Sample Size and Statistical Theory Test Bank Docx Chapter 15

Test Bank

CHAPTER 15 Sample Size and Statistical Theory

True-False

1. The two population characteristics of most interest to researchers

are μ and σ.

2. If the population variance is 4, the sample standard deviation must

be 2.

3. The population variance is a measure of the population dispersion,

based on the degree to which a response differs from the population mean response.

4. The equation for the sample variance is

S² =

5. The sample variance, S2, can be used to estimate the population

variation.

7. Two samples are drawn from the same population with a known

variance, σ². If the sample sizes are 100 and 400, then the

standard error of the mean of the former sample will be half that of the latter.

8. The sample mean is not known, but it can be estimated from the

population characteristics.

9. A 95 percent confidence interval will be smaller than a 90 percent

confidence interval.

10. If we wanted to be 95 percent confident that we wouldn't make an

error of more than 1 unit in our estimate of μ (for σ = 2), we

would need a sample size of approximately 16.

11. If the allowed error is increased, the required sample size would

decrease.

12. When a complex multi-stage design is involved, a common approach is

to replicate the entire sampling plan and get various independent

estimates of the sample mean.

13. If two random samples of equal size are drawn from the same

population, we should expect the mean of both samples to be identical.

14. The size of a sample can be determined only by using statistical

techniques.

15. A researcher takes a modest sample and, if results are worthwhile,

opts for a larger sample. This type of sampling is called sequential sampling.

16. If all members of the population have identical opinions on an issue,

sample of one is unsatisfactory.

17. It does not make sense to use disproportionate sampling if one of the

subgroups of the population is a relatively small percentage of the population

Multiple Choice

1. Bo Derek calculates that she needs a sample size of 200, given the estimated mean,

accuracy variance and required Z-score. However, she learns that the variance is

actually one-half of what she originally believed. Her required sample size will now be

1. 100
2. 800
3. 400
4. 50
5. none of the above

2. The distribution which we say always assume to be “normal” or bell shaped is the

1. sampling distribution
2. population distribution
3. sample distribution
4. variance distribution
5. standard distribution

3. The standard error of the mean is the same as the

1. standard deviation of the population
2. standard deviation of the normal distribution
3. standard deviation of the sampling distribution
4. none of the above

4. As the sample size becomes larger, the mean of the sampling distribution becomes

1. greater than the population mean
2. smaller than the population mean
3. equal to the population mean
4. double that of the population mean

5. Which of the following factors does the interval estimate depends on?

sample size?

1. sample size
2. confidence level
3. population standard deviation
4. all of the above

6. Consider the following:

For the given data:

a. μ= .6, σ² = 0, A = 0, B = 1, C = .3.

b. μ= 0, σ² =.6, A = 0, B = -1, C = .3.

c. μ= 0, σ² = .6, A = 0,B = 1, C= .3.

d. μ= 0, σ² = .6, A = .4,B = -1, C = .3.

e. μ= .6, σ² = 0, A = .4,B = 1,C = .3.

7. Two samples are drawn from the same population:

Which of the following statements is true?

1. The distribution of X₁ will be taller than that of X₂.

2. The standard deviations of the two means are equal.

3. μ₁ ≠ μ₂

4. The two population variances are equal.

a. 1

b. 2

c. 3

d. 4

e. 1 and 4

8. Consider the following data:

_

X = 10 σ² = 4 n = 25

The lower and upper limits of

1. a 90 percent confidence interval will be 9.36 and 10.64,

respectively.

2. a 95 percent confidence interval will be 9.216 and

10.784, respectively.

3. a 90 percent confidence interval cannot be determined

for the given data.

4. a 95 percent confidence interval indicates the range

within which μ must lie.

a. 1

b. 2

c. 3

d. 4

e. 1 and 2

9. Which of the following statements is true?

a. The sample size required to ensure 95 percent confidence limits

will be smaller than that for 90 percent confidence limits.

b. The sample size required to ensure a smaller confidence interval will also be smaller.

c. The only way to reduce the width of a confidence interval is to reduce the confidence requirements.

d. The required sample size is dependent on å only.

e. The required sample size for given error and confidence levels is independent of the population size.

