# 1. Does taking garlic tablets twice a day provide significant health benefits? To investigate this issue, a researcher conducted a study with fifty adult subjects who took garlic tables twice a day for a period of six months. At the end of the study, 100 variables related to the health of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Four of these variables were significantly better (in the sense of statistical significance) at the 5% level for the group taking the garlic tablets as compared to the population as a whole, and one variable was significantly better at the 1% level for the group taking the garlic tablets as compared to the population as a whole. It would be correct to conclude A. That there is good statistical evidence that taking garlic tablets twice a day provides some health benefits. B. That there is good statistical evidence that taking garlic tablets twice a day provides benefits for the variable that was significant at the 1% level. We should be somewhat cautious about making claims for the variables that were significant at the 5% level C. None of the above. 2. National opinion polls are normally conducted by telephone over a short time span, usually 2 or 3 days. A “Presidential Approval Poll” may state that the president’s approval rating is x%, “with a margin of error ± 3%.” What they usually don’t say is that this is really a 95% confidence interval estimate. Can we truly be 95% confident that the true percentage for all adult Americans is within 3% of the estimate? A. Yes. B. No. C. There is not enough information to be able to tell. 3. There is an old saying “Statistics means never having to say you’re certain.” Why might this be true? A. It isn’t — if I get a p-value of 0.0000001 (or less) I’m sure the null hypothesis is wrong. B. It’s just a joke on statisticians. C. To be totally confident, I would need a z* of ∞ . 4. Students at my university wanted to move spring break later in the semester, so that weather at the beach would be warmer. They stood outside a campus eatery and tried to flag down passersby to sign their petition. The student newspaper later announced that “an overwhelming proportion of those contacted were in favor of moving spring break.” It wasn’t moved. Why did the administration reject the plan? A. The sample wasn’t random. B. The administration just didn’t care. C. The sample wasn’t large enough. Criminal jury trials are analogous to statistical tests of hypotheses. If we consider the decision as a test of H : not guilty versus Ha : guilty, and the jury must find “beyond a reasonable doubt” that the defendant is guilty, what might be considered an appropriate α level for the test? A. 10%. B. 5%. C. 1%. 6. The amount of time customers at a “Quick-Change” motor oil store spend waiting for their cars to be serviced has the Normal distribution with mean μ and standard deviation σ = 4 minutes. It is company policy that the customer wait time should be 20 minutes (or less). The manager of a particular store selects a random sample of 150 customer wait times and observes a mean wait time of 21 minutes. Reference: Ref 15-1 If the manager had selected more customers for her sample, the margin of error in the 95% confidence interval computed in problem 3 above would A. become larger. B. not change. C. become smaller. D. become larger or smaller, depending upon the sample taken. 7. To assess the accuracy of a laboratory scale, a standard weight (with known value) is repeatedly weighed a total of n times, and the mean x of the sample weights is computed. Suppose the scale readings are Normally distributed, with unknown mean μ and standard deviation σ = 0.01 g. How large should n be, so that a 95% confidence interval for μ has a margin of error of ± 0.001? A. 10 B. 20 C. 100 D. 385 8. Twenty-five seniors from a large metropolitan area school district volunteer to allow their math SAT test scores to be used in a study. These 25 seniors had a mean math SAT score of = 450. Suppose we know that the standard deviation of the population of math SAT scores for seniors in the district is σ = 100. Assuming the population of math SAT scores for seniors in the district is approximately Normally distributed, a 90% confidence interval for the mean math SAT score μ for the population of seniors computed from these data is A. 450 ± 32.9. B. 450 ± 39.2. C. 450 ± 164.5. D. not trustworthy. 9. The records of the 100 postal employees at a postal station in a large city showed that the average time these employees had worked for the postal service was = 6 years. Assume that we know that the standard deviation of the population of times U.S. Postal Service employees have spent with the postal service is approximately Normal, with standard deviation σ = 5 years. A 95% confidence interval for the mean time μ the population of U.S. Postal Service employees have spent with the postal service is A. 6 ± 0.82. B. 6 ± 0.98. C. 6 ± 9.80. D. not trustworthy. 10. You plan to construct a 99% confidence interval for the mean μ of a Normal population with (known) standard deviation σ. By using a sample size of 400 rather than 100, you can reduce the margin of error by a factor of A. 2 (the new margin of error will be half that of the one based on 100 observations). B. 4 (the new margin of error will be 1/4 that of the one based on 100 observations). C. 16 (the new margin of error will be 1/16 that of the one based on 100 observations). D. None of the above–it depends on the value of σ. 11. A snack food producer produces bags of peanuts labeled as containing 3 ounces. The actual weight of peanuts packaged in individual bags is Normally distributed with mean μ and standard deviation σ = 0.2 ounces. As part of quality control, n bags are selected randomly and their contents are weighed. The hypotheses of interest are H: μ = 3 ounces, Ha : μ≠ 3 ounces. Reference: Ref 15-3 If the inspector samples n = 25 bags and observes a sample mean weight of = 3.01 ounces, the P-value is then A. 0.5987. B. about 0, since we’ve observed the same sample mean as in Problem 15, above. C. 0.1974. D. 0.8026. 12. A snack food producer produces bags of peanuts labeled as containing 3 ounces. The actual weight of peanuts packaged in individual bags is Normally distributed with mean μ and standard deviation σ = 0.2 ounces. As part of quality control, n bags are selected randomly and their contents are weighed. The hypotheses of interest are H: μ = 3 ounces, Ha : μ≠ 3 ounces. Reference: Ref 15-3 Together, the two preceding problems demonstrate that A. when we take a large enough random sample, any effect (however small) will be deemed statistically significant. B. the measure of statistical significance (a P-value) depends heavily on the sample size. C. practical significance is not the same thing as statistical significance. D. all of the statements above are true. 13. Which of the following will increase the value of the power in a statistical test of hypotheses? A. Increase the Type II error probability. B. Increase the sample size. C. Reject the null hypothesis only if the P-value is smaller than the level of significance. D. All of the above. 14. A researcher plans to conduct a test of hypotheses at the α = 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value of the parameter of interest. Reference: Ref 15-4 The probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power is A. 0.05. B. 0.20. C. 0.80. D. equal to the 1 – (P-value) and cannot be determined until the data have been collected. 15. The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels over 200 is Normally distributed, with mean (in milligrams) μ and standard deviation σ = 3. Although the brand advertises that eating their oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels over 200, you believe that the mean decrease in cholesterol is actually less than advertised. To explore this, you test the hypotheses H: μ = 10, Ha: μ at the α = 0.05 level. You will do this by randomly selecting 100 people with cholesterol levels over 200 and, after eating this brand of oatmeal for one month, computing the mean decrease x in cholesterol levels of the subjects. The probability of a Type II error for your test at μ = 7 is A. 0.7995. B. 0.8413. C. 0.95. D. less than 0.001.

