Question 1

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

Accepted Answer

The distribution of all possible sample means for all samples of size $$n$$ taken from the same population.

Question 2

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

Accepted Answer

It approaches a normal distribution, regardless of the population's shape.

Question 3

Suppose the population mean is $$\mu = 50$$ and the population standard deviation is $$\sigma = 10. $$What is the standard deviation of the sampling distribution of the sample mean for samples of size $$n = 25$$?

Accepted Answer

$$\$$\sigma_{\bar{x}} = \frac{10}{\sqrt{25}} = 2$

Question 4

A ski gondola has a maximum capacity of 15 people or 2625 lb. If the mean weight of passengers is $$\mu = 169.6 lb $$and $$\sigma = 43.1 lb$$, what is the mean weight per person that would exceed the gondola's limit?

Accepted Answer

$$\175$ lb

Question 5

If the original population is normally distributed, what can be said about the sampling distribution of the sample mean for any sample size $$n$$?

Accepted Answer

It will be normally distributed for any sample size.

Question 6

In John Kerrich’s coin-tossing experiment, he tossed a coin 10,000 times and grouped the results into samples of 100 tosses each. What does the distribution of the number of heads in each group illustrate?

Accepted Answer

The sampling distribution of the sample mean.

Question 7

Given a population with mean $$\mu = 63.1 in $$and standard deviation $$\sigma = 2.6 in$$, what is the standard error of the mean for a sample of $$n = 31$$ women?

Accepted Answer

$$\$$\frac{2.6}{\sqrt{31}} \approx 0.467$ in

Question 8

If the probability that a single passenger weighs more than 175 lb is 41.2%, but the probability that the mean weight of 15 randomly selected passengers exceeds 175 lb is 31.2%, what statistical concept explains the difference?

Accepted Answer

Central Limit Theorem

Question 9

Which of the following is a practical rule for applying the Central Limit Theorem?

Accepted Answer

If $$n \u003e 30$$, the distribution of sample means can be approximated by a normal distribution.

Question 10

A researcher wants to find the probability that the mean height of 31 randomly selected women is less than 64 in, given $$\mu = 63.1 in $$and $$\sigma = 2.6 in. $$Which $$z-$$score formula should be used?

Accepted Answer

$$\z$$ = \frac{64 - 63.1}{2.6/\sqrt{31}}$

Question 11

Why is it important to use the standard error of the mean when calculating probabilities for sample means?

Accepted Answer

Because it reflects the variability of sample means around the population mean.

Question 12

If a population is not normally distributed, what sample size is generally considered sufficient for the sampling distribution of the sample mean to be approximately normal?

Accepted Answer

$$n \u003e 30$$

Question 13

A population has mean $$\mu = 100$$ and standard deviation $$\sigma = 20. $$What is the probability that a random sample of $$n = 36$$ has a sample mean greater than 105? (Assume normality.)

Accepted Answer

First, calculate the standard error: $$\$$\sigma_{\bar{x}} = \frac{20}{\sqrt{36}} = 3.33$$. Then, z$$ = \frac{105 - 100}{3.33} = 1.5$$. The probability is 1 - 0.9332 = 0.0668$.

Question 14

Which of the following statements about the Central Limit Theorem is FALSE?

Accepted Answer

The Central Limit Theorem only applies if the population is normal.

Question 15

In the context of the gondola example, why is the probability that 15 people will exceed the weight limit lower than the probability that a single person exceeds 175 lb?

Accepted Answer

Because the mean of a sample is less variable than individual observations.

Sampling Distributions and the Central Limit Theorem

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