Question 1

Cluster sampling is
A) a nonprobability sampling method
B) the same as convenience sampling
C) a probability sampling method
D) None of these alternatives is correct.

Accepted Answer

Cluster sampling is a probability sampling method where the population is divided into clusters or groups, and then a random sample of clusters is selected. The selected clusters are then sampled exhaustively or randomly to obtain the final sample. Therefore, cluster sampling has a known probability of inclusion for each cluster.

Question 2

The following data was collected from a simple random sample of a population. 13 15 14 16 12 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 3

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 as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution regardless of the shape of the population distribution. Therefore, option A is the correct choice. Option B is incorrect because the existence of tables for the normal distribution does not affect whether the population distribution is normal or not. Option C is also incorrect because the assumption of normality is not necessary for the central limit theorem to apply. Option D is incorrect because the central limit theorem is a well-established statistical concept.

Question 4

As a rule of thumb, the sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever
A) np ≥ 5
B) n1 - p) ≥ 5 and n ≥ 30
C) n ≥ 30 and 1 - p) = 0.5
D) None of these alternatives is correct.

Accepted Answer

This is known as the normal approximation to the binomial distribution. The condition n(p)(1−p)≥10 should also be met for better accuracy. Choice A only considers the value of np, while choice C only considers the value of (1−p). Choice D is incorrect because there is a valid alternative.

Question 5

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

Question 6

The following data was collected from a simple random sample of a population. 13 15 14 16 12 The point estimate of the population standard deviation is
A) 2.500
B) 1.581
C) 2.000
D) 1.414

Accepted Answer

The point estimate of the population standard deviation can be found using the sample standard deviation formula: 
s = √(Σ(x - x̄)²/(n-1))

Plugging in the values from the sample: 
s = √((13-14.0)² + (15-14.0)² + (14-14.0)² + (16-14.0)² + (12-14.0)²)/(5-1)
s = √(10/4)
s = √2.5

Therefore, the best choice is B.

Question 7

A subset of a population selected to represent the population is
A) a subset
B) a sample
C) a small population
D) a parameter

Accepted Answer

A sample is a subset of a larger population that is selected to represent the population as a whole for research or statistical analysis.

Question 8

The following data was collected from a simple random sample of a population. 13 15 14 16 12 The point estimate of the population mean
A) cannot be determined, since the population size is unknown
B) is 14
C) is 4
D) is 5

Accepted Answer

The point estimate of the population mean can be calculated by finding the average of the sample, which is (13+15+14+16+12)/5 = 14. Therefore, the best choice is B.

Question 9

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 the probability distribution that shows all possible values of the sample mean for all possible samples of a given size taken from a population.

Question 10

The set of all elements of interest in a study is
A) set notation
B) a set of interest
C) a sample
D) a population

Accepted Answer

The set of all elements of interest in a study is known as the population. A sample is a subset of the population, so it is not the best choice for this question. Set notation refers to the way of representing sets using symbols, so it is not a valid answer choice. "A set of interest" is not a commonly used term in statistics and is not the best answer.

Question 11

A population of size 1,000 has a proportion of 0.5. Therefore, the proportion and the standard deviation of the sample proportion for samples of size 100 are
A) 500 and 0.047
B) 500 and 0.050
C) 0.5 and 0.047
D) 0.5 and 0.050

Accepted Answer

The proportion remains 0.5. The standard deviation of the sample proportion is calculated using the formula $\sqrt{\frac{p(1-p)}{n}}$, where $p = 0.5$ and $n = 100$. This gives $\sqrt{\frac{0.5(1-0.5)}{100}} = \sqrt{\frac{0.25}{100}} = \sqrt{0.0025} = 0.05$. However, there seems to be a mistake in my calculation as the correct standard deviation should be 0.05, aligning with option D, not C. My explanation incorrectly supports C due to a miscalculation. The correct answer, based on the correct calculation, should be D, not C.

Question 12

The number of different simple random samples of size 5 that can be selected from a population of size 8 is
A) 40
B) 336
C) 13
D) 56

Accepted Answer

The formula to find the number of different simple random samples of size k that can be selected from a population of size n is given by nCk or (n choose k), which is also written as C(n, k). In this case, we need to find the number of different simple random samples of size 5 that can be selected from a population of size 8. Therefore, the answer is C(8, 5) = 8!/(5!3!) = 56. Hence, the best choice is (D).

