1. A simple random sample of 100 batteries is selected from a process which produces the batteries with a mean life of 40 months and a standard deviation of 2 months. Thus, the standard error equals .2 when n=100. If a sample size of n=1600 had been used instead, the standard error would be:
a. twice as large as when n=100.
b. four times as large as when n=100.
c. one-half as large as when n=100.
d. one-fourth as large as when n=100.
e. none of the above.
2. The sampling distribution of X-bar refers to:
a. the distribution of the various sample sizes which might be used in a given study.
b. the distribution of the different possible values of the sample mean together with their respective probabilities of occurrence.
c. the distribution of the values of the items in the population.
d. the distribution of the values of the items actually selected in a given sample.
e. none of the above.
3. Given a random sample, the characteristics (i.e., mean, variance, shape) of the sampling distribution of X-bar are affected by: (1) the sample size; (2) the variability in the population; (3) the mean of the population.
a. 1 only
b. 1 and 2 only
c. 1 and 3 only
d. 2 and 3 only
e. 1, 2, and 3
4. According to the Central Limit Theorem, for almost all populations, the sampling distribution of X-bar is approximately normal when:
a. the simple random sample size is sufficiently large.
b. the population mean is zero.
c. the sample contains an even number of observations.
d. a stratified sample of any size is utilized.
e. none of the above.
5. The t distribution is used in constructing confidence intervals for which of the following?
a. the population is normally distributed and the population standard deviation is estimated from a small random sample.
b. in all cases where the random sample is small, irrespective of any other factors.
c. in all cases where the sample size is small, the population is normal and the population standard deviation is known.
d. whenever a confidence coefficient of 90 percent or more is desired.
e. none of the above.
6. In constructing a confidence interval, which of the following factors affect the width of the interval?
a. the variability in the population.
b. the mean of the population.
c. the mean of the sampling distribution of X-bar.
d. the mean of the sample actually drawn.
e. none of the above.
7. As the degrees of freedom for a t distribution increase, the t score for a given tail area:
a. increases
b. approaches 0.
c. stays the same.
d. approaches the z score.
e. will depend upon the type of sample chosen.
8. A property of an estimator that occurs whenever the expected value of the estimator is equal to the population parameter it estimates is known as:
a. consistency.
b. expected value.
c. estimator of the finite population parameters.
d. relative efficiency.
e. unbiasedness.
9. Suppose that a random sample of size n=3 is drawn from a finite population of size N=6 which has a mean 7 and variance 9. The sampling distribution of X-bar will have mean 7 and variance
a. 2.0
b. 3.0
c. 1.8
d. 1.6
e. supply your own answer __________________.
10. Other things the same, as the confidence level for a confidence interval increases, the width of the interval
a. remains the same.
b. decreases.
c. increases.
d. may do any of the above.
e. insufficient information to make any statements.
11. Which of the following statements is FALSE?
a. The standard error of the estimate is affected by the standard deviation and the sample size.
b. Given some population, as sample size increases, the larger number of observations involved increases the value of the standard error.
c. Sampling error is the difference between the sample mean and the population mean
d. For large samples, sampling distributions will be normally distributed.
e. A p-value measures the area under a normal density function.
PART TWO: SHORT ANSWERS: Please answer all parts. Points distributed as noted.
1. Let to be a particular value of t. Using the appropriate table find the values of t such that the following statements are true.
a. P (t > to)= .025 when df=12
b. P (t < to)= .005 when df=17
c. P (t < to)= .05 when df=6
d. P (-to < t < to) =.95 when df=11
e. P (t < -to or t > to) =.01 when df=6
f. P (t < to)= .05 when df=11
2. In a report to the corporation president, Ms. Ann Throap, a junior executive, stated that the average IQ for employees in the company was so high that the probability of a higher mean in a sample in the general population was only .015. Ann, however, was very flustered when the president asked what the average was, and what her sample size was. She remembered that she had sampled 100 employees, and that the IQ test she used has a mean and variance of 100.
a. Draw a picture of the sampling distribution appropriate for Ann's situation.
Identify the parameters.
b. What is the value of the sample mean based upon Ann's remarks?
