Worksheet on the Sampling Distribution of the Sample Mean

Use the following to answer questions 1–3:

The scores of individual students on the American College Testing (ACT) Program Composite College

Entrance Examination have a Normal distribution with mean 18.6 and standard deviation 6.0. At

Northside High, 36 seniors take the test. Assume the scores at this school have the same distribution as

national scores.

1. What is the mean of the sampling distribution of the sample mean score for a random sample of

36 students?

A) 1.0 B) 3.1 C) 6.0 D) 18.6

2. What is the standard deviation of the sampling distribution of the sample mean score for a

random sample of 36 students?

A) 1.0 B) 3.1 C) 6.0 D) 18.6

3. What is the sampling distribution of the sample mean score for a random sample of 36 students?

A) Approximately Normal, but the approximation is poor.

B) Approximately Normal, and the approximation is good.

C) Exactly Normal.

D) Neither Normal nor non-Normal. It depends on the particular 36 students selected.

4. A sample of size n is selected at random from a population that has mean and standard

deviation

. The sample mean X will be determined from the observations in the sample.

Which of the following statements about the sample mean, X, is (are) TRUE?

A) The mean of X is the same as the population mean, i.e.,

.

B)

The variance of X is n/2.

C) The standard deviation of X decreases as the sample size grows larger.

D) All of the above are true.

E) Only A and B are true.

5. A gasoline tank for a certain car is designed to hold 15 gallons of gas. Suppose that the actual

capacity of a randomly selected tank has a distribution that is approximately Normal with a mean of 15.0

gallons and a standard deviation of 0.15 gallons. If four tanks are randomly selected, what is the

probability that their average capacity will be between 14.75 and 15.10 gallons?

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A) 0.2397 B) 0.6808 C) 0.9084 D) 0.9962

Use the following to answer questions 6 and 7: During the summer months, the prices of nonsmoking

rooms with a king-size bed in hotels in a certain area are roughly Normally distributed with a mean of

$131.80 and a standard deviation of $29.12.

6. What percentage of nonsmoking rooms with a king-size bed cost more than $150?

A) 0.77% B) 11.25% C) 26.60% D) 36.78%

7. A travel agent randomly selects prices of nonsmoking rooms with a king-size bed from 15 hotels

in the area. What is the probability that their average cost will be more than $150?

A) 0.77% B) 11.25% C) 26.60% D) 36.78%

8. The following histogram shows the distribution of 1000 sample observations from a population

with mean = 4 and variance 2 = 8:

010

Suppose a simple random sample of 100 observations is to be selected from the population and the

sample average, x, calculated. Which of the following statements about the distribution of x is (are)

FALSE?

A) The distribution of x will have a mean of 4.

B) The distribution x will be approximately Normal.

C) Because the distribution shown in the histogram above is clearly skewed to the right, the shape of

the distribution of xwill show as much skewness to the right as the population variable.

D) Even though the distribution of the population variable appears to be skewed to the right, the

distribution of x will be approximately symmetric around = 4.

E) The standard deviation of the distribution of x will be 0.283.9.

Use the following to answer questions 9–11: The length of time it takes to get through the security checks

at a very large urban airport is a random variable with mean 6.20 minutes and standard deviation

4.8. A simple random sample of 36 airline passengers is to be observed going through security.

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9. What is the mean,

x, of the sampling distribution of the sample mean, x

?

A) 8.4 minutes

B) 20.6 minutes

C) 1.4 minutes

D) Close to 20.6 minutes but we can’t be certain without having the data to determine it.

E) Without knowing the distribution, we don’t know what it will be.

10. What is the standard deviation, x

, of the sampling distribution of the sample mean, x

?

A) 8.4 minutes B) 20.6 minutes

B) 1.4 minutes D) None of these

11. For the passengers in the SRS, what is the approximate probability that the mean length of time

for them to get through security will be less than 18 minutes?

A) 0.378 B) 0.186 C) 0.031 D) 0.314

12.

Suppose a simple random sample is selected from a population with mean and variance 2.

The central limit theorem tells us that

A) the sample mean xgets closer to the population mean as the sample size increases.

B) if the sample size n is sufficiently large, the sample will be approximately Normal.

C) the mean of x will be if the sample size n is sufficiently large.

D) if the sample size is sufficiently large, the distribution of x will be approximately Normal with

mean and standard deviation, n.

E) the distribution of xwill be Normal only if the population from which the sample is selected is

also Normal.

13. From the central limit theorem, we know that if we draw a SRS from any population the sampling

distribution of the sample mean will be EXACTLY Normal. TRUE or FALSE

14. In order to reduce the standard deviation from the sampling distribution of the sample

mean, you should ________.

A) take larger samples

B) take smaller samples

C) None of the above.

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