Question 1

Which of the following best describes a sampling distribution?

Accepted Answer

The probability distribution of a statistic computed from all possible samples of a fixed size.

Question 2

Suppose you are interested in the proportion of left-eye dominant individuals in a population. You take all possible samples of size 5 from a population of 20 and calculate the sample proportion for each. What does the histogram of these proportions represent?

Accepted Answer

The sampling distribution of the sample proportion.

Question 3

Which formula gives the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$ for a population proportion $$p$$ and sample size $$n$$?

Accepted Answer

$$\sqrt{\frac{p(1-p)}{n}}$$

Question 4

A researcher wants to estimate the mean blood pressure in a population. She takes many random samples of size $$n$$ and calculates the sample mean for each. According to the Central Limit Theorem, what will the distribution of these sample means look like as $$n$$ becomes large?

Accepted Answer

It will become approximately normal, regardless of the population distribution.

Question 5

Which of the following is a necessary condition for using the normal approximation for the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$np \geq 15$$ and $$n(1-p) \geq 15$$

Question 6

If the population has mean $$\mu$$ and standard deviation $$\sigma$$, what is the mean and standard deviation of the sampling distribution of the sample mean $$\bar{x}$$ for samples of size $$n$$?

Accepted Answer

Mean $$= \mu$$, Standard deviation $$= \frac{\sigma}{\sqrt{n}}$$

Question 7

Which of the following statements about bootstrapping is TRUE?

Accepted Answer

Bootstrapping requires sampling with replacement from the original sample.

Question 8

A scientist measures the enzyme activity in 18 randomly selected yeast cultures. She wants to estimate the sampling distribution of the mean enzyme activity. Which method can she use if the population distribution is unknown?

Accepted Answer

Bootstrap sampling distribution.

Question 9

Which of the following best describes the Central Limit Theorem (CLT)?

Accepted Answer

For large $$n$$, the sampling distribution of the sample mean is approximately normal with mean $$\mu$$ and standard deviation $$\frac{\sigma}{\sqrt{n}}.$$

Question 10

Why is the standard deviation of the sampling distribution of the sample mean smaller than the population standard deviation?

Accepted Answer

Because averaging over $$n$$ observations reduces variability, as shown by $$\frac{\sigma}{\sqrt{n}}.$$

Question 11

A bootstrap sample is:

Accepted Answer

A sample of size $$n$$ drawn with replacement from the original sample.

Question 12

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

Accepted Answer

It has the same standard deviation as the population.

Question 13

In a study of blood types, the sample proportion of type O donors in a sample of size $$n = 50 is$$ $$\hat{p} = 0.45. $$What is the expected value of $$\hat{p}$$?

Accepted Answer

$$p$$

Question 14

Which of the following scenarios would most likely violate the assumptions of the Central Limit Theorem?

Accepted Answer

Sampling from a highly skewed population with a very small sample size.

Question 15

Why is bootstrapping particularly useful for complex statistics such as the median?

Accepted Answer

Because the sampling distribution of the median is difficult to derive analytically.

Question 16

Which of the following best explains why the sample mean is an unbiased estimator of the population mean?

Accepted Answer

The expected value of the sample mean equals the population mean.

Sampling Distributions: Concepts, Properties, and Applications

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