z-score for this observation?

a) 18.750

b) 2.500

c) 1.875

d) 0.750

28. The owner of a fish market has an assistant who has determined that the weights of catfish are

normally distributed, with mean of 3.2 pounds and standard deviation of 0.8 pound. If a sample of

64 fish yields a mean of 3.4 pounds, what is probability of obtaining a sample mean this large or

larger?

a) 0.0001

b) 0.0013

c) 0.0228

d) 0.4987

Sampling Distributions 7-9

29. The owner of a fish market has an assistant who has determined that the weights of catfish are

normally distributed, with mean of 3.2 pounds and standard deviation of 0.8 pound. What

percentage of samples of 4 fish will have sample means between 3.0 and 4.0 pounds?

a) 84%

b) 67%

c) 29%

d) 16%

30. For sample size 16, the sampling distribution of the mean will be approximately normally

distributed

a) regardless of the shape of the population.

b) if the shape of the population is symmetrical.

c) if the sample standard deviation is known.

d) if the sample is normally distributed.

31. The standard error of the mean for a sample of 100 is 30. In order to cut the standard error of the

mean to 15, we would

a) increase the sample size to 200.

b) increase the sample size to 400.

c) decrease the sample size to 50.

d) decrease the sample to 25.

32. Which of the following is true regarding the sampling distribution of the mean for a large sample

size?

a) It has the same shape, mean, and standard deviation as the population.

b) It has a normal distribution with the same mean and standard deviation as the population.

c) It has the same shape and mean as the population, but has a smaller standard deviation.

d) It has a normal distribution with the same mean as the population but with a smaller

standard deviation.

7-10 Sampling Distributions

33. For sample sizes greater than 30, the sampling distribution of the mean will be approximately

normally distributed

a) regardless of the shape of the population.

b) only if the shape of the population is symmetrical.

c) only if the standard deviation of the samples are known.

d) only if the population is normally distributed.

34. For sample size 1, the sampling distribution of the mean will be normally distributed

a) regardless of the shape of the population.

b) only if the shape of the population is symmetrical.

c) only if the population values are positive.

d) only if the population is normally distributed.

35. The standard error of the population proportion will become larger

a) as population proportion approaches 0.

b) as population proportion approaches 0.50.

c) as population proportion approaches 1.00.

d) as the sample size increases.

36. True or False: As the sample size increases, the standard error of the mean increases.

37. True or False: If the population distribution is symmetric, the sampling distribution of the mean

can be approximated by the normal distribution if the samples contain 15 observations.

Sampling Distributions 7-11

38. True or False: If the population distribution is unknown, in most cases the sampling distribution

of the mean can be approximated by the normal distribution if the samples contain at least 30

observations.

39. True or False: If the amount of gasoline purchased per car at a large service station has a

population mean of 15 gallons and a population standard deviation of 4 gallons, then 99.73% of

all cars will purchase between 3 and 27 gallons.

40. True or False: If the amount of gasoline purchased per car at a large service station has a

population mean of 15 gallons and a population standard deviation of 4 gallons and a random

sample of 4 cars is selected, there is approximately a 68.26% chance that the sample mean will be

between 13 and 17 gallons.

41. True or False: If the amount of gasoline purchased per car at a large service station has a

population mean of 15 gallons and a population standard deviation of 4 gallons and it is assumed

that the amount of gasoline purchased per car is symmetric, there is approximately a 68.26%

chance that a random sample of 16 cars will have a sample mean between 14 and 16 gallons.

42. True or False: If the amount of gasoline purchased per car at a large service station has a

population mean of 15 gallons and a population standard deviation of 4 gallons and a random

sample of 64 cars is selected, there is approximately a 95.44% chance that the sample mean will

be between 14 and 16 gallons.

7-12 Sampling Distributions

43. True or False: As the sample size increases, the effect of an extreme value on the sample mean

becomes smaller.

44. True or False: If the population distribution is skewed, in most cases the sampling distribution

of the mean can be approximated by the normal distribution if the samples contain at least 30

observations.

45. True or False: A sampling distribution is a distribution for a statistic.

46. True or False: Suppose

μ

= 50 and

σ

= 10 for a population. In a sample where n = 100 is

randomly taken, 95% of all possible sample means will fall between 48.04 and 51.96.

