Question 1

Which of the following best describes the sampling distribution of a statistic?

Accepted Answer

It is the distribution of all values of the statistic when all possible samples of the same size are taken from the same population.

Question 2

According to the Central Limit Theorem, what happens to the shape of the sampling distribution of the sample mean as the sample size $$n$$ increases?

Accepted Answer

It becomes more normal, regardless of the shape of the population distribution.

Question 3

Given a population with mean $$\mu = 25$$ and standard deviation $$\sigma = 14.4309$$, what is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 16$$?

Accepted Answer

$$\frac{14.4309}{\sqrt{16}}$$

Question 4

If $$X \sim N(98.2, 0.6)$$, what is the $$z-$$score for a single person with a body temperature of $$99^{\circ}F$?

Accepted Answer

$$z = \frac{99 - 98.2}{0.6}$$

Question 5

A sample of 16 healthy adults is taken. If their body temperatures are normally distributed with mean $$98.2^{\circ}F$ and standard deviation 0.6$$^{\circ}$$F$$, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{0.6}{\sqrt{16}}$$

Question 6

Which of the following statements about the Central Limit Theorem is FALSE?

Accepted Answer

The sampling distribution of the sample mean always matches the shape of the population distribution, regardless of $$n.$$

Question 7

Suppose Tylenol pill weights are normally distributed with $$\mu = 605.1 mg $$and $$\sigma = 5.96 mg. $$What is the probability that a single pill weighs less than 600 mg?

Accepted Answer

Find $$z = \frac{600 - 605.1}{5.96}$$ and use the standard normal table.

Question 8

If you take a random sample of $$n = 20$$ Tylenol pills, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{5.96}{\sqrt{20}}$$

Question 9

Which of the following is the correct formula for the $$z-$$score of a sample mean $$\bar{x}$$ from a population with mean $$\mu$$ and standard deviation $$\sigma$$, for sample size $$n$$?

Accepted Answer

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

Question 10

A population has mean $$\mu = 25$$ and standard deviation $$\sigma = 14.4309. $$For samples of size $$n = 100$$, what is the expected value of the sample mean?

Accepted Answer

$$25$$

Question 11

Which of the following best describes the effect of increasing sample size $$n on $$the spread of the sampling distribution of the sample mean?

Accepted Answer

The spread decreases as $$n$$ increases.

Question 12

If the population distribution is highly skewed, what can you say about the sampling distribution of the sample mean for large $$n$$?

Accepted Answer

It will be approximately normal.

Question 13

A researcher collects 1000 samples of size $$n = 4$$ from a population with $$\sigma = 14.4309. $$The standard deviation of the sample means is closest to:

Accepted Answer

$$\frac{14.4309}{2}$$

Question 14

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

Accepted Answer

Its standard deviation increases as $$n$$ increases.

Question 15

Suppose the average body temperature for a group of 16 people is $$99^{\circ}F$. If X$$ \sim N(98.2, 0.6)$$, what is the z$-score for this sample mean?

Accepted Answer

$$z = \frac{99 - 98.2}{0.6 / \sqrt{16}}$$

Question 16

Which of the following best explains why the Central Limit Theorem is considered the 'fundamental theorem of statistics'?

Accepted Answer

It allows us to make inferences about population parameters using sample statistics, regardless of the population's distribution, as long as $$n is $$large.

The Central Limit Theorem and Sampling Distributions

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