Question 13

A sample of 66 observations will be taken from an infinite population. The population proportion equals 0.12. The probability that the sample proportion will be less than 0.1768 is
A) 0.0568
B) 0.0778
C) 0.4222
D) 0.9222

Accepted Answer

The answer of A sample of 66 observations will be...

Question 14

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 decreases as the sample size increases. Specifically, it decreases proportionally to the square root of the sample size. Therefore, doubling the sample size will result in a square root of 2 (√2) increase in sample size, which is approximately 1.41. Therefore, the standard error of the mean will be reduced to approximately 70% (1/√2) of its current value.

Question 15

A sample of 25 observations is taken from an infinite population. The sampling distribution of is
A) not normal since n < 30
B) approximately normal because is always normally distributed
C) approximately normal if np ≥ 5 and n1-P) ≥ 5
D) approximately normal if np > 30 and n1-P) > 30

Accepted Answer

According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normal if the sample size is large enough and if certain conditions are met. These conditions include np ≥ 5 and n(1-P) ≥ 5, where n is the sample size and p is the probability of success. In this case, we do not have information about the value of p, but we can assume that the population is infinite. Therefore, we can use the conservative estimate of p = 0.5 to calculate the minimum sample size that satisfies the conditions. Plugging in the values, we get np = 25(0.5) = 12.5 and n(1-P) = 25(0.5) = 12.5, which both exceed the minimum threshold of 5. Therefore, we can conclude that the sampling distribution of the sample mean is approximately normal.

Question 16

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

As the sample size increases, the sample means tend to become more representative of the population mean, resulting in decreased variability among the sample means. This is due to the law of large numbers, which states that as the sample size increases, the sample mean approaches the population mean.

Question 17

A finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means
A) whenever the population is infinite
B) whenever the sample size is more than 5% of the population size
C) whenever the sample size is less than 5% of the population size
D) The correction factor is not necessary if the population has a normal distribution

Accepted Answer

The answer of A finite population correction factor is needed...

Question 18

The following data was collected from a simple random sample of a population. 13 15 14 16 12 If the population consisted of 10 elements, how many different random samples of size 6 could be drawn from the population?
A) 60
B) 210
C) 3024
D) 362880

Accepted Answer

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

Question 19

A sample of 51 observations will be taken from an infinite population. The population proportion equals 0.85. The probability that the sample proportion will be between 0.9115 and 0.946 is
A) 0.8633
B) 0.6900
C) 0.0819
D) 0.0345

Accepted Answer

The sample size is greater than 30, which means we can use the normal distribution to approximate the sampling distribution of the sample proportion. The mean of the sampling distribution is equal to the population proportion, which is 0.85. The standard deviation of the sampling distribution is equal to the square root of [(population proportion x (1 - population proportion)) / sample size], which is sqrt[(0.85 x 0.15) / 51] = 0.0617.

To find the probability that the sample proportion will be between 0.9115 and 0.946, we need to standardize these values using the sampling distribution.

Z1 = (0.9115 - 0.85) / 0.0617 = 1.00
Z2 = (0.946 - 0.85) / 0.0617 = 1.55

Using a standard normal distribution table or calculator, we can find the probability that Z is between 1.00 and 1.55, which is 0.0819. Therefore, the answer is C.

Question 20

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...

Quiz 7: Sampling and Sampling Distributions

Cluster sampling is

The following data was collected from a simple random sample of a population. 13 15 14 16 12 The mean of the population

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

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

The following data was collected from a simple random sample of a population. 13 15 14 16 12 The point estimate of the population standard deviation is

A subset of a population selected to represent the population is

The following data was collected from a simple random sample of a population. 13 15 14 16 12 The point estimate of the population mean

The sampling distribution of the sample means

The set of all elements of interest in a study is

A population of size 1,000 has a proportion of 0.5. Therefore, the proportion and the standard deviation of the sample proportion for samples of size 100 are

The number of different simple random samples of size 5 that can be selected from a population of size 8 is

A sample of 66 observations will be taken from an infinite population. The population proportion equals 0.12. The probability that the sample proportion will be less than 0.1768 is

Doubling the size of the sample will

A sample of 25 observations is taken from an infinite population. The sampling distribution of is

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

A finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means

The following data was collected from a simple random sample of a population. 13 15 14 16 12 If the population consisted of 10 elements, how many different random samples of size 6 could be drawn from the population?

A sample of 51 observations will be taken from an infinite population. The population proportion equals 0.85. The probability that the sample proportion will be between 0.9115 and 0.946 is

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?