AP Statistics / Mr. Hansen

Name: _________________________________

Test #6D (3/12/1999)

General Instructions: INDICATE ALL YOUR ANSWERS TO QUESTIONS ON THE SEPARATE ANSWER SHEET ENCLOSED. No credit will be given for anything written on this test form, but you may use the booklet for notes or scratchwork. After you have decided which of the suggested answers is best, COMPLETELY fill in the corresponding oval on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely.

Example:

What is the arithmetic mean of the numbers 1,
3, and 6 ?

(A) 1
(B) 7/3
(C) 3
(D) 10/3
(E) 7/2

Answer:

A

B

C

D

E

O

O

O

·

O

Many students wonder whether or not to guess the answers to questions about which they are not certain. In this test, as a correction for haphazard guessing, one-fourth of the number of questions you answer incorrectly will be subtracted from the number of questions you answer correctly. It is improbable, therefore, that mere guessing will improve your score significantly; it may even lower your score, and it does take time. If, however, you are not sure of the best answer but have some knowledge of the question and are able to eliminate one or more of the answer choices as wrong, your chance of answering correctly is improved, and it may be to your advantage to answer such a question.

Use your time effectively, working as rapidly as you can without losing accuracy. Do not spend too much time on questions that are too difficult. Go on to other questions and come back to the difficult ones later if you have time.

 

 

 

 

 

 

 

 

 

 

 

1. The one-sample t test can actually be used to make inferences about the difference between two means. For example, the method can be used to assess the statistical significance of the mean difference between "before" and "after" data points, where the data points are organized as . . .

(A) matched pairs
(B) a placebo group
(C) an experimental group
(D) degrees of freedom
(E) none of the above

2. Which of these is a criterion for choosing a t-test rather than a z-test when making an inference about the mean of a population?

(A) The standard deviation of the population is unknown.
(B) The mean of the population is unknown.
(C) The standard deviation of the population is known.
(D) The population has skewness or outliers.
(E) The population is not normally distributed.

3. Which of the following means has the largest standard error?

I: A sample mean of size 100, with xbar=48.2 and s=15.

II. A sample mean of size 10, with xbar=48.2 and s=12.

III. A sample mean of size 10, with xbar=79.9 and s=12.

(A) I has the largest standard error.
(B) II has the largest standard error.
(C) III has the largest standard error.
(D) I and II are tied for the largest standard error.
(E) II and III are tied for the largest standard error.

4. Ten students were randomly selected from a high school to take part in a program designed to raise their reading comprehension. Each student took a test before and after completing the program. The mean of the differences between the score after the program and the score before the program is 16. It was decided that all students in the school would take part in this program during the next school year. Let m A denote the mean score after the program and m B denote the mean score before the program for all students in the school. The 95 percent confidence interval estimate of the true mean difference for all students is 16 ± 7. Which of the following statements is a correct interpretation of this confidence interval?

(A) If the test were repeated, the probability is 50% that the new sample mean difference would be less than 16.
(B) If the test were repeated, the probability is 50% that the new sample mean difference would be less than or equal to 16.
(C) m A is around 23 and m B is around 9

(D) We are 95% confident that the sample mean difference is between 9 and 23.
(E) None of the above.

5. A large study involving 22,000 male physicians attempted to determine whether aspirin could help prevent heart attacks. In this study, one group of about 11,000 physicians took an aspirin tablet every other day, while a control group took a placebo every other day. Is it important that the size of the control group and experimental group be equal?

(A) Yes, unless the choice of placebo vs. experimental group was made randomly by subject.
(B) Yes, because otherwise the standard error cannot be computed correctly.
(C) No, since there is no value to having the sample sizes be equal.
(D) No, because when you have samples as large as this you don’t even need a control group.
(E) No, although the two-sample t procedures are especially robust in the case where we have approximately equal sample sizes.

6. Referring again to #5, what is meant by a conclusion that "there was statistically significant evidence that physicians in the placebo group had more heart attacks on average"?

