Deck 8: Sampling Distributions

Question 1

If the sampled population is exactly normally distributed, then the sampling distribution of is also expected to be normal, regardless of the sample size.

Answer 1

True

Question 2

As the sample size increases, the standard deviation of the sampling distribution increases.

Answer 2

False

Question 3

The sampling distribution of the sample mean is developed by repeatedly taking samples of size n and computing the sample means and reporting the resulting sample means in the form of a probability distribution.

Answer 3

True

Question 4

The standard deviation of the sampling distribution of the sample mean is σ.

Question 5

The Central Limit Theorem states that as sample size increases, the population distribution more closely approximates a normal distribution.

Question 6

A sample statistic is an unbiased point estimate of a population parameter if the mean of the populations of all possible values of the sample statistic equals the population parameter.

Question 7

The reason sample variance has a divisor of n − 1 rather than n is that it makes the variance an unbiased estimate of the population variance.

Question 8

If a population is known to be normally distributed, then the single sample mean must equal the population mean.

Question 9

For any sampled population, the population of all sample means is approximately normally distributed.

Question 10

A minimum-variance unbiased point estimate has a variance that is as small as or smaller than the variances of any other unbiased point estimate.

Question 11

If the sampled population distribution is skewed then, in most cases, the sampling distribution of the mean can be approximated by the normal distribution if the sample size, n, is at least 30.

Question 12

If a population is known to be normally distributed, then the sample standard deviation must equal σ.

Question 13

The mean of the sampling distribution of is always equal to the mean of the sampled population.

Question 14

The standard deviation of all possible sample proportions increases as the sample size increases.

Question 15

If p = .8 and n = 50, then we can conclude that the sampling distribution of is approximately a normal distribution.

Question 16

If p = .9 and n = 40, then we can conclude that the sampling distribution of is approximately a normal distribution.

Question 17

If we have a sample size of 100 and the estimate of the population proportion is .10, we can estimate the sampling distribution of with a normal distribution.

Question 18

The sampling distribution of must be a normal distribution with mean = 0 and standard deviation = 1.

Question 19

A sample size of 500 is sufficiently large to conclude that the sampling distribution of is a normal distribution, when the estimate of the population proportion is .995.

Question 20

The sampling distribution of a sample statistic is the probability distribution of the population of all possible values of the sample statistic.

Question 21

The quantity is called the finite population multiplier.

Question 22

The mean of the sampling distribution of the sample proportion is equal to ________ when the sample size is sufficiently large.

A) μ
B) p
C) p × n
D) (1 − p)

Question 23

The population of all sample proportions has a normal distribution if the sample size (n) is sufficiently large. The rule of thumb for ensuring that n is sufficiently large is

A) np ≥ 5.
B) n(1 − p) ≥ 5.
C) np ≤ 5.
D) n(1 − p) ≤ 5 and np ≤ 5.
E) np ≥ 5 and n(1 − p) ≥ 5.

Question 24

The notation for the standard deviation of the sampling distribution of the sample mean is ________.

A) or

B) σx
C) ^σ
/n
D) μ

Question 25

________ says that if the sample size is sufficiently large, then the sample means are approximately normally distributed.

A) Cluster sampling
B) Sampling error
C) Sampling distribution of the mean
D) The Central Limit Theorem

Question 26

The sample standard deviation s is an unbiased estimator of the population standard deviation σ.

Question 27

If the sampled population has a mean of 48 and standard deviation of 16, then respectively the mean and the standard deviation for the sampling distribution of for n = 16 are

A) 4 and 1.
B) 12 and 4.
C) 48 and 4.
D) 48 and 1.
E) 48 and 16.

Question 28

As the sample size ________, the variation of the sampling distribution of ________.

A) decreases, decreases
B) increases, remains the same
C) decreases, remains the same
D) increases, decreases
E) None of these answers is correct.

Question 29

According to the Central Limit Theorem, if a sample size is at least ________, then for most sampled populations, we can conclude that the sample means are approximately normal.

A) 25
B) 20
C) 30
D) 50

Question 30

If the sample size n is infinitely large, then s² is an unbiased estimator of σ².

Question 31

Consider a sampling distribution formed based on n = 3. The standard deviation of the population of all sample means is ________ less than the standard deviation of the population of individual measurements σ.

A) always
B) sometimes
C) never

Question 32

The Central Limit Theorem states that as the sample size increases, the distribution of the sample ________ approaches the normal distribution.

