Question 1

As the size of the sample is increased, the mean of  increases.

Accepted Answer

The size of the sample does not necessarily have an effect on the mean, it depends on the distribution of the data. However, as the sample size increases, the sample mean is likely to become a better estimate of the population mean.

Question 2

When a great many simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample mean is normal regardless of the sample size n.

Accepted Answer

This is a property of the Central Limit Theorem, which states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the underlying population distribution.

Question 3

The standard deviation of the sampling distribution of  is also called the:
A) central limit theorem.
B) population standard deviation.
C) finite population correction factor.
D) standard error of the sample mean.

Accepted Answer

The standard deviation of the sampling distribution of a sample mean is known as the standard error of the sample mean. It represents the variability in the means of all possible samples of a given size that can be drawn from a population. The other options are not the standard deviation of the sampling distribution of a sample mean.

Question 4

In inferential statistics, the standard error of the sample mean assesses the uncertainty or error of estimation.

Accepted Answer

The standard error of the sample mean is a measure of the variability or uncertainty of the sample mean as an estimator of the population mean. It quantifies the error in estimating the population mean based on the sample data.

Question 5

If all possible samples of size n are drawn from a normal population, the probability distribution of the sample mean  is an normal distribution.

Accepted Answer

This is a well-known property of the sample mean for normal populations, known as the Central Limit Theorem. As the sample size increases, the distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.

Question 6

A sample of size n is selected at random from an infinite population.As n increases, the standard error of the sample mean increases.

Accepted Answer

As n increases, the sample mean becomes a better and more representative estimate of the population mean, resulting in a decrease in the standard error.

Question 7

If the sample size increases, the standard error of the mean also increases.

Accepted Answer

If the sample size increases, the standard error of the mean decreases. This is because as the sample size increases, the sample mean becomes more representative of the population mean, resulting in less variability in the means of different samples.

Question 8

A sampling distribution is a probability distribution for a statistic, not a parameter.

Accepted Answer

A sampling distribution shows the distribution of a statistic (such as the mean or standard deviation) obtained from multiple samples of a population, whereas a parameter is a fixed value that describes a population (such as the population mean or standard deviation). Therefore, a sampling distribution is a probability distribution for a statistic, not a parameter.

Question 9

Consider an infinite population with a mean of 100 and a standard deviation of 20.A random sample of size 64 is taken from this population.The standard deviation of the sample mean equals 2.50.

Accepted Answer

This is true according to the Central Limit Theorem. The standard deviation of the sample mean equals the standard deviation of the population divided by the square root of the sample size, which is (20/sqrt(64)) = 2.5. This means that the sample mean will have a standard deviation of 2.5, regardless of the shape of the population distribution, as long as the sample size is large enough.

Question 10

The amount of bleach a machine pours into bottles has a mean of 50 ounces with a standard deviation of 0.25 ounces.We take a random sample of 36 bottles filled by this machine.The sampling distribution of the sample mean has a mean of 50 ounces.

Accepted Answer

The statement is true. According to the central limit theorem, the sampling distribution of the sample mean will have a mean equal to the population mean, which is 50 ounces in this case.

Question 11

The amount of bleach a machine pours into bottles has a mean of 50 ounces with a standard deviation of 0.25 ounces.Suppose we take a random sample of 36 bottles filled by this machine.The sampling distribution of the sample mean has a standard error of 0.25 ounces.

Accepted Answer

The standard error of the sample mean is given by the formula standard deviation/√sample size. Therefore, the standard error in this case would be 0.25/√36 = 0.0417, not 0.25.

Question 12

As the size of the sample is increased, the standard error of  decreases.

Accepted Answer

This statement is true. As the sample size increases, the standard error decreases because there is more data to calculate the standard deviation from, resulting in a more precise estimate of the population's standard deviation.

Question 13

A sample of size 25 is selected from a population of size 75.The finite population correction needed to find the standard error of  .

Accepted Answer

The finite population correction is needed when the sample size is more than 5% of the population. In this case, the sample size of 25 is less than 5% of the population size of 75, so the finite population correction is not needed.

Question 14

The amount of time it takes to complete a final examination is negatively skewed distribution with a mean of 70 minutes and a standard deviation of 8 minutes.If 64 students were randomly sampled, the probability that the sample mean of the sampled students exceeds 73.5 minutes is approximately 0.

Accepted Answer

Since the sample size is greater than 30, we can use the Central Limit Theorem to approximate the sampling distribution of the sample mean as a normal distribution with a mean of 70 and a standard deviation of (8/sqrt(64)) = 1. Thus, we can standardize the sample mean as (73.5-70)/1 = 3.5, and calculate the probability of the sample mean being greater than 73.5 as the probability of a standard normal variable being greater than 3.5, which is an extremely small probability close to 0. Therefore, the statement is true.

Question 15

The Central Limit Theorem permits us to draw conclusions about a population based on a sample alone, without having any knowledge about the distribution of that population.And this works no matter what the sample size is.

Accepted Answer

The Central Limit Theorem applies when the sample size is sufficiently large (usually n ≥ 30), not for any sample size. It states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 16

A sampling distribution is defined as the probability distribution of means from all possible sample sizes that are taken from a given population.

Accepted Answer

A sampling distribution is defined as the probability distribution of sample means, not necessarily from all possible sample sizes, but from all possible samples of a given size that can be taken from a population.

Question 17

A sample of size 25 is selected from a population of size 500.The finite population correction is needed to find the standard error of  .

Accepted Answer

The answer of A sample of size 25 is selected...

Question 18

If all possible samples of size n are drawn from an infinite population with standard deviation 8, then the standard error of the sample mean equals 1.0 if the sample size is 64.

Accepted Answer

This statement is true. The formula for standard error of the sample mean is standard deviation divided by the square root of sample size. If the standard deviation is 8 and the sample size is 64, then the standard error is 8/sqrt(64) = 1. Therefore, the statement is true.

Question 19

If the population distribution is skewed, in most cases the sampling distribution of the sample mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Accepted Answer

According to the Central Limit Theorem, the sampling distribution of the sample mean becomes approximately normal regardless of the population distribution as long as the sample size is large enough (typically n >= 30). Therefore, the statement is true.

Question 20

If the population distribution is unknown, in most cases the sampling distribution of the mean can be approximated by the normal distribution if the samples contain at least 30 observations.

Accepted Answer

The central limit theorem states that the sampling distribution of the mean will approach a normal distribution as the sample size increases, regardless of the population distribution, if the sample size is large enough (at least 30 observations).

Quiz 9: Sampling Distributions

As the size of the sample is increased, the mean of increases.