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AMS315/576, First Practice Examination, Spring Semester, 1998

This is the start of the examination.

1. The observed significance level (p-value) reported in a computer printout for a statistical test was 0.0771. Which of the following is a correct decision for this result?
2.

a. Reject at the .01 level of significance.

b. Accept at the .01 level of significance and reject at the .05 level.

c. Accept at the .05 level of significance and reject at the .10 level.

d. Accept at the .10 level of significance.

3. The values of a random sample of size 4 drawn from the random variable Y were 1160, 1230, 1170, and 1160. What is the value of 4?
4.

5. The values of a random sample of size 4 from the random variable Y were 1450, 1340, 1370, and 1360? What is the value of the unbiased estimate of the variance of Y?
6.

7. A research team will run a clinical trial on male patients that has a probability of Type II error 0.10. They will also run a clinical trial on female patients; the probability of Type II error in the female clinical trial is 0.10. What is the probability that the research team makes at least one Type II error?
8.

Common Information for Questions 5-7.

The faculty of a statistics department is considering using an eight question true-false test to determine whether a student is a random guesser or is knowledgeable about statistics. That is, they will present a student with eight true-false questions of equal difficulty in random order. They will use the observed value of E8, the number of errors that the student makes, as the basis for accepting or rejecting the null hypothesis H0 that their student is a random guesser. They will reject H0 when they observe E8 £2 and accept H0 otherwise.

They computed F0, the cumulative distribution function of E8 under the null hypothesis. They also computed F1, the cumulative distribution function of E8 for a student who had a 0.90 chance of correctly answering each question. Table 1 contains these values.

Table 1

Cumulative Distribution Function of E8

under H0 and for Knowledgeable Student

s F0(s) F1(s)

0 0.0039 0.4305

1 0.0352 0.8131

2 0.1445 0.9619

3 0.3633 0.9950

4 0.6367 0.9996

5 0.8555 1.0000

6 0.9648 1.0000

7 0.9961 1.0000

8 1.0000 1.0000

9. What is α, the probability of a Type I error, for this test of the null hypothesis?
10.

11. What is β, the probability of a Type II error, for this eight question examination administered to a student who has a 0.90 probability of correctly answering each question?
12.

13. What is the p-value for a student with e8=3?
14.

End of Group

Common Information for Questions 8 and 9

A research team plans to test the null hypothesis that the random variable Y is normally distributed with E(Y)=0 and standard deviation 200 using a random sample of size 100. The alternative hypothesis is that Y is normally distributed with E(Y)>0. They observe _100=56.

15. What is the observed significance level (p-value)?
16.

17. Which of the following is a correct decision for this result?
18.

a. Reject at the .01 level of significance.

b. Accept at the .01 level of significance and reject at the .05 level.

c. Accept at the .05 level of significance and reject at the .10 level.

d. Accept at the .10 level of significance.

End of Group of Questions

19. A random variable Y has unknown expected value E(Y) and standard deviation σY=50. What is the least sample size n needed so that the 99% margin of sampling error is 4? That is, what is the sample size is needed so that the 99% confidence interval for E(Y) is of the form n±4?
20.

Background Information for Questions 11-13.

A research team will test the null hypothesis that E(Y)=500 at the 0.05 level of significance against the alternative that E(Y)<500. When the null hypothesis is true, Y has a normal distribution with standard deviation 100. They will take a random sample of 225 observations and use the statistic  225, the sample average of the random sample of size 225, to test the null hypothesis.

21. What is the critical value for this test?
22.

23. What is the probability of a Type II error when E(Y)=450, σY=100, n=225, and α=.05?
24.

25. What is the smallest value of n, the sample size so that the probability of a Type II error is no more than .05 when E(Y)=450, σY=100, and α=.05?
26.

End of a Common Problem Section

Background Information for Questions 14 and 15

A normally distributed random variable Y has standard deviation 800. A random sample of 400 observations was taken from Y, and the sample average observed was _400=87.2.

27. What is the right endpoint of the symmetric 95% confidence interval for E(Y)?
28.

29. Based on the information of the common section, which actions are correct for testing situation I and testing situation II below:
30.