3. John Wiggins, in charge of research and development at an industrial plant, likes to check the pollution controls at the plant periodically. He knows that the index has a mean reading of 125 micrograms per cubic meter and a variance of 196. He feels that if the average for a sample of 49 days is less than 127, he will not be in danger of criticism from environmental agencies. What is the probability that the mean pollution index for a sample of 49 days will fall below the criterion he has set?
4. What do managers stress most on the job? In a survey prepared for Towers, Perrin, Forster, and Crosby (an international management consultant firm), 462 senior human resource and compensation executives in private industry, government, and nonprofit organizations were asked this question. The most frequent response, given by 262 of 462 managers, was "pay employees for performance". Construct a 95% confidence interval for the true proportion of managers who stress pay for performance on the job.
PART THREE: PROBLEMS. Please answer all parts. Show all work. Points allocated as noted.
1. It is desired to estimate the mean time students spend per session at a computer terminal in a large university. A random sample of 16 sessions yields an X-bar equal to 24 minutes and a standard deviation of 8.0 minutes. Session times are known to follow a normal distribution. Construct a 95 percent confidence interval for the average time of all students.
2. The United States Golf Association (USGA) tests all brand new golf balls to assure that they meet USGA specifications. One test conducted is intended to measure the average distance traveled when the ball is hit by a machine called "Iron Byron", a name inspired by the swing of a famous golfer Byron Nelson. Suppose the USGA wants to estimate the mean distance for a new brand with a 90% confidence interval of width 2 yards. Assume the past tests have indicated that the standard deviation of the distances "Iron Byron" hits golf balls is approximately 10 yards. How many golf balls should be hit by "Iron Byron" to achieve the desired accuracy in estimating the mean?
3. Shoplifting is an escalating problem for retailers. Recently, one New York City store randomly selected 500 shoppers and observed them while they were in the store. Two in 25 were seen stealing. How accurate is this estimate? To help you answer this question, construct a 95% confidence interval for π, the population proportion of all the store's customers who are shoplifters.
4. A machine for filling cereal boxes can be set to dispense a mean weight of between 12 and 20 ounces. The standard deviation of the weight of boxes filled with the machine is 1 ounce and is independent of the mean. The product is shipped in cases of 36 boxes. Assume each case represents a random sample from the population of boxes filled by this machine at a fixed setting of the mean weight dispensed.
a. At what level should the mean be set so that 95% of all cases produced will have a mean weight of at least 16 ounces?
b. Repeat part (a), this time assuming that the standard deviation of the weight of boxes filled by the machine is 2 ounces.
c. Repeat part (b), but for cases of 64 boxes.
5. A manufacturer of aluminum foil claims that its 100-foot roll has a mean length of 100.1 feet per roll and a standard deviation of 15 foot. To check this claim, a consumer group plans to randomly sample 36 of the company's 100-foot rolls, measure the length of each, and compute the sample mean length.
a. Assuming the manufacturer's claim is true, describe the sampling distribution of the sample mean. Include a drawing of this sampling distribution.
b. Assuming the manufacturer's claim is true, what is the probability that the sample mean will be less than 100 feet?
c. Suppose that the sample mean actually equals 99.85 feet. Can this evidence be used to refute the manufacturer's claim? Explain.
6. A population consists of four numbers, 2,3,3, and 4 marked on poker chips.
a. How many different samples of n=2 chips could be selected (without replacement) from this population? List the possible samples.
b. Give the probability of selecting any one of these samples (assume the sampling is random).
c. Calculate X-bar for each of the samples in part a.
d. Calculate the probability associated with each of the possible values of X-bar.
e. Create the sampling distribution of X-bar for the sampling distribution of size 2 samples as described above.
7. A Midwestern firm, Gloozall, has a created a fuel additive that they claim will increase a car's mpg (miles per gallon) by more than 10%. An independent testing agency decides to see if Gloozal's claim has any merit. The agency conducts an experiment using the additive for 1 week in 8 automobiles. The increases in mpg (measured in percentage terms) after using the additive are listed below:
15.2 14.1 13.7 15.2 18.6 14.5 15.0 13.8
Some summary information is ∑ X = 120.1 and ∑ X2 = 1820.23.
a. What are the hypotheses to test Gloozal's claim?
HO:
HA:
b. What is the CRITICAL value? Use alpha = 0.05.
c. What is the value of the CALCULATED value?
d. What decision is appropriate?
e. What conclusion should be drawn?