Question 1

The purpose of statistical inference is to provide information about the
A)sample based upon information contained in the population
B)population based upon information contained in the sample
C)population based upon information contained in the population
D)mean of the sample based upon the mean of the population

Accepted Answer

Statistical inference uses information from a sample to make inferences about the larger population. Therefore, the purpose of statistical inference is to provide information about the population based upon information contained in the sample.

Question 2

From a population of 500 elements, a sample of 225 elements is selected. It is known that the variance of the population is 900. The standard error of the mean is approximately
A)1.1022
B)2
C)30
D)1.4847

Accepted Answer

The formula for standard error of the mean is:
SE = (population standard deviation) / √(sample size)
We are given the population variance, which is the square of the standard deviation, so the population standard deviation is √900 = 30.
Plugging in the values, we get:
SE = 30 / √225 = 30 / 15 = 2
Therefore, the standard error of the mean is approximately 2, so the answer is (B).

Question 3

A simple random sample of 64 observations was taken from a large population. The sample mean and the standard deviation were determined to be 320 and 120 respectively. The standard error of the mean is
A)1.875
B)40
C)5
D)15

Accepted Answer

The answer of A simple random sample of 64 observations...

Question 4

A random sample of 150 people was taken from a very large population. Ninety of the people in the sample were female. The standard error of the proportion is
A)0.0016
B)0.2400
C)0.1600
D)0.0400

Accepted Answer

The standard error of the proportion is calculated using the formula SE = sqrt[p(1-p)/n], where p is the sample proportion and n is the sample size. Here, p = 90/150 = 0.6, and n = 150. So, SE = sqrt[0.6(1-0.6)/150] = 0.0400.

Question 5

The sample statistic, such as  , s, or  , that provides the point estimate of the population parameter is known as
A)a point estimator
B)a parameter
C)a population parameter
D)a population statistic

Accepted Answer

A point estimator is a sample statistic that is used to estimate an unknown population parameter. This includes examples like the ones given in the question. The other answer options are incorrect.

Question 6

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 200 and 18, respectively. The distribution of the population is unknown. The mean and the standard error of the mean are
A)200 and 18
B)81 and 18
C)9 and 2
D)200 and 2

Accepted Answer

The mean of the population is 200, which will also be the mean of the sampling distribution of the mean.
The standard error of the mean (SEM) will be the standard deviation of the sampling distribution, which is equal to the population standard deviation divided by the square root of the sample size:
SEM = 18 / sqrt(81) = 2.

Therefore, the answer is D - 200 and 2.

Question 7

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 the
A)approximation theorem
B)normal probability theorem
C)central limit theorem
D)central normality theorem

Accepted Answer

The Central Limit Theorem states that for large sample sizes, the sampling distribution of the sample mean or sample proportion will have a normal distribution. This theorem is a foundational concept in statistics and allows us to make inferences about population parameters based on sample statistics.

Question 8

Random samples of size 49 are taken from a population that has 200 elements, a mean of 180, and a variance of 196. The distribution of the population is unknown. The mean and the standard error of the mean are
A)180 and 24.39
B)180 and 28
C)180 and 2
D)180 and 1.74

Accepted Answer

The answer of Random samples of size 49 are taken...

Question 9

The number of random samples (without replacement) of size 3 that can be drawn from a population of size 5 is
A)15
B)10
C)20
D)125

Accepted Answer

The number of ways to choose 3 samples from a population of 5 without replacement is given by the combination formula C(n, k) = n! / (k!(n-k)!), where n is the population size and k is the sample size. Here, C(5, 3) = 5! / (3!2!) = 10.

Question 10

A simple random sample of size n from an infinite population of size N is to be selected. Each possible sample should have
A)the same probability of being selected
B)a probability of 1/n of being selected
C)a probability of 1/N of being selected
D)a probability of N/n of being selected

Accepted Answer

In a simple random sample, each possible sample of size n has an equal probability of being selected from the population.

Question 11

Random samples of size 525 are taken from an infinite population whose population proportion is 0.3. The standard deviation of the sample proportions (i.e., the standard error of the proportion) is
A)0.0004
B)0.2100
C)0.3000
D)0.0200

Accepted Answer

The standard deviation of the sample proportions (i.e., the standard error of the proportion) is given by the formula sqrt(pq/n), where p is the population proportion, q is the complementary proportion (q = 1-p), and n is the sample size. Plugging in the values, we get sqrt(0.3*0.7/525) = 0.0199, which rounds up to 0.0200. Therefore, the best choice is D.

Question 12

A single numerical value used as an estimate of a population parameter is known as
A)a parameter
B)a population parameter
C)a mean estimator
D)a point estimate

Accepted Answer

A point estimate is a single numerical value used as an estimate of a population parameter. Choices A, B, and C are incorrect because they either refer to a specific type of estimate (such as a population parameter) or do not accurately describe the concept of a point estimate.

Question 13

There are 6 children in a family. The number of children defines a population. The number of simple random samples of size 2 (without replacement) which are possible equals
A)12
B)15
C)3
D)16

Accepted Answer

The number of simple random samples of size 2 from a population of 6 children is calculated using combinations: C(6,2) = 6! / (2!(6-2)!) = 15.

Question 14

Random samples of size 36 are taken from an infinite population whose mean and standard deviation are 20 and 15, respectively. The distribution of the population is unknown. The mean and the standard error of the mean are
A)36 and 15
B)20 and 15
C)20 and 0.417
D)20 and 2.5

Accepted Answer

The mean of the population is represented by μ = 20, and the standard deviation of the population is represented by σ = 15. The standard error of the mean is represented as σ/√n, where n is the sample size. In this case, n = 36. Therefore, the standard error of the mean is:

σ/√n = 15/√36 = 15/6 = 2.5

So, the mean of the sample is 20, and the standard error of the mean is 2.5. Therefore, the best choice is D.