(A) The mean number of heart attacks per patient in the control group exceeded that of the experimental group by an amount that would often be produced by chance alone if aspirin had no effect.
(B) The mean number of heart attacks per patient in the experimental group exceeded that of the control group by an amount that would often be produced by chance alone if aspirin had no effect.
(C) The physicians in the placebo group had many more heart attacks than the physicians who received aspirin.
(D) The physicians in the placebo group had a much higher rate of heart attacks than the physicians who received aspirin.
(E) None of the above

7. When the minimal assumptions of the t procedures cannot be met, one can . . .

(A) never compute a p-value, since that would be wrong methodology
(B) use the z procedures instead
(C) try a nonparametric method, such as the sign test
(D) discard outliers to improve the p-value
(E) use pooled procedures

8. A random sample of the costs of dry cleaning orders at a large dry cleaning establishment produces a mean of $16.50 with a standard deviation of $10.05. If there were 20 orders in this sample, with no indication of outliers or strong skewness, which of these gives a 95 percent confidence interval for the average revenue per customer transaction?

(A) $16.50 ± $2.25
(B) $16.50 ± $3.70
(C) $16.50 ± $4.48
(D) $16.50 ± $4.70
(E) None of the above

9. The scores on the last AP Statistics test were approximately normally distributed with a mean of 75 and a standard deviation of 12. The scores on the U.S. History test were also approximately normally distributed, with a mean of 78 and a standard deviation of 8. Milton scored 81 on the statistics test and 84 on the history test. Relative to the students in each respective course, in which subject did Milton do better?

(A) Statistics
(B) History
(C) Equally well in each
(D) There is no basis for comparison (different subjects, different departments)
(E) Cannot be solved without knowing the number of students in each course

10. Referring again to #9, change the problem to assume now that Milton is at a school where all statistics students take U.S. history and vice versa. In other words, the courses have identical student lists. If you add Milton’s two scores to get 165, what is his score of 165 terms of a z-score? (To answer this question, you may make the quite unrealistic assumption that students’ test scores in the two courses are independent.)

(A) 0.500
(B) 0.693
(C) 0.702
(D) 0.832
(E) None of the above.

11. In a test of the null hypothesis H0: m =10 against the alternative hypothesis Ha: m >10, a sample from a normal population produces a mean of 13.4. The z-score for the sample is 2.12 and the p-value is 0.017. Based on these statistics, which of the following conclusions could be drawn?

(A) There is reason to conclude that m >10.
(B) Due to random fluctuation, a sample mean exceeds 10 about 48.3 percent of the time.
(C) The error of rejecting the alternative hypothesis occurs about 1.7 percent of the time.
(D) The probability that m >10 is 0.017.
(E) The probability that m <10 is 0.983.

12. The distribution of the weights of loaves of bread from a certain bakery follows approximately a normal distribution. Based on a very large sample, it was found that 10 percent of the loaves weighed less than 15.34 ounces, and 20 percent of the loaves weighed more than 16.31 ounces. Which of the following is true concerning the distribution of the weights of the loaves of bread?

(A) 16.31 corresponds to a z-score of 0.8
(B) 16.31 corresponds to a z-score of 0.2
(C) 16.31 corresponds to a z-score of 0.842
(D) 16.31 corresponds to a z-score of 0.248
(E) None of the above.

13. The manufacturing process for a certain pharmaceutical yields capsules with varying amounts of the active ingredient. The manufacturer claims that the average amount of active ingredient per capsule is at least 200 mg. In a random sample of 70 capsules, the mean content of the active ingredient is found to be 194.3 mg, with a standard deviation of 21 mg. Which test is most appropriate to see whether there is evidence that the manufacturer is consistently underfilling the capsules?

