Question 1

Which statement best describes the Central Limit Theorem (CLT)?

Accepted Answer

For any random variable, the distribution of the sample mean approaches a normal distribution as the sample size increases.

Question 2

A researcher collects 10 samples of 20 students each to study the average number of siblings at a university. According to the Central Limit Theorem, what is the value of $$n$$ for the sampling distribution?

Accepted Answer

$$n = 20$$

Question 3

Suppose the population mean play time for a video game is 36.7 hours. A retailer takes 100 random samples of 40 players each. What is the mean of the sampling distribution?

Accepted Answer

$$36.7$$

Question 4

If the population mean $$bc = 3.2$$ and population standard deviation $$c = 0.95$$, what is the standard error of the mean for a sample size of $$n = 60$$?

Accepted Answer

$$\frac{0.95}{\sqrt{60}}$$

Question 5

A marketing team takes 80 samples of 10 recent clients to create a sampling distribution of sample means for customer satisfaction. According to the Central Limit Theorem, what is the sample size $$n$$ used to determine the sampling distribution?

Accepted Answer

$$n = 10$$

Question 6

Which formula is used to calculate the $$z-$$score for a sample mean in a sampling distribution?

Accepted Answer

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

Question 7

A die is rolled 30 times, and the mean of the results is recorded for each trial. What does the Central Limit Theorem predict about the distribution of these sample means?

Accepted Answer

The distribution of sample means will be approximately normal.

Question 8

In a movie rating scenario, the average rating is $$4.2$$ with a standard deviation of $$0.25. $$What is the probability that a random sample of 8 moviegoers will give the film an average rating of $$4.3 or $$greater?

Accepted Answer

Calculate $$z = \frac{4.3 - 4.2}{\frac{0.25}{\sqrt{8}}}$$ and use the standard normal table.

Question 9

Why does the sampling distribution of the sample mean become approximately normal as $$n$$ increases, regardless of the population's distribution?

Accepted Answer

Because the Central Limit Theorem states that for large $$n$$, the sample mean's distribution approaches normality.

Question 10

A pet owner is interested in the average number of pets owned by households in America. They take 30 samples of 8 households each. What is the distribution of the sample means likely to look like?

Accepted Answer

It will be more normal than the distribution of the original data.

Question 11

If the population mean is $$\mu = 3.2$$ and the population standard deviation is $$\sigma = 0.95$$, what is the probability of getting a sample mean above $$3.5 in a $$sample of $$n = 60$$?

Accepted Answer

Find $$z = \frac{3.5 - 3.2}{\frac{0.95}{\sqrt{60}}}$$ and use the standard normal table.

Question 12

Which of the following best describes a sampling distribution?

Accepted Answer

It is the distribution of a statistic (such as the mean) calculated from many samples of the same size from a population.

Question 13

A sample mean is said to be 'off' from the population mean. What does the sampling distribution help us understand about this?

Accepted Answer

It helps us understand how much sample means vary from the population mean.

Question 14

If you increase the sample size $$n$$ when constructing a sampling distribution, what happens to the standard error?

Accepted Answer

It decreases.

Question 15

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

Accepted Answer

The sample size must be very small.

Question 16

A company wants to estimate the average time employees spend on a task. They take 50 samples of 20 employees each. What is the mean of the sampling distribution if the population mean is 16.7 minutes?

Accepted Answer

$$16.7$$

Question 17

Which scenario best illustrates the application of the Central Limit Theorem in a real-world context?

Accepted Answer

A hospital averages the blood pressure readings from 100 randomly selected patients to estimate the average blood pressure of all patients.

Question 18

If the sample size $$n is $$large, which of the following is true about the distribution of the sample mean?

Accepted Answer

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

Question 19

A researcher wants to find the probability that the sample mean is less than 2.5 for a sample of size $$n = 30$$ when rolling a die. Which formula should they use?

Accepted Answer

$$z = \frac{2.5 - \mu}{\frac{\sigma}{\sqrt{30}}}$$

Sampling Distributions and the Central Limit Theorem: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards