Question 1

Which of the following best describes a sampling distribution of the sample mean?

Accepted Answer

The distribution of all possible means calculated from samples of size n taken from a population.

Question 2

If a population has mean $$\mu = 100$$ and standard deviation $$\sigma = 20$$, what is the mean of the sampling distribution of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$100$$

Question 3

What is the standard error of the mean for a population with $$\sigma = 30$$ and a sample size of $$n = 36$$?

Accepted Answer

$$5$$

Question 4

According to the Central Limit Theorem, what happens to the shape of the sampling distribution of the sample mean as the sample size increases?

Accepted Answer

It becomes more normally distributed, regardless of the population's distribution.

Question 5

A population is normally distributed with $$\mu = 50$$ and $$\sigma = 10. $$What is the probability that the mean of a random sample of $$n = 16 is $$less than $$47$$? (Use the z-score formula and standard normal table.)

Accepted Answer

0.0668

Question 6

Which formula correctly calculates the z-score for a sample mean $$\bar{x}$$ from a population with mean $$\mu$$ and standard deviation $$\sigma$$ for sample size $$n$$?

Accepted Answer

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

Question 7

A researcher samples 100 people from a population with $$\mu = 70$$ and $$\sigma = 15. $$What is the standard deviation of the sampling distribution of the sample mean?

Accepted Answer

$$1.5$$

Question 8

If the population distribution is not normal, what sample size is generally considered large enough for the Central Limit Theorem to apply?

Accepted Answer

$$n \geq 30$$

Question 9

A population has $$\mu = 200$$ and $$\sigma = 40. If a $$sample of $$n = 64 is $$taken, what is the probability that the sample mean is greater than $$210$$? (Use the z-score formula and standard normal table.)

Accepted Answer

0.1056

Question 10

Which of the following statements is TRUE regarding the mean and standard error of the sampling distribution of the sample mean as sample size increases?

Accepted Answer

The mean stays the same, but the standard error decreases.

Question 11

A company knows the lifespans of its light bulbs are normally distributed with $$\mu = 1200$$ hours and $$\sigma = 100$$ hours. What is the probability that a random sample of 25 bulbs has a mean lifespan between 1180 and 1220 hours?

Accepted Answer

0.9544

Question 12

If the population mean is $$\mu = 60$$ and the population standard deviation is $$\sigma = 12$$, what is the variance of the sampling distribution of the sample mean for samples of size $$n = 16$$?

Accepted Answer

$$9$$

Question 13

A random sample of 36 students is taken from a university where the average GPA is $$2.8$$ with a standard deviation of $$0.6. $$What is the probability that the sample mean GPA is less than $$2.7$$?

Accepted Answer

0.2525

Question 14

Which of the following is NOT a property of the sampling distribution of the sample mean?

Accepted Answer

The mean of the sampling distribution is always less than the population mean.

Question 15

A population has $$\mu = 500$$ and $$\sigma = 80. If a $$sample of $$n = 64 is $$taken, what is the z-score for a sample mean of $$520$$?

Accepted Answer

$$2$$

Question 16

Why is the Central Limit Theorem important in inferential statistics?

Accepted Answer

It allows us to use the normal distribution to make inferences about population means, even if the population is not normal, provided the sample size is large enough.

Sampling Distributions and the Central Limit Theorem: Study Guide

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