Question 1

According to the Central Limit Theorem, what is the mean and standard deviation of the sampling distribution of the sample mean $$\bar{x}$$ for samples of size $$n$$ drawn from a population with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

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

Question 2

A population has a mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$2$$

Question 3

Which of the following statements best describes when the Central Limit Theorem can be applied to the sampling distribution of the sample mean?

Accepted Answer

When the population is normal or the sample size $$n \u003e 30$$

Question 4

Suppose the weights of adult males are normally distributed with $$\mu = 189 lb $$and $$\sigma = 39 lb. $$What is the probability that a randomly selected male weighs more than 148 lb?

Accepted Answer

$$0.853$$

Question 5

If a sample of $$n = 27$$ adult males is selected from the population in the previous question, what is the standard deviation of the sample mean?

Accepted Answer

$$\frac{39}{\sqrt{27}} \approx 7.51$$

Question 6

A sample of $$n = 27$$ adult males is selected. What is the probability that their mean weight is greater than 148 lb? ($$\mu = 189$$, $$\sigma = 39)$$

Accepted Answer

$$0.99999998$$

Question 7

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

Accepted Answer

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

Question 8

A population of body temperatures has $$\mu = 98.6^\circF $$and $$\sigma = 0.62^\circF. $$For a sample of $$n = 106$$, what is the standard deviation of the sample mean?

Accepted Answer

$$0.0602$$

Question 9

Why does the probability of the mean weight of 27 randomly selected adult males being greater than 148 lb approach 1, even though the probability for a single male is much lower?

Accepted Answer

Because the sampling distribution of the mean is much narrower, making extreme sample means highly unlikely.

Question 10

If you are given a sample mean $$\bar{x}$$ from a sample of size $$n \u003e 30$$ and the population standard deviation $$\sigma$$, which distribution should you use to find probabilities involving $$\bar{x}$$?

Accepted Answer

A normal distribution with mean $$\mu$$ and standard deviation $$\frac{\sigma}{\sqrt{n}}$$

Question 11

A researcher wants to estimate the probability that the average body temperature of a random sample of 106 people is less than $$98.2^\circF. $$Which Excel formula should they use?

Accepted Answer

=NORM.DIST(98.2, 98.6, 0.62/SQRT(106), 1)

Question 12

Which of the following best describes the 'standard error of the mean'?

Accepted Answer

The standard deviation of the sampling distribution of the sample mean

Question 13

If the original population is not normally distributed, under what condition can the sampling distribution of the sample mean be approximated by a normal distribution?

Accepted Answer

If the sample size $$n \u003e 30$$

Question 14

A sample of size $$n = 50 is $$drawn from a population with mean $$\mu = 100$$ and standard deviation $$\sigma = 20. $$What is the probability that the sample mean is less than 95? (Assume normality.)

Accepted Answer

First, calculate $$z = \frac{95 - 100}{20/\sqrt{50}} = \frac{-5}{2.828} \approx -1.77. $$The probability is $$P(z \u003c -1.77) \approx 0.038$$

Question 15

Which of the following is NOT a requirement for applying the Central Limit Theorem to the sample mean?

Accepted Answer

The population must be finite.

Central Limit Theorem and Sampling Distributions: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards