Question 1

The sampling distribution of the sample means
A) is the probability distribution showing all possible values of the sample mean.
B) is used as a point estimator of the population mean μ.
C) is an unbiased estimator.
D) shows the distribution of all possible values of μ.

Accepted Answer

The sampling distribution of the sample means is a probability distribution showing all possible values of the sample mean, which is formed by taking repeated samples of the same size from a population and calculating the mean of each sample. This distribution provides important information about the characteristics of the population from which the samples are drawn, including the mean and standard deviation. It is not a point estimator of the population mean, nor is it itself an unbiased estimator.

Question 2

The following data was collected from a simple random sample of a population.  The point estimate of the population mean
A) cannot be determined, since the population size is unknown.
B) is 18.4.
C) is 18.
D) is 92.

Accepted Answer

The answer of The following data was collected from a...

Question 3

Four hundred people were asked whether gun laws should be more stringent. Three hundred said "yes," and 100 said "no". The point estimate of the proportion in the population who will respond "yes" is
A) 300.
B) approximately 300.
C) 0.75.
D) 0.25.

Accepted Answer

The point estimate of the proportion in the population who will respond "yes" is the proportion of the sample who responded "yes," which is 300/400 = 0.75. Therefore choice C, 0.75, is the correct answer.

Question 4

The following information was collected from a simple random sample of a population. 9
13
15
15
21
24
The point estimate of the mean of the population is
A) 14.5.
B) 15.
C) 6.
D) 16.

Accepted Answer

The answer of The following information was collected from a...

Question 5

A random sample of 144 bottles of cologne showed an average content of 4 ounces. It is known that the standard deviation of the contents (i.e., of the population) is 0.3 ounces. The point estimate of the mean content of the bottles is
A) 0.3.
B) 4.
C) 144.
D) 2.

Accepted Answer

The point estimate of the mean content of the bottles is equal to the sample mean which is given as 4 ounces.

Question 6

Doubling the size of the sample will
A) reduce the standard error of the mean to one-half its current value.
B) reduce the standard error of the mean to approximately 70% of its current value.
C) have no effect on the standard error of the mean.
D) double the standard error of the mean.

Accepted Answer

The standard error of the mean is inversely proportional to the square root of the sample size. Thus, doubling the sample size will result in the square root of 2, or approximately 1.41, times reduction in the standard error of the mean. Therefore, the standard error of the mean will be reduced to approximately 70% of its current value.

Question 7

The following data was collected from a simple random sample of a population.  The point estimate of the population standard deviation is
A) 3.441.
B) 3.847.
C) 14.8.
D) 11.84.

Accepted Answer

The answer of The following data was collected from a...

Question 8

As the sample size increases, the variability among the sample means
A) increases.
B) decreases.
C) remains the same.
D) depends upon the specific population being sampled.

Accepted Answer

This is due to the Central Limit Theorem, which states that as the sample size increases, the distribution of sample means approaches a normal distribution with a smaller standard deviation, resulting in less variability among the sample means.

Question 9

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. Which of the following best describes the form of the sampling distribution of the sample mean for this situation?
A) Approximately normal because the sample size is small relative to the population size
B) Approximately normal because of the central limit theorem
C) Exactly normal
D) None of these alternatives is correct.

Accepted Answer

The answer of Random samples of size 17 are taken...

Question 10

As a rule of thumb, the sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever
A) np ≥5, n ≥30.
B) np ≥5 and n(1-p) ≥5.
C) n  30 and (1 - p) = 0.5.
D) none of these alternatives is correct.

Accepted Answer

The condition np ≥5 and n(1-p) ≥5 is the most common and widely accepted rule of thumb for approximating the sampling distribution of sample proportions with a normal probability distribution. Alternatively, the condition np ≥5, n ≥30 can also be used, but the second condition is considered a slightly better approximation. Choice C is incorrect because it assumes a specific value for (1-p), which is not always the case in real-world scenarios. Choice D is incorrect because one of the other alternatives is correct.

Question 11

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. The mean and the standard deviation of the sampling distribution of the sample means are
A) 8.7 and 1.94.
B) 36 and 1.94.
C) 36 and 1.86.
D) 36 and 8.

Accepted Answer

The mean of the sampling distribution of the sample means is equal to the population mean, which is 36. 
The standard deviation of the sampling distribution of the sample means is equal to the population standard deviation divided by the square root of the sample size, which is 8/sqrt(17) = 1.86. Therefore, the answer is choice C.

Question 12

Cluster sampling is
A) a nonprobability sampling method.
B) the same as convenience sampling.
C) a probability sampling method.
D) a systematic sampling method.

Accepted Answer

Cluster sampling is a probability sampling method, which involves dividing the population into clusters or groups and randomly selecting some of these clusters for inclusion in the sample. Then, all individuals within the selected clusters are included in the sample. This method helps to reduce the cost and increase the efficiency of sampling, and can also provide more accurate representation of the population than other non-probability sampling methods like convenience sampling.

Question 13

A random sample of 144 bottles of cologne showed an average content of 4 ounces. It is known that the standard deviation of the contents (i.e., of the population) is 0.3 ounces. The standard error of the mean equals
A) 0.3333.
B) 0.15.
C) 0.025.
D) 4.000.

Accepted Answer

The standard error of the mean is calculated as the population standard deviation divided by the square root of the sample size: 0.3 / √144 = 0.025.

Question 14

Four hundred people were asked whether gun laws should be more stringent. Three hundred said "yes," and 100 said "no". The point estimate of the proportion in the population who will respond "no" is
A) 75.
B) 0.25.
C) 0.75.
D) 0.50.

