Question 1

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

Accepted Answer

The CLT states that, for a sufficiently large sample size ($$n \geq 30$$), the distribution of sample means will be approximately normal, regardless of the shape of the population distribution.

Question 2

Given a population with mean $$\mu = 25$$ and standard deviation $$\sigma = 3$$, what is the standard error of the mean for samples of size $$n = 20$$?

Accepted Answer

$$\$$\sigma_{\bar{x}} = \frac{3}{\sqrt{20}} = 0.67\$

Question 3

A study finds that the average age of a boat is 96 months with a standard deviation of 16 months. If a random sample of 36 boats is selected, what is the probability that the sample mean is between 90 and 100 months?

Accepted Answer

First, calculate the standard error: $$\$$\sigma_{\bar{x}} = \frac{16}{\sqrt{36}} = 2\$ months. Then, convert 90 and 100 to z-scores and use the standard normal table to find the probability.

Question 4

Which formula should be used to convert a sample mean $$\bar{x} to a z-$$score when using the Central Limit Theorem?

Accepted Answer

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

Question 5

If the average number of pounds of pork consumed per year is 218.4 with a standard deviation of 25, what is the probability that a randomly selected person consumes less than 224 pounds per year?

Accepted Answer

Find the z-score: $$\Z$$ = \frac{224 - 218.4}{25} = 0.224\$; use the standard normal table to find the probability.

Question 6

A sample of 40 individuals is selected from a population where the mean pork consumption is 218.4 pounds and the standard deviation is 25 pounds. What is the probability that the sample mean is less than 224 pounds?

Accepted Answer

Calculate the standard error: $$\$$\sigma_{\bar{x}} = \frac{25}{\sqrt{40}} = 3.95\$; then find the z-score and use the normal table.

Question 7

Which of the following is NOT a property of the distribution of sample means?

Accepted Answer

The standard deviation of the sample means increases as sample size increases.

Question 8

If a population is not normally distributed, what is the minimum sample size required for the sampling distribution of the mean to be approximately normal according to the Central Limit Theorem?

Accepted Answer

$$n \geq 30$$

Question 9

Which scenario best illustrates the use of the Central Limit Theorem in a real-world context?

Accepted Answer

Estimating the average number of hours college students sleep per night by taking the mean of a large random sample.

Question 10

Why is the standard error of the mean smaller than the population standard deviation?

Accepted Answer

Because it is divided by the square root of the sample size: $$\$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\$

Question 11

A population has a mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the probability that a sample mean from $$n = 25$$ observations is greater than 54?

Accepted Answer

First, calculate $$\$$\sigma_{\bar{x}} = \frac{10}{\sqrt{25}} = 2\$$; then $$\$$Z = \frac{54 - 50}{2} = 2\$$; use the normal table to find the probability.

Question 12

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

Accepted Answer

The distribution becomes narrower and more concentrated around the population mean.

Question 13

If the original population is normal, what can be said about the sampling distribution of the mean for any sample size?

Accepted Answer

It will also be normal, regardless of sample size.

Question 14

A researcher wants to estimate the average time spent on social media by teenagers. The population standard deviation is 4 hours. If she wants the standard error to be less than 1 hour, what is the minimum sample size she should use?

Accepted Answer

$$n \u003e (\frac{4}{1})^2$ = 16$$

Question 15

Which of the following is the correct formula for the standard error of the mean?

Accepted Answer

$$\$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\$

The Normal Distribution, Sampling Distribution, and the Central Limit Theorem: Study Guide

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