A) medians
B) means
C) standard deviations
D) variances

Question 33

There is no easy way to calculate an unbiased point estimate of σ.

Question 34

The spread of the sampling distribution of is ________ the spread of the corresponding population distribution sampling distribution.

A) larger than
B) smaller than
C) the same as
D) exactly 1/2

Question 35

For nonnormal populations, as the sample size (n) ________, the distribution of sample means approaches a(n) ________ distribution.

A) decreases, uniform
B) increases, normal
C) decreases, normal
D) increases, uniform
E) increases, exponential

Question 36

For large samples, the sampling distribution of is approximately normal with a mean of ________.

A) μ
B) μ/
C)
D) z

Question 37

Consider two population distributions labeled A and B. Distribution A is highly skewed and nonnormal, while distribution B is slightly skewed and near normal. In order for the sampling distributions of A and B to achieve the same degree of normality,

A) population A will require a larger sample size.
B) population B will require a larger sample size.
C) populations A and B will require the same sample size.
D) None of these answers is correct.

Question 38

If a population distribution is known to be normal, then it follows that

A) the sample mean must equal the population mean.
B) the sample mean must equal the population mean for large samples.
C) the sample standard deviation must equal the population standard deviation.
D) the sample mean must equal the population mean, the sample mean must equal the population mean for large samples, and the sample standard deviation must equal the population standard deviation.
E) None of these answers is correct.

Question 39

Whenever the population has a normal distribution, the sampling distribution of is a normal, or near normal, distribution

A) for only large sample sizes.
B) for only small sample sizes.
C) for any sample size.
D) for only samples of size 30 or more.

Question 40

If the sampled population is finite and at least ________ times larger than the sample size, we treat the population as infinite.

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

Question 41

An unbiased estimate of σ² is ________.

A) s
B) s²
C)
D) σ

Question 42

Packages of sugar bags for Sweeter Sugar Inc. have an average weight of 16 ounces and a standard deviation of .2 ounces. The weights of the sugar packages are normally distributed. What is the probability that 16 randomly selected packages will have a weight in excess of 16.075 ounces?

A) .0500
B) .3520
C) .9332
D) .0668

Question 43

Find P( < 25) if μ = 16 and = 4.

A) 1.000
B) 2.25
C) .9878
D) .0122

Question 44

The population of all ________ proportions is described by the sampling distribution of .

A) population
B) random
C) observed
D) sample

Question 45

Find P( > 2,510) if μ = 2,500 and = 7.

A) .0764
B) .9998
C) .9236
D) .0001

Question 46

The sampling distribution of the sample mean is a normal distribution for ________ sample sizes, regardless of the shape of the corresponding population distribution.

A) random
B) large
C) small

Question 47

A random sample of size 36 is taken from a population with a mean of 50 and a standard deviation of 5. The sampling distribution of ________.

A) cannot be determined
B) is skewed to the left
C) is approximately normal
D) is skewed to the right

Question 48

Find P( < 402), if μ = 400, σ = 200, and n = 100.

A) .8413
B) .4602
C) .5398
D) .1587

Question 49

As the sample size ________, the standard deviation of the population of all sample proportions increases.

A) increases
B) stays the same
C) is variable
D) decreases

Question 50

A theorem that allows us to use the normal probability distribution to approximate the sampling distribution of sample means and sample proportions whenever the sample size is large is known as ________.

A) cluster sampling
B) sampling error
C) sampling distribution of the mean
D) the Central Limit Theorem

Question 51

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of .3 inches. What is the probability that the average length of a steel sheet from a sample of 9 units is more than 29.95 inches long?

A) .8413
B) .6293
C) .3707
D) .1587

Question 52

A golf tournament organizer is attempting to determine whether hole (pin) placement has a significant impact on the average number of strokes for the 13th hole on a given golf course. Historically, the pin has been placed in the front right corner of the green and the historical mean number of strokes for the hole has been 4.25, with a standard deviation of 1.6 strokes. On a particular day during the most recent golf tournament, the organizer placed the hole (pin) in the back left corner of the green. Sixty-four golfers played the hole with the new placement on that day. Determine the probability of the sample average number of strokes exceeding 4.75 using the historical mean and standard deviation.

A) .9938
B) .4013
C) .0062
D) .3783

Question 53

Find P( < 35) if μ = 40, σ_x = 16, n = 16.

A) .9944
B) .4483
C) .5517
D) .1056

Question 54

The ________ is the probability distribution of the population of all possible sample means that could be obtained from all possible samples of the same size.

