Question 1

Which statistical concept is best illustrated by the image, where different population distributions result in a normal distribution of sample means as sample size increases?

Accepted Answer

Central Limit Theorem

Question 2

According to the Central Limit Theorem, what happens to the distribution of sample means as the sample size $$n$$ increases (assuming $$n \u003e 30$$)?

Accepted Answer

It approaches a normal distribution, regardless of the population's shape.

Question 3

If a population has a mean $$\mu = 50$$ and standard deviation $$\sigma = 10$$, what is the standard deviation of the sampling distribution of the mean for samples of size $$n = 25$$?

Accepted Answer

$$\sigma_{\bar{x}} = 2$$

Question 4

A researcher collects a simple random sample of $$n = 40$$ from a population with unknown distribution, mean $$\mu = 100$$, and standard deviation $$\sigma = 20. $$What is the mean of the sampling distribution of the sample mean?

Accepted Answer

$$\mu_{\bar{x}} = 100$$

Question 5

Which of the following is the correct formula for the z-score of a sample mean $$\bar{x}$$ when using the Central Limit Theorem?

Accepted Answer

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

Question 6

Suppose the weights of adult males are normally distributed with $$\mu = 165 lb $$and $$\sigma = 31 lb. If 16 $$males are randomly selected, what is the standard error of the mean?

Accepted Answer

$$\frac{31}{\sqrt{16}} = 7.75$$

Question 7

Why can the normal distribution be used to approximate the sampling distribution of the mean for a sample size of $$n = 23$$ from a normally distributed population?

Accepted Answer

Because the original population is normal, regardless of sample size.

Question 8

A population is heavily skewed right. What must be true for the sampling distribution of the sample mean to be approximately normal?

Accepted Answer

The sample size $$n$$ must be greater than 30.

Question 9

If the mean hip width of adult males is $$14.3 in. $$with a standard deviation of $$0.9 in$$., what is the probability that a randomly selected male has a hip width greater than $$16.0 in$$.? Which formula should you use?

Accepted Answer

Use $$z = \frac{x - \mu}{\sigma}$$

Question 10

In the context of airline seat design, why is the probability that the mean hip width of all 126 male passengers exceeds the seat width more relevant than the probability for a single passenger?

Accepted Answer

Because the mean determines if the group as a whole will fit comfortably.

Question 11

A sample of $$n = 9$$ adult females is selected from a population with $$\mu = 77$$ and $$\sigma = 12. $$What is the standard deviation of the sample mean?

Accepted Answer

$$\frac{12}{\sqrt{9}} = 4$$

Question 12

If the original population is not normally distributed and $$n \leq 30$$, what can be said about the sampling distribution of the mean?

Accepted Answer

It cannot be well-approximated by a normal distribution.

Question 13

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 14

A population has mean $$\mu = 204 cm $$and standard deviation $$\sigma = 8.5 cm. $$What is the probability that the mean of 23 randomly selected values is greater than 216 cm? Which formula should you use for the z-score?

Accepted Answer

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

Question 15

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

Accepted Answer

The sample must be biased.

Central Limit Theorem and Sampling Distributions: Study Guide

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