Question 1

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

Accepted Answer

False

Question 2

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 uncertainty in estimating the population mean based on a sample mean. It takes into account the variability within the sample and the sample size, and is a key component in hypothesis testing and confidence interval estimation in inferential statistics.

Question 3

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

This is known as the Central Limit Theorem, which states that for large enough sample sizes (typically n > 30), the sampling distribution of the sample mean will be approximately normal regardless of the shape of the population distribution.

Question 4

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

Accepted Answer

The answer of As the size of the sample is...

Question 5

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

The answer of The amount of time it takes to...

Question 6

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

Accepted Answer

The answer of If all possible samples of size n...

Question 7

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 8

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 standard error of the sample mean actually decreases, because the sample mean becomes a better estimator of the population mean.

Question 9

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

Accepted Answer

A sampling distribution is the distribution of all possible values of a statistic from all possible samples of a certain size from a population. It is not a distribution of values for a parameter, as the parameter is fixed and unknown in a particular population.

Question 10

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

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 population distribution, as long as the samples are independent and identically distributed. Therefore, the statement is true for normal populations as well.

Question 11

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

The formula for the standard error of the sample mean is given by SE = σ/√n, where σ is the population standard deviation and n is the sample size. Plugging in the given values, we get SE = 8/√64 = 1. Therefore, the statement is true.

Question 12

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 a statistic (not just means) obtained from a large number of samples drawn from a specific population. It does not vary with sample sizes but is based on a particular sample size chosen for the study.

Question 13

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 based on the formula for standard error of the mean, which is equal to the population standard deviation divided by the square root of the sample size. So, in this case, the standard error of the mean would be 20/sqrt(64) = 2.5.

Question 14

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), and it does not guarantee accurate population inferences for small sample sizes, especially if the population distribution is significantly non-normal.

Question 15

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

Question 16

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 calculated as the standard deviation divided by the square root of the sample size (n). Therefore, for a sample size of 36, the standard error would be $0.25 / \sqrt{36} = 0.25 / 6 = 0.0417$ ounces, not 0.25 ounces.

Question 17

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

This is known as the Central Limit Theorem, which states that as sample size increases, the distribution of sample means approaches a normal distribution regardless of the distribution of the population. The general rule of thumb is that if the sample size is at least 30, the sampling distribution of the mean can be approximated by a normal distribution.

Question 18

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 a larger sample size provides more precise estimates of the population mean, resulting in less variability and a smaller standard error.

Question 19

The mean of the sampling distribution of the sample proportion  , when the sample size n = 100 and the population proportion p = 0.92, is 92.0.

Accepted Answer

The answer of The mean of the sampling distribution of...

Question 20

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

This is true since the mean of the sampling distribution of the sample mean is equal to the mean of the population (50 ounces).

Quiz 8: Sampling Distributions

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