10. Let the variance of a normally distributed random sample be 16. A

point that is 8 units away from the mean is

1. a point such that the probability of getting any other

point further from the mean is less than 0.025.

2. a point such that the probability of getting any other

point further from the mean is less than or equal to .95.

3. a point such that the probability of getting any other

point further from the mean is greater than 0.025.

4. two standard deviations away from the mean.

a. 1

b. 2

c. 3

d. 4

e. 1 and 4

11. It is estimated that the proportion of Californians in favor of

Proposition 5 is between 0.4 and 0.6. The sponsors of the proposition would like to sample Californians and get a 95 percent confidence interval for the true proportion of people in favor of Proposition 5. They want to restrict the error level to 5 percentage points. The required sample size is then

a. 400.

b. 40.

c. 100.

d. 1,000.

e. none of the above.

12. ABC Incorporated wants to find out the proportion of customers who

would purchase their new product. They have decided to take a modest

sample, look at the results, and then decide if more information, in

the form of a larger sample, is needed. ABC is using

a. simple random sampling.

b. biased random sampling.

c. stratified sampling.

d. sequential sampling.

e. none of the above.

Use the following information for questions 15 through 17

A soap factory produces 500-pound boxes of soap for industries. However, the weight of the boxes varies somewhat due to the process. Assume that the weight of each box is distributed normally with

mean = μ = 500 pounds variance =σ² = 25 pounds²

Quality control takes a random sample of 25 boxes per day. Let the weight of these boxes be denoted as X₁, X₂, ..., X₂₅.

13. The probability that X8 is between 495 and 505 pounds is closest to

a. .0228

b. .3174

c. .9772

d. .6826

e. .3830

14. The probability that the sample mean lies between 498 and 502 is

closest to

a. .3174

b. .9772

c. .9544

d. .3108

e. .6826

15. What happens to the standard deviation of the sample mean, if we

change the sample size from 25 to 100?

a. It is decreased by a factor of 4.

b. It is increased by a factor of 4.

c. It is decreased by a factor of 2.

d. It is decreased by a factor of 16.

e. None of the above.

16. A market research firm has been engaged to do a study of dishwashing

detergents preferred by homemakers in San Francisco (800,000

population) as compared to those preferred in New York (8,000,000

population). In order to obtain results of about equal reliability

in each city, the number of homemakers polled in New York should be

a. ten times the number polled in San Francisco.

b. 10 or 3.16 times the number polled in San Francisco.

c. about the same as in San Francisco.

d. half the number polled in San Francisco.

17. A college dean wants to determine the average time it takes

student to get from one class to the next. Preliminary studies have

shown that it is reasonable to let σ_t = 1.5 minutes (the standard

deviation of class travel times). What is the sample size which

should be examined if the dean wants to assert with a probability of

95 percent that his estimate, the mean time of the random sample,

will not be "off" by more than .25 minutes (in either direction)?

Choose the closest answer.

a. 72 or more, but not less

b. 81 or more, but not less

c. 139 or more, but not less

d. 200 or more, but not less

e. None of the above

18. Assume that μ is distributed normally with μ = 0, σ2 = 1. What is

P(z = 1)?

a. 0.8413

b. 0.1587

c. 0.3413

d. 0.0025

e. None of the above

Use the following information for questions 26 through 28

Consider a meat packing plant which produces packages of frozen

steak. Let X be the weight (ounces) and assume that X is normally

distributed with μ_x = 8.0 and σ_x = .5, n (sample size) = 16.

19. The standard error of the sample mean is

a. 0.5

b. 0.5 divided by 16

c. 0.25 divided by 16

d. 0.5 divided by 4

e. 0.25

20. Before the sample is drawn, the probability that the sample mean

will exceed 9 ounces is closest to

a. 0.3085

b. 0.1587

c. 0.1056

d. 0.0256

e. 0.0100

21. The probability that X1 will exceed 9 ounces is closest to

a. 0.3085

b. 0.1587

c. 0.1056

d. 0.0256

e. 0.0100

22. Which of the following factors does sample size depend on?

1. size of population, number of groups and subgroups within the sample
2. accuracy required of the results
3. cost of the sample
4. variability of the population
5. all of the above