Question 1

Which of the following statements best describes the Central Limit Theorem?

Accepted Answer

It states that the distribution of sample means will approach a normal distribution as the sample size increases, regardless of the population's distribution.

Question 2

A population has a mean $$\mu = 50$$ and a standard deviation $$\sigma = 10. If $$you take random samples of size $$n = 40$$, what is the standard error of the mean?

Accepted Answer

$$\sigma_x = \frac{10}{\sqrt{40}} = 1.58$$

Question 3

If the original population is normally distributed, what is true about the distribution of sample means for any sample size $$n$$?

Accepted Answer

The sample means will be normally distributed for any sample size $$n.$$

Question 4

A researcher is studying the average reaction time of drivers. The population distribution is unknown, but she takes samples of size $$n = 50. $$According to the Central Limit Theorem, what can she conclude about the distribution of sample means?

Accepted Answer

The distribution of sample means will be approximately normal.

Question 5

Which formula represents the standard error of the mean?

Accepted Answer

$$\sigma_x = \frac{\sigma}{\sqrt{n}}$$

Question 6

A population has a mean $$\mu = 100$$ and standard deviation $$\sigma = 20. If $$you take samples of size $$n = 25$$, what is the mean of the sampling distribution of sample means?

Accepted Answer

$$\mu_x = 100$$

Question 7

Why is the Central Limit Theorem important in real-world applications such as medical research?

Accepted Answer

It allows researchers to use normal probability methods even when the population distribution is unknown, provided the sample size is large.

Question 8

If a population is not normally distributed, what is the minimum sample size $$n$$ commonly recommended for the sample means to be approximately normal?

Accepted Answer

$$n \u003e 30$$

Question 9

A population has $$\mu = 60$$ and $$\sigma = 12. $$You take samples of size $$n = 36. $$What is the probability that the sample mean is within one standard error of the population mean?

Accepted Answer

Approximately 68% (since one standard error corresponds to one standard deviation in a normal distribution)

Question 10

Which of the following is NOT a conclusion of the Central Limit Theorem?

Accepted Answer

The sample means will always be identical to the population values.

Question 11

A population of test scores is heavily skewed. If you take samples of size $$n = 100$$, what can you say about the distribution of the sample means?

Accepted Answer

The distribution of sample means will be approximately normal.

Question 12

In a manufacturing process, the weights of widgets are not normally distributed. If quality control samples $$n = 35$$ are taken, what does the Central Limit Theorem allow you to do?

Accepted Answer

Use normal probability calculations for the sample mean.

Question 13

Which of the following best describes the relationship between the sample size $$n$$ and the standard error of the mean?

Accepted Answer

As $$n$$ increases, the standard error decreases.

Question 14

A population has $$\mu = 80$$ and $$\sigma = 16. If $$you take samples of size $$n = 64$$, what is the standard error of the mean?

Accepted Answer

$$\sigma_x = \frac{16}{\sqrt{64}} = 2$$

Question 15

A population is normally distributed with $$\mu = 70$$ and $$\sigma = 14. If $$you take samples of size $$n = 4$$, what is the distribution of the sample means?

Accepted Answer

The sample means will be normally distributed.

Question 16

A population has $$\mu = 55$$ and $$\sigma = 11. If $$you take samples of size $$n = 49$$, what is the mean and standard error of the sample means?

Accepted Answer

Mean: $$\mu_x = 55$$, Standard error: $$\sigma_x = \frac{11}{\sqrt{49}} = 1.57$$

Question 17

Which scenario best illustrates the Central Limit Theorem in action?

Accepted Answer

A scientist measures the average blood pressure from many random samples of patients, and finds the distribution of sample means is nearly normal, even though individual blood pressures are not.

Question 18

If the sample size is small and the population is not normal, what can be said about the distribution of sample means?

Accepted Answer

The distribution of sample means may not be normal.

Question 19

A population has $$\mu = 90$$ and $$\sigma = 18. If $$you take samples of size $$n = 81$$, what is the standard error of the mean?

Accepted Answer

$$\sigma_x = \frac{18}{\sqrt{81}} = 2$$

The Central Limit Theorem and Sampling Distributions

Study Guide - Practice Questions

Study Guide - Flashcards