1. Does taking garlic tablets twice a day provide significant health benefits? To investigate this issue, a researcher conducted a study with fifty adult subjects who took garlic tables twice a day for a period of six months. At the end of the study, 100 variables related to the health of the subjects were measured on each subject and the means compared to known means for these variables in the population of all adults. Four of these variables were significantly better (in the sense of statistical significance) at the 5% level for the group taking the garlic tablets as compared to the population as a whole, and one variable was significantly better at the 1% level for the group taking the garlic tablets as compared to the population as a whole. It would be correct to conclude

A.

That there is good statistical evidence that taking garlic tablets twice a day provides some health benefits.

B.

That there is good statistical evidence that taking garlic tablets twice a day provides benefits for the variable that was significant at the 1% level. We should be somewhat cautious about making claims for the variables that were significant at the 5% level

C.

None of the above.

2. National opinion polls are normally conducted by telephone over a short time span, usually 2 or 3 days. A “Presidential Approval Poll” may state that the president’s approval rating is x%, “with a margin of error ± 3%.” What they usually don’t say is that this is really a 95% confidence interval estimate. Can we truly be 95% confident that the true percentage for all adult Americans is within 3% of the estimate?

 A. Yes. B. No. C. There is not enough information to be able to tell.

3. There is an old saying “Statistics means never having to say you’re certain.” Why might this be true?

 A. It isn’t — if I get a p-value of 0.0000001 (or less) I’m sure the null hypothesis is wrong. B. It’s just a joke on statisticians. C. To be totally confident, I would need a z* of ∞ .

4. Students at my university wanted to move spring break later in the semester, so that weather at the beach would be warmer. They stood outside a campus eatery and tried to flag down passersby to sign their petition. The student newspaper later announced that “an overwhelming proportion of those contacted were in favor of moving spring break.” It wasn’t moved. Why did the administration reject the plan?

 A. The sample wasn’t random. B. The administration just didn’t care. C. The sample wasn’t large enough.
Criminal jury trials are analogous to statistical tests of hypotheses. If we consider the decision as a test of H : not guilty versus Ha : guilty, and the jury must find “beyond a reasonable doubt” that the defendant is guilty, what might be considered an appropriate α level for the test?
 A. 10%. B. 5%. C. 1%.
6. The amount of time customers at a “Quick-Change” motor oil store spend waiting for their cars to be serviced has the Normal distribution with mean μ and standard deviation σ = 4 minutes. It is company policy that the customer wait time should be 20 minutes (or less). The manager of a particular store selects a random sample of 150 customer wait times and observes a mean wait time of 21 minutes.