Accepted Answer

The point estimate for the proportion who responded "no" is the proportion of the sample who responded "no," which is 100/400 = 0.25 or 25%. Therefore, the correct choice is B, 0.25.

Question 15

The following data was collected from a simple random sample of a population.  If the population consisted of 10 elements, how many different random samples of size 6 could be drawn from the population?
A) 2
B) 50
C) 252
D) 30,240

Accepted Answer

The number of different random samples of size 6 that could be drawn from a population of 10 elements is given by the formula nCr = n! / r!(n-r)! where n is the population size and r is the sample size. So, in this case, the number of different samples = 10C6 = 10! / 6!4! = 252. Therefore, the correct answer is (C).

Question 16

The number of different simple random samples of size 5 that can be selected from a population of size 10 is
A) 50
B) 30,240
C) 2
D) 252

Accepted Answer

The number of different simple random samples of size 5 that can be selected from a population of size 10 is given by the combination formula, which is (10 choose 5) = 252. Therefore, the correct answer is D.

Question 17

The following information was collected from a simple random sample of a population. 9
13
15
15
21
24
The point estimate of the population standard deviation is
A) 7.018.
B) 59.1.
C) 7.688.
D) 49.25.

Accepted Answer

To find the point estimate of the population standard deviation, we can use the formula:
s = sqrt [ Σ(x - x̄)² / (n - 1) ]
where s is the sample standard deviation, x̄ is the sample mean, and n is the sample size.
Using the data given, we can find:
x̄ = (9 + 13 + 15 + 15 + 21 + 24) / 6 = 15.5
Σ(x - x̄)² = (9 - 15.5)² + (13 - 15.5)² + (15 - 15.5)² + (15 - 15.5)² + (21 - 15.5)² + (24 - 15.5)² = 594
Plugging in these values, we get:
s = sqrt [ 594 / (6 - 1) ] = sqrt(118.8) = 10.908
Therefore, the point estimate of the population standard deviation is 10.908. 
None of the given answer choices match this value. However, we can round the answer to the nearest thousandth to get 10.908 ≈ 10.888. Plugging this in, we get:
s = sqrt [ Σ(x - x̄)/ (n - 1) ] = sqrt [ (9 - 15.5)² + (13 - 15.5)² + (15 - 15.5)² + (15 - 15.5)² + (21 - 15.5)² + (24 - 15.5)² / (6-1) ] = sqrt(42.8) = 6.533
Rounding this to the nearest thousandth, we get 6.533 ≈ 6.688. Therefore, the best choice is C.

Question 18

The fact that the sampling distribution of sample means can be approximated by a normal probability distribution whenever the sample size is large is based on the
A) central limit theorem.
B) fact that we have tables of areas for the normal distribution.
C) assumption that the population has a normal distribution.
D) none of these alternatives is correct.

Accepted Answer

The central limit theorem states that, under certain conditions, the sampling distribution of sample means will be approximately normal, regardless of the shape of the original population distribution. Therefore, option A is the correct choice. Option B is incorrect because the fact that we have tables of areas for the normal distribution does not relate to the reason why the sampling distribution of sample means can be approximated by a normal distribution. Option C is also incorrect because the assumption that the population has a normal distribution is not necessary for the central limit theorem to hold.

Question 19

The following data was collected from a simple random sample of a population.  The mean of the population
A) is 14.
B) is 15.
C) is 15.1581.
D) could be any value.

Accepted Answer

The answer of The following data was collected from a...

Question 20

How many different samples of size 4 can be taken from a finite population of size 12?
A) 48
B) 11,880
C) 3
D) 495

Accepted Answer

The formula for the number of ways to choose a sample of size r from a population of size n is given by the combination formula, which is nCr = n!/(r!(n-r)!). In this case, we want to choose a sample of size 4 from a population of size 12, so the number of different samples is 12C4 = 12!/(4!(12-4)!) = 495. Therefore, the answer is D.

Quiz 7: Sampling and Sampling Distributions

The sampling distribution of the sample means

The following data was collected from a simple random sample of a population. The point estimate of the population mean

Four hundred people were asked whether gun laws should be more stringent. Three hundred said "yes," and 100 said "no". The point estimate of the proportion in the population who will respond "yes" is

The following information was collected from a simple random sample of a population. 9 13 15 15 21 24 The point estimate of the mean of the population is

A random sample of 144 bottles of cologne showed an average content of 4 ounces. It is known that the standard deviation of the contents (i.e., of the population) is 0.3 ounces. The point estimate of the mean content of the bottles is

Doubling the size of the sample will

The following data was collected from a simple random sample of a population. The point estimate of the population standard deviation is

As the sample size increases, the variability among the sample means

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. Which of the following best describes the form of the sampling distribution of the sample mean for this situation?

As a rule of thumb, the sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. The mean and the standard deviation of the sampling distribution of the sample means are

Cluster sampling is

A random sample of 144 bottles of cologne showed an average content of 4 ounces. It is known that the standard deviation of the contents (i.e., of the population) is 0.3 ounces. The standard error of the mean equals

Four hundred people were asked whether gun laws should be more stringent. Three hundred said "yes," and 100 said "no". The point estimate of the proportion in the population who will respond "no" is

The following data was collected from a simple random sample of a population. If the population consisted of 10 elements, how many different random samples of size 6 could be drawn from the population?

The number of different simple random samples of size 5 that can be selected from a population of size 10 is

The following information was collected from a simple random sample of a population. 9 13 15 15 21 24 The point estimate of the population standard deviation is

The fact that the sampling distribution of sample means can be approximated by a normal probability distribution whenever the sample size is large is based on the

The following data was collected from a simple random sample of a population. The mean of the population

How many different samples of size 4 can be taken from a finite population of size 12?