A) sample mean
B) sampling Distribution of Sample Mean
C) probability
D) observations

Question 55

Find σ_x, if μ = 400, P( < 396) = .0228, and n = 100.

A) .20
B) 200
C) 20
D) 2.00

Question 56

Find P( > 172), if μ = 175 and = 9.

A) .1587
B) .8413
C) .6293
D) .3707

Question 57

As the sample size increases, the variability of the sampling distribution of the mean ________.

A) increases
B) stays the same
C) is variable
D) decreases

Question 58

A random sample of size 1,000 is taken from a population where p = .20. Describe the sampling distribution of .

A) cannot be determined
B) approximately normal
C) skewed to the left
D) skewed to the right

Question 59

Find P(395.4 < < 404.6), if the population mean = 400, σ = 20, and n = 100.

A) .9786
B) .9999
C) .0214
D) .9893

Question 60

The ________ is a minimum-variance unbiased point estimate of the mean of a normally distributed population.

A) sample mean
B) sample variance
C) sample standard deviation
D) observed mean

Question 61

A random sample of size 1,000 is taken from a population where p = .20. Find P( < .22).

A) .2643
B) .9431
C) .9207
D) .0571

Question 62

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5. Find P( < 51.5).

A) .9641
B) .0359
C) .1389
D) .9999

Question 63

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of .2 inches. A sample of 4 metal sheets is randomly selected from a batch. What is the probability that the average length of a sheet is between 30.25 and 30.35 inches long?

A) .9773
B) .0227
C) .0386
D) .0214

Question 64

The number of defectives in 10 different samples of 50 observations each is the following: 5, 1, 1, 2, 3, 3, 1, 4, 2, 3. What is the estimate of the population proportion of defectives?

A) .25
B) .50
C) .05
D) .42

Question 65

A random sample of size 1,000 is taken from a population where p = .20. Find P( > .21).

A) .2146
B) .0239
C) .9761
D) .7852

Question 66

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of .3 inches. What is the probability that the average length of a steel sheet from a sample of 9 steel sheets is more than 29.95 inches long?

A) .4602
B) .8413
C) .1587
D) .5397

Question 67

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5. Find P( < 48).

A) .0082
B) .8330
C) .0999
D) .1389

Question 68

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of .2 inches. What is the probability that a randomly selected sample of 4 sheets will have an average length of less than 29.9 inches long?

A) .0668
B) .9332
C) .0014
D) .4404

Question 69

The number of defectives in 10 different samples of 100 observations each is the following: 1, 2, 1, 0, 2, 3, 1, 4, 2, 1. What is the estimate of the population proportion of defectives?

A) .017
B) .17
C) .016
D) .16

Question 70

A random sample of size 1,000 is taken from a population where p = .20. What is ?

A) .006
B) .20
C) .02
D) .16

Question 71

Packages of sugar bags for Sweeter Sugar Inc. have an average weight of 16 ounces and a standard deviation of .3 ounces. The weights of the sugar packages are normally distributed. What is the probability that 9 randomly selected packages will have an average weight in excess of 16.025 ounces?

A) .5987
B) .0062
C) .4013
D) .9938

Question 72

Suppose that 60 percent of the voters in a particular region support a candidate. Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 4 percentage points of the actual proportion.

A) .0155
B) .9952
C) .9484
D) .9902

Question 73

A random sample of size 1,000 is taken from a population where p = .20. Find P( > .175).

A) .9759
B) .0239
C) .0392
D) .9999

Question 74

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5. Find P( > 50.5).

A) .0002
B) .7257
C) .2743
D) .1389

Question 75

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5. Find P( > 49).

A) .8331
B) .1151
C) .8849
D) .1389

Question 76

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5. What is μ_x?

A) 50
B) 5
C) 8.33
D) .833

Question 77

A random sample of size 36 is taken from a population with mean 50 and standard deviation 5. What is ?

A) .1389
B) 5
C) 8.33
D) .833

Question 78

Suppose that 60 percent of the voters in a particular region support a candidate. Find the probability that a sample of 1,000 voters would yield a sample proportion in favor of the candidate within 2 percentage points.

A) .9015
B) .8033
C) .0155
D) .7939

Question 79

A random sample of size 1,000 is taken from a population where p = .20. Find P( < .18).

A) .9429
B) .0569
C) .2643
D) .0793

Question 80

A random sample of size 1,000 is taken from a population where p = .20. What is ?

A) .0051
B) .03162
C) .01414
D) .01265