Question 1

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

Accepted Answer

The standard error of the mean is calculated as the population standard deviation divided by the square root of the sample size: 120 / √64 = 120 / 8 = 15.

Question 2

A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of $\bar { X }$ is _____.
A) approximately normal because $\bar { X }$ is always approximately normally distributed
B) approximately normal because the sample size is large in comparison to the population size
C) approximately normal because of the central limit theorem
D) normal if the population is normally distributed

Accepted Answer

The answer of A sample of 24 observations is taken...

Question 3

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 mean from that sample will be larger than 82 is _____.
A) .5228
B) .9772
C) .4772
D) .0228

Accepted Answer

To solve this problem, we will use the central limit theorem to approximate the sampling distribution of the sample mean. According to the central limit theorem, the sampling distribution of the sample mean will be approximately normal with a mean of 80 and a standard deviation of 7/√49 = 1.

We want to find the probability that the sample mean will be larger than 82:

P(X > 82) = P(Z > (82-80)/1) = P(Z > 2)

Looking at the standard normal distribution table, we can see that the probability of getting a Z-score larger than 2 is approximately 0.0228.

Therefore, the answer is D: 0.0228.

Question 4

From a population of 200 elements, the standard deviation is known to be 14. A sample of 49 elements is selected. It is determined that the sample mean is 56. The standard error of the mean is _____.
A) 3
B) 2
C) greater than 2
D) less than 2

Accepted Answer

The answer of From a population of 200 elements, the...

Question 5

Random samples of size 36 are taken from a process (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 distribution of sample 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 distribution of the sample mean is the same as the population mean, which is 20. The standard error of the sample mean is the population standard deviation divided by the square root of the sample size, which is 15/√36 = 2.5.

Question 6

For a population with an unknown distribution, the form of the sampling distribution of the sample mean is _____.
A) approximately normal for all sample sizes
B) exactly normal for large sample sizes
C) exactly normal for all sample sizes
D) approximately normal for large sample sizes

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normal for large sample sizes (typically n > 30) regardless of the population distribution. Thus, option D is the best choice as it reflects this idea.

Question 7

A sample of 92 observations is taken from a process (an infinite population). The sampling distribution of $\bar { X }$ is approximately normal because _____.
A) $\bar { x }$ is always approximately normally distributed
B) the sample size is small in comparison to the population size
C) of the central limit theorem
D) None of the answers is correct.

Accepted Answer

The central limit theorem states that the sampling distribution of the sample mean will be approximately normal if the sample size is sufficiently large, regardless of the population's distribution. In this case, a sample of 92 observations is considered large enough for the sampling distribution of $\bar{X}$ to be approximately normal.

Question 8

Whenever the population has a normal probability distribution, the sampling distribution of $\bar { X }$ is a normal probability distribution for _____.
A) only large sample sizes
B) only small sample sizes
C) any sample size
D) only samples of size 30 or greater

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed regardless of the shape of the population distribution, as long as the sample size is sufficiently large. However, if the population itself is normally distributed, then the sampling distribution of the sample mean will be normally distributed for any sample size, not just large ones.

Question 9

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 distribution of sample means are _____.
A) 180 and 24.39
B) 180 and 28
C) 180 and 1.74
D) 180 and 2

Accepted Answer

The mean of the distribution of sample means is equal to the population mean, which is 180. The standard error of the distribution of sample means is calculated using the formula $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Given a population variance of 196, the standard deviation ($\sigma$) is the square root of 196, which is 14. Therefore, the standard error is $\sigma_{\bar{x}} = \frac{14}{\sqrt{49}} = \frac{14}{7} = 2$. However, because the population size is finite and the sample size is more than 5% of the population (49 out of 200), we should apply the finite population correction (FPC) formula: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N - n}{N - 1}}$, where $N$ is the population size. Applying the FPC, we get $\sigma_{\bar{x}} = 2 \times \sqrt{\frac{200 - 49}{200 - 1}}$, which slightly adjusts the standard error. However, the options provided do not account for this correction, and the closest correct calculation without considering the FPC is 2. Therefore, the correct choice based on the options provided and without applying the FPC is 180 for the mean and 2 for the standard error, making option C the best answer given the options.

Question 10

Random samples of size 81 are taken from a process (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 distribution of sample means are _____.
A) 200 and 18
B) 81 and 18
C) 9 and 2
D) 200 and 2

Accepted Answer

The mean of the population, 200, will be the mean of the distribution of sample means. The standard error of the distribution of sample means can be calculated as the standard deviation of the population divided by the square root of the sample size: 18/sqrt(81) = 2. Therefore, the mean and the standard error of the distribution of sample means are 200 and 2, respectively.

Question 11

A population has a mean of 53 and a standard deviation of 21. A sample of 49 observations will be taken. The probability that the sample mean will be greater than 57.95 is _____.
A) 0
B) .0495
C) .4505
D) None of the answers is correct.

Accepted Answer

The standard error of the mean is 21/sqrt(49) = 3. The z-score for a sample mean of 57.95 is (57.95 - 53) / 3 = 1.65. The probability of a z-score greater than 1.65 is approximately 0.0495.

