Question 1

If samples of size n = 16 are selected from a population with μ = 40 and σ = 8,the distribution of sample means will have a standard error of 2 points.

Accepted Answer

This statement is true. The formula for the standard error of the mean is σ/√n. Plugging in the values given, we get 8/√16 = 2.

Question 2

If samples are selected from a normal population,the distribution of sample means will also be normal.

Accepted Answer

This is one of the main principles of the Central Limit Theorem, which states that as the sample size increases, the distribution of sample means will approach a normal distribution even if the population itself is not normally distributed.

Question 3

A mathematical proposition known as the central limit theorem provides a precise description of the distribution that would be obtained if you selected every possible sample,calculated every sample mean,and constructed the distribution of the sample mean.

Accepted Answer

The central limit theorem states that the sample means obtained from a large number of independent, identically distributed random variables will be approximately normally distributed, regardless of the underlying distribution of the variables. This means that if every possible sample was selected and their means were calculated, they would follow a normal distribution.

Question 4

If samples of size n = 16 are selected from a population with μ = 40 and σ = 8,the distribution of sample means will have an expected value of 40.​

Accepted Answer

This is true because the expected value of the distribution of sample means is equal to the population mean, which is 40 in this case.

Question 5

The mean for a sample of n = 4 scores has a standard error σ_M = 5 points.This sample was selected from a population with a standard deviation of σ = 20.

Accepted Answer

The answer of The mean for a sample of n...

Question 6

A sample of n = 25 scores is selected from a population with a mean of µ = 80 and a standard deviation of σ = 20.The standard error for the sample mean is 20.

Accepted Answer

The standard error for the sample mean is given by the formula σ/√n, where σ is the population standard deviation and n is the sample size. Therefore, the standard error for this sample would be 20/√25 = 4, not 20.

Question 7

The smallest possible standard error is obtained when a small sample is taken from a population with a small standard deviation.

Accepted Answer

The standard error decreases as the sample size increases, not when a small sample is taken. A smaller population standard deviation does contribute to a smaller standard error, but the key factor for minimizing standard error is increasing the sample size.

Question 8

On average,a sample of n = 16 scores from a population with s = 10 will provide a better estimate of the population mean than you would get with a sample of n = 16 scores from a population with s = 5.

Accepted Answer

A smaller standard deviation (s = 5) in the population indicates that the scores are closer to the mean, making the sample mean a more accurate estimate of the population mean compared to a population with a larger standard deviation (s = 10).

Question 9

The law of large numbers states that the larger the sample size (n),the more probable it is that the sample mean will be close to the population mean..

Accepted Answer

This is the correct statement of the law of large numbers. As the sample size increases, the sample mean will converge to the population mean.

Question 10

A researcher obtained M = 27 for a sample of n = 36 scores selected from a population with µ = 30 and s = 18.This sample mean corresponds to a z-score of z = -1.00.

Accepted Answer

To find the z-score, we use the formula:

z = (x - µ) / (s / √n)

Substituting the given values, we get:

-1 = (27 - 30) / (18 / √36)

Simplifying this equation gives us:

-1 = -3 / 3

Therefore, the z-score is -1, which is consistent with the given information. So, the answer is A.

Question 11

The distribution of sample means is the collection of sample means for all the possible random samples of a particular size (n)that can be obtained from a population.

Accepted Answer

This is the definition of the distribution of sample means, which is an important concept in statistics.

Question 12

If the sample size is equal to the population standard deviation (n = s),then the standard error for the sample mean is equal to 1.00.

Accepted Answer

the standard error for the sample mean is equal to the population standard deviation divided by the square root of the sample size, which in this case would be s/sqrt(s) = sqrt(s) . Therefore, the standard error would not be equal to 1.00 if n = s.

Question 13

The mean for a sample of n = 9 scores has a standard error of 2 points.This sample was selected from a population with a standard deviation of s = 18.

Accepted Answer

The standard error of the mean (SEM) is calculated as the population standard deviation (s) divided by the square root of the sample size (n), which in this case would be 18 / sqrt(9) = 18 / 3 = 6, not 2.

Question 14

A sample is obtained from a population with s = 20.If the sample mean has a standard error of 5 points,then the sample size is n = 4.

Accepted Answer

The formula for the standard error is SE = s/√n, where s is the sample standard deviation and n is the sample size. Given that s = 20 and SE = 5, we can solve for n:

5 = 20/√n

√n = 20/5

√n = 4

n = (4)^2

n = 16

Therefore, the sample size is not 4, but rather 16. So, the statement is false.

Question 15

As the sample size increases,the standard error also increases.

Accepted Answer

This statement is false. As the sample size increases, the standard error actually decreases.

Question 16

If the standard deviation for a population increases,the standard error for sample means from the population will also increase.

Accepted Answer

The standard error for sample means is calculated by dividing the standard deviation of the population by the square root of the sample size. Therefore, if the standard deviation of the population increases, the standard error will also increase as a larger standard deviation will lead to a less precise estimate of the population mean.

Question 17

According to the central limit theorem,the standard error for a sample mean becomes smaller as the sample size increases.

Accepted Answer

The central limit theorem states that as the sample size increases, the distribution of the sample means approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. This means that the standard error, which is the standard deviation of the sampling distribution of the mean, becomes smaller as the sample size increases.

Question 18

A sample of n = 25 scores is selected from a population with a mean of µ = 80 and a standard deviation of σ = 20.The expected value for the sample mean is 80.

Accepted Answer

The expected value for the sample mean is the same as the population mean, which is 80. Therefore, the statement is true.

Question 19

The sampling error is the natural discrepancy,or amount of error,between a sample statistic and its corresponding population parameter.

Accepted Answer

This statement accurately defines the concept of sampling error. Sampling error arises due to the fact that we are working with a sample of the population, rather than the entire population itself. As a result, any statistic we derive from that sample will likely differ from the true population parameter to some degree. This variation between sample and population values is known as sampling error.

Question 20

Two samples probably will have different means even if they are both the same size and they are both selected from the same population.

Accepted Answer

Two samples selected from the same population are still expected to have different means due to sampling variability. Even if the sample size is the same, random chance can lead to different values for the sample means.

Quiz 7: Probability and Samples: the Distribution of Sample Means