Reference: Ref 15-1
If the manager had selected more customers for her sample, the margin of error in the 95% confidence interval computed in problem 3 above would

 A. become larger. B. not change. C. become smaller. D. become larger or smaller, depending upon the sample taken.

7. To assess the accuracy of a laboratory scale, a standard weight (with known value) is repeatedly weighed a total of n times, and the mean x of the sample weights is computed. Suppose the scale readings are Normally distributed, with unknown mean μ and standard deviation σ = 0.01 g. How large should n be, so that a 95% confidence interval for μ has a margin of error of ± 0.001?

 A. 10 B. 20 C. 100 D. 385

8. Twenty-five seniors from a large metropolitan area school district volunteer to allow their math SAT test scores to be used in a study. These 25 seniors had a mean math SAT score of = 450. Suppose we know that the standard deviation of the population of math SAT scores for seniors in the district is σ = 100. Assuming the population of math SAT scores for seniors in the district is approximately Normally distributed, a 90% confidence interval for the mean math SAT score μ for the population of seniors computed from these data is

 A. 450 ± 32.9. B. 450 ± 39.2. C. 450 ± 164.5. D. not trustworthy.

9. The records of the 100 postal employees at a postal station in a large city showed that the average time these employees had worked for the postal service was = 6 years. Assume that we know that the standard deviation of the population of times U.S. Postal Service employees have spent with the postal service is approximately Normal, with standard deviation σ = 5 years. A 95% confidence interval for the mean time μ the population of U.S. Postal Service employees have spent with the postal service is

 A. 6 ± 0.82. B. 6 ± 0.98. C. 6 ± 9.80. D. not trustworthy.

10. You plan to construct a 99% confidence interval for the mean μ of a Normal population with (known) standard deviation σ. By using a sample size of 400 rather than 100, you can reduce the margin of error by a factor of

A. 2 (the new margin of error will be half that of the one based on 100 observations).
B. 4 (the new margin of error will be 1/4 that of the one based on 100 observations).
C. 16 (the new margin of error will be 1/16 that of the one based on 100 observations).
D. None of the above–it depends on the value of σ.

11. A snack food producer produces bags of peanuts labeled as containing 3 ounces. The actual weight of peanuts packaged in individual bags is Normally distributed with mean μ and standard deviation σ = 0.2 ounces. As part of quality control, n bags are selected randomly and their contents are weighed. The hypotheses of interest are H: μ = 3 ounces, Ha : μ≠ 3 ounces.

Reference: Ref 15-3
If the inspector samples n = 25 bags and observes a sample mean weight of = 3.01 ounces, the P-value is then

 A. 0.5987. B. about 0, since we’ve observed the same sample mean as in Problem 15, above. C. 0.1974. D. 0.8026.

12. A snack food producer produces bags of peanuts labeled as containing 3 ounces. The actual weight of peanuts packaged in individual bags is Normally distributed with mean μ and standard deviation σ = 0.2 ounces. As part of quality control, n bags are selected randomly and their contents are weighed. The hypotheses of interest are H: μ = 3 ounces, Ha : μ≠ 3 ounces.

Reference: Ref 15-3
Together, the two preceding problems demonstrate that

 A. when we take a large enough random sample, any effect (however small) will be deemed statistically significant. B. the measure of statistical significance (a P-value) depends heavily on the sample size. C. practical significance is not the same thing as statistical significance. D. all of the statements above are true.

13. Which of the following will increase the value of the power in a statistical test of hypotheses?

 A. Increase the Type II error probability. B. Increase the sample size. C. Reject the null hypothesis only if the P-value is smaller than the level of significance. D. All of the above.

14. A researcher plans to conduct a test of hypotheses at the α = 0.05 significance level. She designs her study to have a power of 0.80 at a particular alternative value of the parameter of interest.

Reference: Ref 15-4
The probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power is

 A. 0.05. B. 0.20. C. 0.80. D. equal to the 1 – (P-value) and cannot be determined until the data have been collected.

15. The decrease in cholesterol level after eating a certain brand of oatmeal for breakfast for one month in people with cholesterol levels over 200 is Normally distributed, with mean (in milligrams) μ and standard deviation σ = 3. Although the brand advertises that eating their oatmeal for breakfast daily for one month will produce a mean decrease in cholesterol of more than 10 points for people with cholesterol levels over 200, you believe that the mean decrease in cholesterol is actually less than advertised. To explore this, you test the hypotheses

H: μ = 10, Ha: μ

at the α = 0.05 level. You will do this by randomly selecting 100 people with cholesterol levels over 200 and, after eating this brand of oatmeal for one month, computing the mean decrease x in cholesterol levels of the subjects. The probability of a Type II error for your test at μ = 7 is

 A. 0.7995. B. 0.8413. C. 0.95. D. less than 0.001.