Question 12

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 a central limit theorem property. According to the central limit theorem, as the sample size increases, the distribution of sample means approaches a normal distribution, and the variability among the sample means decreases.

Question 13

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 mean from that sample will be between 183 and 186 is _____.
A) 0.1359
B) 0.8185
C) 0.3413
D) 0.4772

Accepted Answer

The sampling distribution of the sample mean will have a mean of 180 and a standard deviation of 24/√64 = 3. The Z-scores for 183 and 186 are (183-180)/3 = 1 and (186-180)/3 = 2, respectively. The probability of a Z-score between 1 and 2 is 0.1359.

Question 14

As the sample size becomes larger, the sampling distribution of the sample mean approaches a _____.
A) binomial distribution
B) Poisson distribution
C) hypergeometric distribution
D) normal probability distribution

Accepted Answer

The Central Limit Theorem states that as the sample size becomes larger, the sampling distribution of the sample mean approaches a normal probability distribution.

Question 15

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 as the sample size gets larger, the distribution of sample means and sample proportions approaches a normal distribution, regardless of the shape of the population distribution. Therefore, it allows us to use the normal probability distribution to approximate the sampling distribution when the sample size is large.

Question 16

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. Doubling the sample size reduces the standard error to 1/sqrt(2) of its original value, which is approximately 70%.

Question 17

The fact that the sampling distribution of the sample mean 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 there are tables of areas for the normal distribution
C) assumption that the population has a normal distribution
D) normal probability theorem

Accepted Answer

The central limit theorem is a fundamental concept in statistics that states that the mean of a sufficiently large sample from any population is approximately normally distributed. This theorem is the reason why the sampling distribution of the sample mean can be approximated by a normal probability distribution when the sample size is large. The other options (B, C, and D) are not correct because the fact that there are tables of areas for the normal distribution or the assumption that the population has a normal distribution does not imply that the sample mean distribution is normal. Also, there is no such thing as the "normal probability theorem."

Question 18

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 = (standard deviation of population) / √(sample size)

In this case, we are given the variance of the population, which is equal to the standard deviation squared. So, the standard deviation of the population is:

SD = √900 = 30

Plugging this into the formula for SE, we get:

SE = 30 / √225 = 30 / 15 = 2

Therefore, the answer is (D) 1.4847.

Question 19

As the sample size increases, the _____.
A) standard deviation of the population decreases
B) population mean increases
C) standard error of the mean decreases
D) standard error of the mean increases

Accepted Answer

As the sample size increases, the standard error of the mean decreases. This is because the standard error of the mean is inversely proportional to the square root of the sample size. As the sample size increases, the square root of the sample size increases as well, resulting in a decrease in the standard error of the mean.

Question 20

A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is _____.
A) .0347
B) .7200
C) .9511
D) None of the answers is correct.

Accepted Answer

To solve this problem, we need to use the formula for the standard error of the mean: SE = σ/√n where σ is the population standard deviation, n is the sample size, and SE is the standard error of the mean. In this case, σ = 12 and n = 36, so: SE = 12/√36 = 2 Next, we need to standardize the sample mean using the z-score formula: z = (x - μ) / SE where x is the sample mean, μ is the population mean, and SE is the standard error of the mean. We want to find the probability that the sample mean falls between 80.54 and 88.9, so we need to standardize those values as well: z1 = (80.54 - 84) / 2 = -1.73 z2 = (88.9 - 84) / 2 = 2.45 Now we can look up these z-scores in the standard normal distribution table (or use a calculator) to find the probabilities: P(z < -1.73) = 0.0422 P(z > 2.45) = 0.0070 Note that we need to use the complement of P(z > 2.45) to find the probability that the sample mean is less than 88.9, so: P(z < 2.45) = 0.9929 P( -1.73 < z < 2.45) = 0.9929 - 0.0422 = 0.9507 Therefore, the probability that the sample mean will be between 80.54 and 88.9 is 0.9507, which is approximately 0.9511. The answer is C.

Quiz 7: Sampling and Sampling Distributions

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

A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of $\bar { X }$ 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 mean from that sample will be larger than 82 is _____.

From a population of 200 elements, the standard deviation is known to be 14. A sample of 49 elements is selected. It is determined that the sample mean is 56. The standard error of the mean is _____.

Random samples of size 36 are taken from a process (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 distribution of sample mean are _____.

For a population with an unknown distribution, the form of the sampling distribution of the sample mean is _____.

A sample of 92 observations is taken from a process (an infinite population) . The sampling distribution of $\bar { X }$ is approximately normal because _____.

Whenever the population has a normal probability distribution, the sampling distribution of $\bar { X }$ is a normal probability distribution for _____.

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 distribution of sample means are _____.

Random samples of size 81 are taken from a process (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 distribution of sample means are _____.

A population has a mean of 53 and a standard deviation of 21. A sample of 49 observations will be taken. The probability that the sample mean will be greater than 57.95 is _____.

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

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 mean from that sample will be between 183 and 186 is _____.

As the sample size becomes larger, the sampling distribution of the sample mean approaches a _____.

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 _____.

Doubling the size of the sample will _____.

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

As the sample size increases, the _____.

A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is _____.