Question 1

Which of the following best describes the sampling distribution of a statistic?

Accepted Answer

It is the distribution of all values of a statistic when all possible samples of the same size n are taken from the same population.

Question 2

Suppose the population proportion of adults uncomfortable with self-driving vehicles is $$p = 0.70. If 50$$,000 surveys of size $$n = 1000$$ are conducted, what will the mean of the sample proportions $$\hat{p} be$$?

Accepted Answer

$$0.70$$

Question 3

Which of the following is TRUE about the behavior of sample proportions as sample size increases?

Accepted Answer

Sample proportions tend to be normally distributed.

Question 4

Given a population with mean $$\mu = 3.5$$, if you repeatedly roll a die 5 times and calculate the sample mean $$\bar{x}$$, what is the expected value of $$\bar{x}$$?

Accepted Answer

$$3.5$$

Question 5

Which of the following statistics is an unbiased estimator of its corresponding population parameter?

Accepted Answer

Sample variance $$s^2$$

Question 6

Why is sampling with replacement often preferred in probability theory?

Accepted Answer

It results in independent events, simplifying analysis and formulas.

Question 7

If the sample standard deviation $$s is $$used to estimate the population standard deviation $$\sigma$$, what is true about $$s$$?

Accepted Answer

$$s$$ tends to underestimate $$\sigma.$$

Question 8

A researcher rolls a die 5 times and records the variance $$s^2 of $$the results. If this process is repeated many times, what is the shape of the sampling distribution of $$s^2$$?

Accepted Answer

It is skewed to the right.

Question 9

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

Accepted Answer

It is always exactly equal to the population parameter in every sample.

Question 10

In a large population, the proportion of people with a certain trait is $$p = 0.4. If $$you take many random samples of size $$n = 100$$, what will the mean of the sample proportions $$\hat{p} be$$?

Accepted Answer

$$0.4$$

Question 11

Which of the following is a reason why the sample mean $$\bar{x} is $$considered an unbiased estimator of the population mean $$\mu$$?

Accepted Answer

The mean of all possible sample means equals the population mean.

Question 12

Which of the following statistics is a biased estimator of its corresponding population parameter?

Accepted Answer

Sample median

Question 13

If you roll a fair die 5 times and calculate the proportion of odd numbers, what is the expected value of the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.50$$

Question 14

Which of the following best describes the distribution of sample means as the sample size increases?

Accepted Answer

It becomes closer to a normal distribution.

Question 15

A sample of size $$n is $$taken from a population. Which formula correctly defines the sample proportion $$\hat{p}$$?

Accepted Answer

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

Question 16

Which of the following is TRUE about the sample variance $$s^2 as an $$estimator?

Accepted Answer

$$s^2 is an $$unbiased estimator of the population variance $$\$$sigma^2$$.

Question 17

Which of the following is NOT a reason for sampling with replacement?

Accepted Answer

It ensures that the sample size is always larger than the population.

Question 18

If the population variance is $$\$$sigma^2$$ = 2.9$$, what is the expected value of the sample variance $$s^2$$ when rolling a die 5 times and repeating the process many times?

Accepted Answer

$$2.9$$

Question 19

Which of the following is a property of the sampling distribution of the sample mean?

Accepted Answer

Its mean equals the population mean.

Question 20

Which of the following is a correct statement about the sample standard deviation $$s$$?

Accepted Answer

$$s is a $$biased estimator but the bias is small for large samples.

6.3 Sampling Distributions and Estimators: Chapter 6 Study Notes

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