Question 1

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

Accepted Answer

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

Question 2

If you roll a fair six-sided die 5 times and calculate the proportion of odd numbers, what is the expected value of the sample proportion as the process is repeated indefinitely?

Accepted Answer

0.50

Question 3

Which formula correctly represents the z-score for a sample mean when the sample size is greater than 30?

Accepted Answer

$$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$

Question 4

Which of the following statistics is NOT an unbiased estimator of its corresponding population parameter?

Accepted Answer

Sample standard deviation $$s$$

Question 5

Why is sampling with replacement often preferred in statistical analysis?

Accepted Answer

It results in independent events that are unaffected by previous outcomes, simplifying calculations and formulas.

Question 6

According to the Central Limit Theorem, what is the standard deviation of the sampling distribution of the sample mean for samples of size n?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Question 7

A population has a mean $$\mu = 10$$ and standard deviation $$\sigma = 2. $$What is the mean of all sample means for samples of size $$n = 50$$?

Accepted Answer

$$10$$

Question 8

Which of the following is a requirement for using the normal distribution to approximate the sampling distribution of the sample mean?

Accepted Answer

The sample size must be greater than 30 or the population must be normally distributed.

Question 9

In the context of airline seat design, why is the probability of an individual male having a hip width greater than the seat width more relevant than the probability that the mean hip width of all passengers exceeds the seat width?

Accepted Answer

Because each seat is occupied by an individual, not by the average of a group.

Question 10

Which of the following is true about the behavior of sample proportions?

Accepted Answer

The mean of all sample proportions equals the population proportion.

Question 11

If a population is not normally distributed and the sample size is less than or equal to 30, what can be said about the sampling distribution of the sample mean?

Accepted Answer

It cannot be approximated well by a normal distribution.

Question 12

Which formula should be used to convert an individual value $$x$$ from a normally distributed population to a z-score?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 13

Which of the following is a biased estimator?

Accepted Answer

Sample range

Question 14

What is the main reason the sample standard deviation $$s is $$used to estimate the population standard deviation $$\sigma$$ even though $$s is a $$biased estimator?

Accepted Answer

Because the bias is relatively small in large samples.

Question 15

A researcher selects a simple random sample of size $$n = 40$$ from a population with mean $$\mu = 100$$ and standard deviation $$\sigma = 20. $$What is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$\frac{20}{\sqrt{40}}$$

Question 16

Which of the following statements about unbiased estimators is correct?

Accepted Answer

An unbiased estimator targets the value of the corresponding population parameter.

Question 17

Suppose you have a population with mean $$\mu$$ and standard deviation $$\sigma. If $$you take all possible samples of size $$n$$ and calculate their means, what is the mean of the sampling distribution of the sample mean?

Accepted Answer

$$\mu$$

Question 18

Which of the following is NOT a requirement for the Central Limit Theorem to apply?

Accepted Answer

The population must be binomial.

Sampling Distributions and the Central Limit Theorem: Study Notes

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