Question 1

Which of the following best describes a sampling distribution of the sample mean?

Accepted Answer

It is the probability distribution of a sample statistic formed when random samples of size $$n$$ are repeatedly taken from a population.

Question 2

Suppose a population has mean $$\mu = 10$$ and standard deviation $$\sigma = 4. If $$you take random samples of size $$n = 16$$, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\sigma_{\bar{x}} = 1$$

Question 3

Which of the following statements about the Central Limit Theorem (CLT) is TRUE?

Accepted Answer

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

Question 4

A population has $$\mu = 50$$ and $$\sigma = 12. If $$you take samples of size $$n = 36$$, what is the variance of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{12^2$}{36} = 4$$

Question 5

If the population of exam scores is not normally distributed, which of the following is TRUE about the sampling distribution of the sample mean for samples of size $$n = 40$$?

Accepted Answer

The sampling distribution will be approximately normal due to the Central Limit Theorem.

Question 6

Given a population with $$\mu = 100$$ and $$\sigma = 20$$, and a sample size $$n = 25$$, what is the probability that the sample mean is less than $$92$$? Assume the population is normal.

Accepted Answer

First, calculate $$\sigma_{\bar{x}} = \frac{20}{\sqrt{25}} = 4. $$Then, $$z = \frac{92 - 100}{4} = -2. $$The probability is $$P(z \u003c -2) = 0.0228.$$

Question 7

A population has mean $$\mu = 3$$ and variance $$\$$sigma^2$$ = \frac{8}{3}. If $$you take all possible samples of size $$n = 3$$ with replacement, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{\sqrt{8/3}}{\sqrt{3}} \approx 0.9428$$

Question 8

The diameters of fully grown white oak trees are normally distributed with mean $$3.5$$ feet and standard deviation $$0.2$$ feet. If you take samples of size $$n = 16$$, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{0.2}{\sqrt{16}} = 0.05$$

Question 9

In a recent year, the mean existing home sales price in the US was $$296,700$$ with a standard deviation of $$50,000. If $$you randomly select $$12$$ homes, what is the probability that the sample mean is more than $$275,000$$? Assume normality.

Accepted Answer

First, $$\sigma_{\bar{x}} = \frac{50,000}{\sqrt{12}} \approx 14,433.76$$. $$z = \frac{275,000 - 296,700}{14,433.76} \approx -1.50$$. $$P(z \u003e -1.50) = 0.9332.$$

Question 10

Which formula should you use to convert a sample mean $$\bar{x} to a$$ $$z-$$score when working with the sampling distribution of the sample mean?

Accepted Answer

$$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$

Question 11

A population is normally distributed with mean $$\mu = 80$$ and standard deviation $$\sigma = 10. $$What is the probability that the mean of a random sample of size $$n = 25 is $$between $$78$$ and $$82$$?

Accepted Answer

First, $$\sigma_{\bar{x}} = \frac{10}{5} = 2$$. $$z_1$$ = \frac{78 - 80}{2} = -1$$, z_2$ = \frac{82 - 80}{2} = 1$$. $$P(-1 \u003c z \u003c 1) = 0.6826.$$

Question 12

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

Accepted Answer

The sampling distribution is always normal, regardless of the population distribution and sample size.

Question 13

If you take all possible samples of size $$n$$ from a population and calculate the sample mean for each, what will the mean of these sample means be?

Accepted Answer

It will be equal to the population mean $$\mu.$$

Question 14

A population is not normally distributed. Which sample size is most likely to result in a sampling distribution of the sample mean that is approximately normal?

Accepted Answer

$$n = 30$$

Question 15

Suppose you have a population with $$\mu = 70$$ and $$\sigma = 21. If $$you take a random sample of size $$n = 49$$, what is the probability that the sample mean is greater than $$75$$? Assume normality.

Accepted Answer

$$\sigma_{\bar{x}} = \frac{21}{7} = 3$$. $$z = \frac{75 - 70}{3} = 1.67$$. $$P(z \u003e 1.67) = 0.0475.$$

Question 16

Which of the following best explains why the standard deviation of the sampling distribution of the sample mean is less than the population standard deviation?

Accepted Answer

Because the sample mean averages out the variability in the population, reducing the spread by a factor of $$\sqrt{n}.$$

Question 17

A researcher wants to estimate the average height of a species of plant. The population is not normal, but the researcher takes a sample of $$n = 50. $$Which statement is most accurate?

Accepted Answer

The sampling distribution of the sample mean will be approximately normal due to the Central Limit Theorem.

Sampling Distributions and the Central Limit Theorem

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