Question 15

A sample of 400 observations will be taken from an infinite population. The population proportion equals 0.8. The probability that the sample proportion will be greater than 0.83 is
A)0.4332
B)0.9332
C)0.0668
D)0.5668

Accepted Answer

We will use the central limit theorem since the sample size is large enough.
The mean of the sampling distribution of proportions is equal to the population proportion, which is 0.8.
The standard deviation of the sampling distribution of proportions is calculated as follows: 
sqrt( (p*(1-p))/n ) = sqrt( (0.8*0.2)/400 ) = 0.026
Now, we need to standardize the value of 0.83:
z = (0.83 - 0.8) / 0.026 = 1.15
Using a standard normal distribution table, we find that the probability of z being greater than 1.15 is 0.0668. Therefore, the answer is C.

Question 16

A simple random sample of 28 observations was taken from a large population. The sample mean equaled 50. Fifty is a
A)population parameter
B)biased estimate of the population mean
C)sample parameter
D)point estimate

Accepted Answer

The sample mean of 50 is a single value that estimates the population mean, making it a point estimate. It is not a population parameter or sample parameter, and whether or not it is biased cannot be determined without additional information about the population.

Question 17

A simple random sample of 5 observations from a population containing 400 elements was taken, and the following values were obtained.12 18
19
20
21
A point estimate of the mean is
A)400
B)18
C)20
D)10

Accepted Answer

The point estimate of the mean is the average of the sample values: (12 + 18 + 19 + 20 + 21) / 5 = 18.

Question 18

A population has a mean of 80 and a standard deviation of 7. A sample of 49 observations will be taken. The probability that the sample mean will be larger than 82 is
A)0.5228
B)0.9772
C)0.4772
D)0.0228

Accepted Answer

We need to use the Central Limit Theorem (CLT) to solve this problem. According to CLT, the sampling distribution of the sample means is approximately normal with mean μ and standard deviation σ/√n, where μ is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, we have μ = 80, σ = 7, and n = 49. Therefore, the sampling distribution of the sample means is approximately normal with mean 80 and standard deviation 7/√49 = 1.

Now, we want to find the probability that the sample mean will be larger than 82. We can standardize the sample mean by using the z-score formula:

z = (x̄ - μ) / (σ/√n)

where x̄ is the sample mean. Plugging in the values, we get:

z = (82 - 80) / (1) = 2

We want to find the probability that z is greater than 2. This is equivalent to finding the area to the right of z = 2 under the standard normal distribution curve.

Using a standard normal distribution table or a calculator, we find that the probability is 0.0228 (rounded to four decimal places).

Therefore, the answer is D) 0.0228.

Question 19

For a population with any distribution, the form of the sampling distribution of the sample mean is
A)sometimes normal for all sample sizes
B)sometimes normal for large sample sizes
C)always normal for all sample sizes
D)always normal for large sample sizes

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will approach a normal distribution as the sample size becomes large, regardless of the shape of the population distribution.

Question 20

A population has a mean of 180 and a standard deviation of 24. A sample of 64 observations will be taken. The probability that the sample mean will be between 183 and 186 is
A)0.1359
B)0.8185
C)0.3413
D)0.4772

Accepted Answer

The probability that the sample mean will be between 183 and 186 is 0.1359. This is because the z-score for a sample mean of 183 is -0.58, and the z-score for a sample mean of 186 is 0.25. The area under the normal curve between these two z-scores is 0.1359.

Quiz 7: Sampling and Sampling Distributions

The purpose of statistical inference is to provide information about the

From a population of 500 elements, a sample of 225 elements is selected. It is known that the variance of the population is 900. The standard error of the mean is approximately

A simple random sample of 64 observations was taken from a large population. The sample mean and the standard deviation were determined to be 320 and 120 respectively. The standard error of the mean is

A random sample of 150 people was taken from a very large population. Ninety of the people in the sample were female. The standard error of the proportion is

The sample statistic, such as , s, or , that provides the point estimate of the population parameter is known as

Random samples of size 81 are taken from an infinite population whose mean and standard deviation are 200 and 18, respectively. The distribution of the population is unknown. The mean and the standard error of the mean are

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 the

Random samples of size 49 are taken from a population that has 200 elements, a mean of 180, and a variance of 196. The distribution of the population is unknown. The mean and the standard error of the mean are

The number of random samples (without replacement) of size 3 that can be drawn from a population of size 5 is

A simple random sample of size n from an infinite population of size N is to be selected. Each possible sample should have

Random samples of size 525 are taken from an infinite population whose population proportion is 0.3. The standard deviation of the sample proportions (i.e., the standard error of the proportion) is

A single numerical value used as an estimate of a population parameter is known as

There are 6 children in a family. The number of children defines a population. The number of simple random samples of size 2 (without replacement) which are possible equals

Random samples of size 36 are taken from an infinite population whose mean and standard deviation are 20 and 15, respectively. The distribution of the population is unknown. The mean and the standard error of the mean are

A sample of 400 observations will be taken from an infinite population. The population proportion equals 0.8. The probability that the sample proportion will be greater than 0.83 is

A simple random sample of 28 observations was taken from a large population. The sample mean equaled 50. Fifty is a

A simple random sample of 5 observations from a population containing 400 elements was taken, and the following values were obtained.12 18 19 20 21 A point estimate of the mean is

A population has a mean of 80 and a standard deviation of 7. A sample of 49 observations will be taken. The probability that the sample mean will be larger than 82 is

For a population with any distribution, the form of the sampling distribution of the sample mean is

A population has a mean of 180 and a standard deviation of 24. A sample of 64 observations will be taken. The probability that the sample mean will be between 183 and 186 is