Q: Suppose you take a simple random sample of 1,000 voters from a population where the true proportion who support candidate A is $$p = 0.538. $$What is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

$$\sqrt{\dfrac{0.538 \times (1-0.538)}{1000}} = 0.015$$

Q: A random sample of 100 USC students found that 42% had attended at least one football game. What is the standard error of the sample proportion?

$$\sqrt{\dfrac{0.42 \times (1-0.42)}{100}} = 0.04935$$

Question 1

Suppose you take a simple random sample of 1,000 voters from a population where the true proportion who support candidate A is $$p = 0.538. $$What is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{0.538 	imes (1-0.538)}{1000}} = 0.015$$

Question 2

Which of the following statements about the sampling distribution of the sample mean $$\bar{x} is $$TRUE according to the Central Limit Theorem?

Accepted Answer

As the sample size $$n$$ increases, the sampling distribution of $$\bar{x}$$ approaches a normal distribution regardless of the population's shape.

Question 3

A random sample of 100 USC students found that 42% had attended at least one football game. What is the standard error of the sample proportion?

Accepted Answer

$$\sqrt{\dfrac{0.42 	imes (1-0.42)}{100}} = 0.04935$$

Question 4

Why is the sample proportion $$\hat{p}$$ considered an unbiased estimator of the population proportion $$p$$?

Accepted Answer

Because $$E(\hat{p}) = p$$, meaning the expected value of $$\hat{p}$$ equals the true population proportion.

Question 5

If you increase the sample size $$n in a $$survey, what happens to the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

It decreases, because the standard deviation is $$\sqrt{\dfrac{p(1-p)}{n}}.$$

Question 6

A random sample of 50 students is selected from a university where 78% of all students attended public high school. What is the probability that the sample proportion is less than 0.80? (Assume normality.)

Accepted Answer

First, calculate the standard error: $$SE = \sqrt{\dfrac{0.78 	imes 0.22}{50}} = 0.0586. $$Then, $$z = \dfrac{0.80 - 0.78}{0.0586} = 0.341. $$The probability $$P(\hat{p} \u003c 0.80) = 0.6331.$$

Question 7

Which of the following is a necessary condition for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

Both $$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 8

A random sample of 36 roller coasters in the US is selected. The average height is 103 feet and the sample standard deviation is 24 feet. What is the probability that a randomly selected sample has a mean height greater than 100 feet?

Accepted Answer

First, $$SE = \dfrac{24}{\sqrt{36}} = 4$$. $$z = \dfrac{100 - 103}{4} = -0.75$$. $$P(X \u003e 100) = 0.7734$$

Question 9

In a population, the mean number of lawyers in big law firms is 532 with a standard deviation of 9.1. If 49 firms are sampled, what is the probability that the sample mean is between 531 and 533?

Accepted Answer

First, $$SE = \dfrac{9.1}{\sqrt{49}} = 1.3$$. $$z_1$$ = \dfrac{531 - 532}{1.3} = -0.769$$, z_2$ = \dfrac{533 - 532}{1.3} = 0.769$$. $$P(-0.769 \u003c Z \u003c 0.769) = 0.554$$

Question 10

Which of the following best describes a sampling distribution?

Accepted Answer

The distribution of a statistic (like $$\hat{p} or$$ $$\bar{x}) $$over all possible samples of a given size from a population.

Question 11

Why is it important that samples are random and representative when using sampling distributions for inference?

Accepted Answer

Because only random, representative samples ensure unbiased estimators and valid inferences.

Question 12

A random sample of 3,000 voters is taken, and 65% support a recall. What is the 95% interval for the sample proportion using the 2-standard deviation rule?

Accepted Answer

First, $$SE = \sqrt{\dfrac{0.65 	imes 0.35}{3000}} = 0.00871. $$The interval is $$0.65 \pm 2 	imes 0.00871 = [0.63258, 0.66742]$$

Question 13

If the population mean is $$\mu = 55$$ seconds and the standard deviation is $$\sigma = 2.2$$ seconds, what is the 68% probability interval for the mean time of a random sample of 50 swimmers?

Accepted Answer

First, $$SE = \dfrac{2.2}{\sqrt{50}} = 0.311. $$The interval is $$55 \pm 0.311 = [54.689, 55.311]$$

Question 14

Which of the following is NOT a property of an unbiased estimator?

Accepted Answer

Its expected value is always less than the true parameter.

Question 15

If you repeatedly take samples of size $$n$$ from a population and calculate the sample mean each time, what will the distribution of these sample means look like as $$n$$ increases?

Accepted Answer

It will become approximately normal, regardless of the population's original distribution.

Question 16

A researcher wants to estimate the proportion of students who have attended at least one game. She takes a random sample of 100 students and finds $$\hat{p} = 0.42. $$What is the correct way to describe the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\hat{p} \sim Normal(0.42, 0.04935)$$

Question 17

Which of the following best explains why larger samples produce less variable sample statistics?

Accepted Answer

Because the standard error decreases as $$n$$ increases, according to $$SE = \dfrac{\sigma}{\sqrt{n}}.$$

Sampling Distributions: Concepts, Properties, and Applications

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드