Question 1

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

Accepted Answer

For any population, the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (n \u003e 30), regardless of the population's distribution.

Question 2

Given a population with 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

$$\sigma_{\bar{x}} = \frac{10}{\sqrt{25}} = 2$$

Question 3

A researcher collects a random sample of $$n = 40$$ from a population with unknown distribution, mean $$\mu = 100$$, and standard deviation $$\sigma = 15. $$What is the mean of the sampling distribution of the sample mean?

Accepted Answer

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

Question 4

Which formula should be used to calculate the z-score for a sample mean $$\bar{x}$$ when the sample size is large ($$n \u003e 30$$)?

Accepted Answer

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

Question 5

In the Boeing 737 seat width example, why is the probability of all 126 males having a mean hip width greater than 16.0 inches so much smaller than the probability for a single individual?

Accepted Answer

Because the standard error of the mean ($$\sigma_{\bar{x}}) is $$much smaller than the population standard deviation, making extreme sample means highly unlikely.

Question 6

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

Accepted Answer

$$\sigma_{\bar{x}} = \frac{0.62}{\sqrt{106}} = 0.0602$$

Question 7

If the original population is not normally distributed and $$n \leq 30$$, what should you do when analyzing the sample mean?

Accepted Answer

Use nonparametric or bootstrapping methods instead of the normal distribution.

Question 8

A sample of 106 body temperatures has a mean of 98.2°F. If the population mean is 98.6°F and $$\sigma = 0.62$$, what is the z-score for this sample mean?

Accepted Answer

$$z = \frac{98.2 - 98.6}{0.0602} = -6.64$$

Question 9

According to the Rare Event Rule for Inferential Statistics, what should you conclude if the probability of an observed sample mean is extremely low under the assumption of a given population mean?

Accepted Answer

The assumption about the population mean is probably not correct.

Question 10

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

Accepted Answer

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

Question 11

A company is considering reducing the width of airplane seats to 16.0 inches. If the mean hip width of adult males is 14.3 inches with a standard deviation of 0.9 inches, what is the probability that a randomly selected male will not fit in the seat?

Accepted Answer

Approximately 0.03 (or 3%)

Question 12

Why is the result for an individual more relevant than the result for the sample mean when designing airplane seats?

Accepted Answer

Because each seat is occupied by an individual, not by a group average.

Question 13

If you are given a sample mean $$\bar{x}$$, population mean $$\mu$$, population standard deviation $$\sigma$$, and sample size $$n$$, which formula would you use to find the probability of observing $$\bar{x} or $$more extreme values?

Accepted Answer

First, calculate $$z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$$, then use the standard normal distribution to find the probability.

Question 14

A sample of size $$n = 20 is $$drawn from a population that is not normally distributed. Which of the following statements is correct?

Accepted Answer

The sampling distribution of the sample mean cannot be well-approximated by a normal distribution.

Question 15

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

Accepted Answer

The population mean must be equal to the sample mean.

6.4 Central Limit Theorem and Normal Probability Distributions: Study Notes

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