Question 1

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

Accepted Answer

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

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

Suppose the distribution of weights of apples in an orchard is highly skewed. If you take a random sample of $$n = 40$$ apples, which statement is true about the distribution of the sample mean?

Accepted Answer

It can be approximated by a normal distribution due to the Central Limit Theorem.

Question 4

Given a population with mean $$\mu = 100$$ and standard deviation $$\sigma = 20$$, what is the $$z-$$score for a sample mean $$\overline{x} = 108$$ from a sample of $$n = 16$$?

Accepted Answer

$$z = 1.6$$

Question 5

Which formula should you use to calculate the $$z-$$score for an individual value $$x$$ from a normal population?

Accepted Answer

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

Question 6

A researcher wants to use the normal distribution to approximate the sampling distribution of the mean. The population is not normally distributed, and the sample size is $$n = 20. $$What should the researcher conclude?

Accepted Answer

The normal approximation is not appropriate because $$n \leq 30$$ and the population is not normal.

Question 7

In the Boeing 737 seat width example, why is the probability that the mean hip width of 126 men exceeds 16.0 inches so much smaller than the probability that a single man exceeds 16.0 inches?

Accepted Answer

Because the standard deviation of the sample mean is much smaller than the standard deviation of individual values.

Question 8

If the population standard deviation is $$\sigma = 0.9$$ inches and the sample size is $$n = 126$$, what is the standard deviation of the sample mean?

Accepted Answer

$$0.1$$

Question 9

A company is considering reducing the width of its 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 $$z-$$score for a randomly selected man with a hip width of 16.0 inches?

Accepted Answer

$$z = 1.89$$

Question 10

Which of the following is a requirement for using the normal distribution to approximate the sampling distribution of the sample mean?

Accepted Answer

The population is normal or the sample size is greater than 30.

Question 11

A sample of $$n = 50 is $$taken from a population with mean $$\mu = 200$$ and standard deviation $$\sigma = 40. $$What is the probability that the sample mean is greater than $$210$$? (Use the $$z-$$score formula for sample means.)

Accepted Answer

First, calculate $$z = \frac{210 - 200}{40/\sqrt{50}} = 1.77. $$The probability is approximately $$0.0384.$$

Question 12

If you are working with a single value $$x$$ from a normally distributed population, which formula should you use to find the $$z-$$score?

Accepted Answer

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

Question 13

Why is the result from part (a) of the Boeing 737 seat width example more relevant for seat design than the result from part (b)?

Accepted Answer

Because individual seats are occupied by individual passengers, not by groups.

Question 14

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

Accepted Answer

It cannot be well approximated by a normal distribution.

Question 15

A sample of $$n = 100 is $$taken from a population with $$\mu = 60$$ and $$\sigma = 20. $$What is the probability that the sample mean is less than $$57$$? (Use the $$z-$$score formula for sample means.)

Accepted Answer

First, calculate $$z = \frac{57 - 60}{20/\sqrt{100}} = -1.5. $$The probability is approximately $$0.0668.$$

Question 16

Which of the following is NOT a correct step when solving a real-world problem involving the sample mean and the normal distribution?

Accepted Answer

Assume the normal distribution applies without checking the population distribution or sample size.

Question 17

A population has $$\mu = 75$$ and $$\sigma = 12. $$What is the probability that a randomly selected individual has a value greater than $$90$$? (Use the $$z-$$score formula for individuals.)

Accepted Answer

First, calculate $$z = \frac{90 - 75}{12} = 1.25. $$The probability is approximately $$0.1056.$$

Question 18

If the mean of a population is $$\mu$$ and the standard deviation is $$\sigma$$, what is the mean of the sampling distribution of the sample mean $$\overline{x}$$ for samples of size $$n$$?

Accepted Answer

$$\mu$$

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