Q: For a population with $$p = 0.4$$ and $$n = 50$$, what is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

$$\sqrt{\dfrac{0.4 \times 0.6}{50}}$$

Question 1

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

Accepted Answer

$$\mu_{\overline{x}} = 50$$

Question 2

Given a population with mean $$\mu = 100$$ and standard deviation $$\sigma = 20$$, what is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 16$$?

Accepted Answer

$$\sigma_{\overline{x}} = 5$$

Question 3

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

Accepted Answer

As the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution.

Question 4

A population is strongly right-skewed. According to the Central Limit Theorem, what can be said about the shape of the sampling distribution of the sample mean for samples of size $$n = 40$$?

Accepted Answer

It will be approximately normal.

Question 5

If the population standard deviation is $$\sigma = 12$$ and the sample size is $$n = 36$$, what is the standard error of the sample mean?

Accepted Answer

$$2$$

Question 6

A researcher collects 200 independent random samples of size $$n = 5$$ from a population and computes the sample mean for each. What is the distribution of these 200 sample means called?

Accepted Answer

The sampling distribution of the sample mean

Question 7

Suppose the population proportion is $$p = 0.3$$ and the sample size is $$n = 100. $$What is the mean of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.3$$

Question 8

For a population with $$p = 0.4$$ and $$n = 50$$, what is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{0.4 	imes 0.6}{50}}$$

Question 9

A population is not normally distributed. Which of the following is true about the sampling distribution of the sample mean as the sample size increases?

Accepted Answer

It becomes approximately normal as $$n$$ increases.

Question 10

If the mean of a population is $$\mu = 80$$ and the standard deviation is $$\sigma = 16$$, what is the probability that a sample mean from a random sample of size $$n = 64 is $$greater than $$82$$? (Assume normality.)

Accepted Answer

$$P(\overline{x} \u003e 82) = P\left(z \u003e \dfrac{82-80}{2}\right) = P(z \u003e 1) \approx 0.1587$$

Question 11

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

Accepted Answer

The population must be normally distributed.

Question 12

A company claims that 15% of its products are defective. In a random sample of 200 products, what is the expected number of defective products?

Accepted Answer

30

Question 13

If the population mean is $$\mu$$ and the standard deviation is $$\sigma$$, what is the formula for the standard deviation of the sampling distribution of the sample mean for samples of size $$n$$?

Accepted Answer

$$\sigma_{\overline{x}} = \dfrac{\sigma}{\sqrt{n}}$$

Question 14

A population has $$\mu = 60$$ and $$\sigma = 12. $$What is the probability that a sample mean from a random sample of size $$n = 36 is $$less than $$57$$? (Assume normality.)

Accepted Answer

$$P(\overline{x} \u003c 57) = P\left(z \u003c \dfrac{57-60}{2}\right) = P(z \u003c -1.5) \approx 0.0668$$

Question 15

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

Accepted Answer

It decreases as sample size increases.

Question 16

A population is bimodal. If you take many samples of size $$n = 50$$, what will the shape of the sampling distribution of the sample mean be?

Accepted Answer

Approximately normal

Question 17

If $$P(\overline{x} \u003e 40) is $$the probability that the sample mean is greater than 40, how does this differ from $$P(x \u003e 40)$$?

Accepted Answer

$$P(\overline{x} \u003e 40)$$ refers to the probability for the sample mean, while $$P(x \u003e 40)$$ refers to the probability for a single observation.

Question 18

A population has $$p = 0.2. $$For a sample of size $$n = 100$$, what is the standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{0.2 	imes 0.8}{100}} = 0.04$$

Question 19

In a study, the sample mean of home runs for 2000 samples of size $$n = 4 is $$found to be 18.2 with a standard deviation of 2.5. What does the standard deviation of 2.5 represent?

Accepted Answer

The standard error of the sample mean

Sampling Distributions: Sample Means and Proportions

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