Question 1

Which of the following best describes a sampling distribution?

Accepted Answer

A distribution of all possible values of a statistic for a given size sample selected from a population.

Question 2

If a population is normal with mean $\mu$ and standard deviation $\sigma$, what is the distribution of the sample mean $\overline{X}$ for samples of size $n$?

Accepted Answer

The sampling distribution of $\overline{X}$ is also normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$.

Question 3

A population has mean $\mu = 100$ and standard deviation $\sigma = 30$. If a sample of size $n = 100$ is drawn, what is the standard error of the sample mean?

Accepted Answer

$3$

Question 4

Which statement about the Central Limit Theorem (CLT) is correct?

Accepted Answer

The CLT states that, for large $n$, the sampling distribution of the sample mean is approximately normal regardless of the population's shape.

Question 5

If the population distribution is unknown and highly skewed, what sample size is generally considered 'large enough' for the sampling distribution of the mean to be approximately normal?

Accepted Answer

$n > 30$

Question 6

Which formula correctly represents the standard deviation of the sampling distribution of the sample mean (standard error)?

Accepted Answer

$\sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}}$

Question 7

A company wants to estimate the average age of its employees. The ages are not normally distributed. If they take a random sample of 50 employees, what can be said about the distribution of the sample mean?

Accepted Answer

It will be approximately normal due to the Central Limit Theorem.

Question 8

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

It is always normal, regardless of sample size or population shape.

Question 9

Why does the Central Limit Theorem not apply to the sampling distribution of the sample variance?

Accepted Answer

Because the sampling distribution of the sample variance is skewed to the right for all sample sizes.

Question 10

A population has mean $\mu = 50$ and standard deviation $\sigma = 10$. If a sample of size $n = 25$ is taken, what is the probability that the sample mean is greater than $54$? (Assume the population is normal.)

Accepted Answer

First, calculate the standard error: $\sigma_{\overline{X}} = \frac{10}{\sqrt{25}} = 2$. The $z$-score is $z = \frac{54 - 50}{2} = 2$. The probability $P(\overline{X} > 54) = 1 - 0.9772 = 0.0228$.

Question 11

Which of the following scenarios best illustrates the application of the Central Limit Theorem?

Accepted Answer

A researcher samples 100 customers from a highly skewed population and finds the sample mean is approximately normally distributed.

Question 12

If the population mean is $\mu = 80$ and the population standard deviation is $\sigma = 16$, what is the mean and standard error of the sampling distribution of the sample mean for samples of size $n = 64$?

Accepted Answer

Mean: $80$, Standard Error: $2$

Question 13

A sample of size $n = 100$ is drawn from a population with unknown shape, mean $\mu = 100$, and standard deviation $\sigma = 30$. What is the probability that the sample mean is less than $105$?

Accepted Answer

The probability is $0.9522$ using $NORMDIST(105, 100, 3, 1)$.

Question 14

Which of the following statements is TRUE regarding the effect of increasing sample size on the sampling distribution of the sample mean?

Accepted Answer

The standard deviation of the sampling distribution decreases.

Question 15

A population has values 18, 20, 22, and 24. What is the mean of the sampling distribution of the sample mean for samples of size 2 (with replacement)?

Accepted Answer

$21$

Sampling Distributions (Chapter 7): Business Statistics Study Notes

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