Question 1

Which of the following best describes the Central Limit Theorem?

Accepted Answer

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

Question 2

Suppose the population mean sleep time for college students is 6.7 hours with a standard deviation of 0.8 hours. If you take a random sample of 100 students, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{0.8}{\sqrt{100}} = 0.08$$

Question 3

A population has a mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the mean of the sampling distribution of the sample means for samples of size $$n = 25$$?

Accepted Answer

$$50$$

Question 4

If the population is normally distributed with mean $$\mu$$ and standard deviation $$\sigma$$, what is the shape of the sampling distribution of the sample means for any sample size $$n$$?

Accepted Answer

Normal

Question 5

A researcher selects a random sample of 36 drivers from a population with mean daily driving distance $$\mu = 20$$ miles and $$\sigma = 5$$ miles. What is the probability that the sample mean is greater than 21 miles? (Assume normality.)

Accepted Answer

First, calculate the standard error: $$\frac{5}{\sqrt{36}} = 0.833. $$Then, $$z = \frac{21 - 20}{0.833} = 1.20. $$The probability to the right of $$z = 1.20 is $$approximately $$0.1151.$$

Question 6

Which formula gives the standard deviation of the sampling distribution of the sample mean (standard error) for samples of size $$n$$?

Accepted Answer

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

Question 7

A population of credit card expenses is normally distributed with mean $$\mu = 2268$$ and $$\sigma = 750. $$What is the probability that a randomly selected student paid less than $$2000$$?

Accepted Answer

Find $$z = \frac{2000 - 2268}{750} = -0.3587. $$The probability is approximately $$0.3594.$$

Question 8

If you take a sample of size $$n = 25$$ from the same population in the previous question, what is the probability that the sample mean is less than $$2000$$?

Accepted Answer

Find standard error: $$\frac{750}{\sqrt{25}} = 150$$. $$z = \frac{2000 - 2268}{150} = -1.79. $$Probability is approximately $$0.0367.$$

Question 9

Why is the probability that a sample mean is less than a certain value usually less than the probability that a single observation is less than that value?

Accepted Answer

Because the standard error of the mean is smaller than the population standard deviation, making extreme sample means less likely.

Question 10

A population is not normally distributed. What is the minimum sample size $$n$$ required for the sampling distribution of the sample mean to be approximately normal according to the Central Limit Theorem?

Accepted Answer

$$n \geq 30$$

Question 11

A population has mean $$\mu = 135$$ and standard deviation $$\sigma = 18. If $$you take samples of size $$n = 4$$, what is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{18}{\sqrt{4}} = 9$$

Question 12

Which of the following statements is TRUE regarding the mean of the sampling distribution of the sample means?

Accepted Answer

It is always equal to the population mean.

Question 13

A sample of size $$n = 9 is $$drawn from a population with mean $$\mu = 13,175$$ and standard deviation $$\sigma = 1,650. $$What is the probability that the sample mean is less than $$13,600$$?

Accepted Answer

Standard error: $$\frac{1650}{\sqrt{9}} = 550$$. $$z = \frac{13,600 - 13,175}{550} = 0.77. $$Probability is approximately $$0.7794.$$

Question 14

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

The standard deviation of the sample means is equal to the population standard deviation.

Question 15

A population has mean $$\mu = 100$$ and standard deviation $$\sigma = 20. If $$you take a sample of size $$n = 16$$, what is the variance of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{20^2$}{16} = 25$$

Sampling Distributions and the Central Limit Theorem: Study Notes

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