47. True or False: Suppose

μ

= 80 and

σ

= 20 for a population. In a sample where n = 100 is

randomly taken, 95% of all possible sample means will fall above 76.71.

48. True or False: Suppose

μ

= 50 and

σ

= 10 for a population. In a sample where n = 100 is

randomly taken, 90% of all possible sample means will fall between 49 and 51.

Sampling Distributions 7-13

49. True or False: The Central Limit Theorem ensures that the sampling distribution of the sample

mean approaches a normal distribution as the sample size increases.

50. True or False: The standard error of the mean is also known as the standard deviation of the

sampling distribution of the sample mean.

51. True or False: A sampling distribution is defined as the probability distribution of possible

sample sizes that can be observed from a given population.

52. True or False: As the size of the sample is increased, the standard deviation of the sampling

distribution of the sample mean for a normally distributed population will stay the same.

53. True or False: For distributions such as the normal distribution, the arithmetic mean is

considered more stable from sample to sample than other measures of central tendency.

54. True or False: The fact that the sample means are less variable than the population data can be

observed from the standard error of the mean.

7-14 Sampling Distributions

55. The amount of tea leaves in a can from a particular production line is normally distributed with

μ

= 110 grams and

σ

= 25 grams. A sample of 25 cans is to be selected. What is the probability

that the sample mean will be between 100 and 120 grams?

56. The amount of tea leaves in a can from a particular production line is normally distributed with

μ

= 110 grams and

σ

= 25 grams. A sample of 25 cans is to be selected. What is the

probability that the sample mean will be less than 100 grams?

57. The amount of tea leaves in a can from a particular production line is normally distributed with

μ

= 110 grams and

σ

= 25 grams. A sample of 25 cans is to be selected. What is the probability

that the sample mean will be greater than 100 grams?

58. The amount of tea leaves in a can from a particular production line is normally distributed with

μ

= 110 grams and

σ

= 25 grams. A sample of 25 cans is to be selected. So, 95% of all sample

means will be greater than how many grams?

59. The amount of tea leaves in a can from a particular production line is normally distributed with

μ

= 110 grams and

σ

= 25 grams. A sample of 25 cans is to be selected. So, the middle 70% of

all sample means will fall between what two values?

Sampling Distributions 7-15

60. The amount of time required for an oil and filter change on an automobile is normally

distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample

of 16 cars is selected. What is the standard error of the mean?

61. The amount of time required for an oil and filter change on an automobile is normally

distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample

of 16 cars is selected. What is the probability that the sample mean is between 45 and 52

minutes?

62. The amount of time required for an oil and filter change on an automobile is normally

distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample

of 16 cars is selected. What is the probability that the sample mean will be between 39 and 48

minutes?

63. The amount of time required for an oil and filter change on an automobile is normally

distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample

of 16 cars is selected. 95% of all sample means will fall between what two values?

64. The amount of time required for an oil and filter change on an automobile is normally

distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample

of 16 cars is selected. 90% of the sample means will be greater than what value?

7-16 Sampling Distributions

65. True or False: The amount of bleach a machine pours into bottles has a mean of 36 oz. with a

standard deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this

machine. The sampling distribution of the sample mean has a mean of 36 oz.

66. True or False: The amount of bleach a machine pours into bottles has a mean of 36 oz. with a

standard deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this

machine. The sampling distribution of the sample mean has a standard error of 0.15.

67. True or False: The amount of bleach a machine pours into bottles has a mean of 36 oz. with a

standard deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this

machine. The sampling distribution of the sample mean will be approximately normal only if the

population sampled is normal.

68. The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard

deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this machine. The

probability that the mean of the sample exceeds 36.01 oz. is __________.

69. The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard

deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this machine. The

probability that the mean of the sample is less than 36.03 is __________.

Sampling Distributions 7-17

70. The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard

deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this machine. The

probability that the mean of the sample is between 35.94 and 36.06 oz. is __________.

71. The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard

deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this machine. The

probability that the mean of the sample is between 35.95 and 35.98 oz. is __________.

72. The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard

deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this machine. So,

the middle 95% of the sample means based on samples of size 36 will be between __________

and __________.

73. A manufacturer of power tools claims that the mean amount of time required to assemble their