Question 1

Suppose the weights of pennies minted after 1982 are normally distributed with mean $$\mu = 2.46$$ grams and standard deviation $$\sigma = 0.02$$ grams. What is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$\frac{0.02}{\sqrt{25}} = 0.004$$

Question 2

Which of the following statements best describes the Central Limit Theorem?

Accepted Answer

As the sample size $$n$$ increases, the sampling distribution of the sample mean becomes approximately normal, regardless of the shape of the population distribution.

Question 3

A random sample of 90 school-aged children results in 24 overweight children. If the population proportion is $$p = 0.188$$, what is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{24}{90} = 0.2667$$

Question 4

If 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?

Accepted Answer

$$n \geq 30$$

Question 5

Given a population with mean $$\mu = 11.4$$ and standard deviation $$\sigma = 3.2$$, what is the mean and standard deviation of the sampling distribution of the sample mean for $$n = 35$$?

Accepted Answer

Mean $$= 11.4$$, Standard deviation $$= \frac{3.2}{\sqrt{35}} = 0.5409$$

Question 6

Which of the following is the formula for the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$

Question 7

A poll finds that 42% of registered voters support a policy. For a random sample of $$n = 50$$, what is the mean of the sampling distribution of the sample proportion?

Accepted Answer

$$0.42$$

Question 8

Which condition must be met for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

$$np(1-p) \geq 10$$

Question 9

If the population standard deviation is $$\sigma = 0.02$$ and the sample size is increased from $$n = 5 to$$ $$n = 20$$, what happens to the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

It decreases by a factor of $$\sqrt{4} = 2$$

Question 10

A random sample of 90 children yields a sample proportion $$\hat{p} = 0.2667$$ for being overweight, while the population proportion is $$p = 0.188. $$What is the $$z-$$score for this sample proportion if the standard deviation is $$0.0412$$?

Accepted Answer

$$z = \frac{0.2667 - 0.188}{0.0412} = 1.91$$

Question 11

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

Accepted Answer

The distribution becomes more normal and less variable.

Question 12

If a population is normally distributed, what can be said about the sampling distribution of the sample mean for any sample size $$n$$?

Accepted Answer

It is also normally distributed.

Question 13

A sample of size $$n = 100 is $$drawn from a population with unknown distribution. Which statement is true about the sampling distribution of the sample mean?

Accepted Answer

It will be approximately normal due to the Central Limit Theorem.

Question 14

Which of the following is the formula for the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{x}{n}$$

Question 15

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

Accepted Answer

$$\frac{10}{\sqrt{25}} = 2$$

Question 16

Which of the following is NOT a requirement for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

$$n \geq 30$$

Sampling Distributions: Sample Means and Sample Proportions

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