(A) one-sided one-sample t
(B) two-sided one-sample t
(C) one-sided two-sample t
(D) two-sided two-sample t
(E) two-sample F test

14. What are the null and alternative hypotheses for the test described in #13?

(A) H0: m < 200 mg; Ha: m ³ 200 mg
(B) H0: m £ 200 mg; Ha: m > 200 mg
(C) H0: m = 200 mg; Ha: m > 200 mg
(D) H0: m = 200 mg; Ha: m < 200 mg
(E) H0: m = 200 mg; Ha: m ¹ 200 mg

15. Suppose that student heights are known to be normally distributed at STA and NCS. Random samples are drawn and heights (in inches) are recorded. STA sample heights are 66, 69, 70, 70, 71, 72, 72, 73, and 75. NCS sample heights are 62, 64, 64, 65, 66, 68, and 69. What can we say?
I. STA sample mean height exceeds that of NCS.
II. STA sample variance for height exceeds that of NCS.
III. There is good evidence that STA’s population standard deviation for height exceeds that of NCS.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

16. The F statistic for #15, computed according to the standard convention, is . . .

(A) 1.000
(B) 1.054
(C) 1.111
(D) 1.222
(E) 1.333

17. The numerator degrees of freedom for the F test in #15 would be . . .

(A) 6
(B) 7
(C) 8
(D) 9
(E) 16

18. In a sample of size 20, the t score corresponding to the 50th percentile would be . . .

(A) 0
(B) 0.686
(C) 0.687
(D) 0.688
(E) None of the above

19. One of the most robust tools in the professional statistician’s repertoire is the . . .

(A) 1-sample t test
(B) 2-sample t test
(C) 1-sample z test
(D) 1-sample variance test
(E) None of the above

20. Referring back to #15, the pooled estimator of standard deviation would equal . . .

(A) 2.418
(B) 2.516
(C) 2.614
(D) 2.712
(E) 2.810

21. The conservative degrees of freedom associated with performing a two-sample t test on the data in problem #15 would be . . .

(A) 4
(B) 6
(C) 8
(D) 10
(E) 16

22. The degrees of freedom used in a pooled two-sample t test for #15 would be . . .

(A) 12
(B) 13
(C) 14
(D) 15
(E) 16

23. Why do pooled procedures often produce lower p-values and narrower confidence intervals than non-pooled procedures?

(A) In the pooled procedures, df will be higher, which increases the t critical value.
(B) In the pooled procedures, df will be higher, which reduces the t critical value.
(C) The pooled estimator of standard deviation is always less than either of the sample standard deviations.
(D) The pooled estimator of standard deviation is always greater than either of the sample standard deviations.
(E) None of the above

24. When is use of pooled procedures justified?
I. Whenever the F test reveals no evidence that the two samples have different standard deviations.
II. Whenever the non-pooled p-value slightly exceeds a (so that publication can be made more likely).
III. Whenever there is strong, credible reason to believe that the two populations have the same s .

(A) I only
(B) II only
(C) III only
(D) Either I or III will by itself provide sufficient justification.
(E) Any one of I, II, or III will constitute sufficient justification for using pooled procedures.

25. The F test differs from the z and t tests by being . . .

(A) very robust
(B) a test for inference about standard deviations, not means
(C) very non-robust
(D) both A and B
(E) both B and C

 

 

AP Statistics / Mr. Hansen

Name: _________________________________

Test #6D (3/12/1999)

 

ANSWER SHEET

No.

A

B

C

D

E

1

O

O

O

O

O

2

O

O

O

O

O

3

O

O

O

O

O

4

O

O

O

O

O

5

O

O

O

O

O

6

O

O

O

O

O

7

O

O

O

O

O

8

O

O

O

O

O

9

O

O

O

O

O

10

O

O

O

O

O

11

O

O

O

O

O

12

O

O

O

O

O

13

O

O

O

O

O

14

O

O

O

O

O

15

O

O

O

O

O

16

O

O

O

O

O

17

O

O

O

O

O

18

O

O

O

O

O

19

O

O

O

O

O

20

O

O

O

O

O

21

O

O

O

O

O

22

O

O

O

O

O

23

O

O

O

O

O

24

O

O

O

O

O

25

O

O

O

O

O

 

Answer Key (no fair peeking until you’ve tried on your own!):

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