Question 1

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

Accepted Answer

The CLT states that the sampling distribution of the sample mean will be approximately normal for large sample sizes, regardless of the population's distribution.

Question 2

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

Accepted Answer

$$\mu = 50$$

Question 3

What is the formula for the standard deviation of the sampling distribution of the sample mean (standard error)?

Accepted Answer

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

Question 4

A researcher collects 100 samples of size $$n = 30$$ from a population with mean $$\mu = 80$$ and standard deviation $$\sigma = 12. $$What is the expected mean and standard error of the sampling distribution of the sample mean?

Accepted Answer

Mean: $$80$$, Standard Error: $$\frac{12}{\sqrt{30}} \approx 2.19$$

Question 5

If the population distribution is highly skewed, which of the following is true about the sampling distribution of the sample mean as sample size increases?

Accepted Answer

It becomes approximately normal as sample size increases.

Question 6

Which statement is correct regarding the mean and standard deviation of the sampling distribution of the sample mean?

Accepted Answer

The mean equals the population mean, and the standard deviation equals $$\frac{\sigma}{\sqrt{n}}.$$

Question 7

A hospital is studying the average recovery time for a certain surgery. The population mean is 7 days, and the population standard deviation is 2 days. If they take a random sample of 16 patients, what is the standard error of the mean?

Accepted Answer

$$\frac{2}{\sqrt{16}} = 0.5$$

Question 8

Which of the following is NOT a way to determine the sampling distribution of the sample mean?

Accepted Answer

Using only the population mean without considering sample size.

Question 9

If a population is normally distributed, what is the shape of the sampling distribution of the sample mean for any sample size?

Accepted Answer

It is always normal.

Question 10

A study finds that the mean weight gain during pregnancy is 30 pounds with a standard deviation of 12.5 pounds. If a random sample of 25 underweight women has a mean weight gain of 36.2 pounds, what is the $$z-$$score for this sample mean?

Accepted Answer

$$z = \frac{36.2 - 30}{12.5 / \sqrt{25}} = \frac{6.2}{2.5} = 2.48$$

Question 11

Which of the following best explains why the standard error decreases as sample size increases?

Accepted Answer

Because $$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$, so increasing $$n$$ makes the denominator larger.

Question 12

If the population mean is $$\mu = 100$$ and the population standard deviation is $$\sigma = 15$$, what is the probability that a sample mean from a sample of $$n = 36 is $$greater than 105? (Assume normality.)

Accepted Answer

First, calculate $$\sigma_{\bar{x}} = \frac{15}{\sqrt{36}} = 2.5. $$Then, $$z = \frac{105 - 100}{2.5} = 2$$. $$P(\bar{x} \u003e 105) = 0.0228.$$

Question 13

Which of the following is a correct statement about the effect of increasing sample size on the sampling distribution of the sample mean?

Accepted Answer

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

Question 14

For a non-normal population, what is the minimum sample size generally recommended for the sampling distribution of the sample mean to be approximately normal?

Accepted Answer

$$n \geq 25$$

Question 15

Which of the following is the correct formula for the mean of the sampling distribution of the sample mean?

Accepted Answer

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

Question 16

A pharmaceutical company wants to estimate the average effect of a new drug. The population standard deviation is unknown, but the sample size is large ($$n = 50). $$Which theorem allows them to use the normal distribution to approximate the sampling distribution of the sample mean?

Accepted Answer

Central Limit Theorem

Question 17

If a particular sample is obtained, can it be obtained a second time in the same sampling process?

Accepted Answer

No, once a particular sample is obtained, it cannot be obtained a second time.

Question 18

Which of the following best describes the standard error of the mean?

Accepted Answer

It is the standard deviation of the sampling distribution of the sample mean.

Question 19

A population is not normally distributed. As the sample size increases, what happens to the shape of the sampling distribution of the sample mean?

Accepted Answer

It becomes approximately normal.

Question 20

Which of the following is a real-world example of using the sampling distribution of the sample mean?

Accepted Answer

Estimating the average blood pressure of adults in a city by taking multiple random samples.

Section 8.1 Distribution of the Sample Mean: Central Limit Theorem

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