I. H0: E(Y)=50, α=0.05, H1: E(Y) ¹ 50.

II. H0: E(Y)=250, α=0.05, H1: E(Y) ¹ 250.

a. Accept H0 in both situations I and II.

b. Accept H0 in situation I, and reject H0 in situation II.

c. Reject H0 in situation I, and accept H0 in situation II.

d. Reject H0 in both situations I and II.

End of a Common Problem Section

31. A research team took a random sample of 6 observations from a normally distributed random variable Y with unknown expected value E(Y) and unknown standard deviation σY. Their problem was to test the null hypothesis H0: E(Y)=100 against the alternative hypothesis H1: E(Y)<100. The sample statistics were _6=62.4, and the value of the estimated standard deviation was 57.6. Which of the following is the correct decision for accepting or rejecting the null hypothesis based on the sample average and standard deviation given?
32.

a. Reject at the .01 level.

b. Reject at the .05 level and accept at the .01 level.

c. Reject at the .10 level and accept at the .05 level.

d. Accept at the .10 level.

Background Information for Questions 17 and 18

A random sample of size 3 was taken from a normally distributed random variable Y with unknown standard deviation. The observed sample average was 548, and the observed standard deviation was 78.1.

33. What is the left endpoint of the symmetric 99% confidence interval for E(Y)?
34.

35. Based on the information of the common section, which actions are correct for testing situation I and testing situation II below:
36.

I. H0: E(Y)=0, α=0.01, H1: E(Y) ¹ 0.

II. H0: E(Y)=1000, α=0.01, H1: E(Y) ¹ 1000.

a. Accept H0 in both situations I and II.

b. Accept H0 in situation I, and reject H0 in situation II.

c. Reject H0 in situation I, and accept H0 in situation II.

d. Reject H0 in both situations I and II.

End of Common Problem Section

Common Information for Questions 19-21

In a clinical trial, 400 patients suffering from an illness will be randomly assigned to one of two groups so that 200 receive an experimental treatment and 200 receive the best available treatment. The random variable X is the response of a patient to the experimental medicine, and the random variable B is the response of a patient to the best currently available treatment. Both X and B are normally distributed with σXB=500. The null hypothesis to be tested is that E(X)-E(B)=0 against the alternative that E(X)-E(B)>0 at the 0.01 level of significance, and the test statistic to be used is  200200.

37. What is the critical value for this test?
38.

39. What is the probability of a Type II error for the test specified in the common section when E(X)-E(B)=100 and σXB=500?
40.

41. What is the least number n in each group that would have to be taken so that the probability of a Type II error for the test of the null hypothesis specified in the common section is 0.01 when E(X)-E(B)=100 and σXB=500?
42.

End of Common Group

Common Information for Problems 22-24.

Each patient in a study will take a specified medicine, and the patient’s response to that medicine will be measured. Eight patients were randomly assigned to one of two groups each containing four patients. Group 1 will receive an experimental medicine. The random variable X denotes a patient’s response to the experimental medicine and is normally distributed with unknown expected value E(X) and unknown standard deviation σ. Group 2 will receive the best currently available medicine. The random variable B denotes a patient's response to this medicine and is normally distributed with unknown expected value E(B) and unknown standard deviation σ. The experiment was run. The observed sample averages were  4 = 2312.8 and  4=141.2. The observed standard deviations were sX=666.8 and sB=573.5. The resulting pooled estimate of the standard deviation σ was 621.8.

43. What is the standard deviation of 4-4?
44.

45. What is the right endpoint of the 99% confidence interval for E(X-B)?
46.

47. Based on the information of the common section, which actions are correct for testing situation I and testing situation II below:
48.

I. H0: E(X-B)=1000, α=0.01, H1: E(X-B) ¹ 1000.

II. H0: E(X-B)=4000, α=0.01, H1: E(X-B) ¹ 4000.

a. Accept H0 in both situations I and II.

b. Accept H0 in situation I, and reject H0 in situation II.

c. Reject H0 in situation I, and accept H0 in situation II.

d. Reject H0 in both situations I and II.

End of Group of Questions

49. A research team collected information on ten null hypotheses in a study. They will test the ten null hypotheses: H0,1: E(Y1)=0 at level of significance α, ¼, H0,10: E(Y10)=0 at level of significance α. Let αO be the probability that the researcher rejects at least one of H0,1, ¼, H0,10. What is the largest value of α that guarantees that αO £ 0.05?

End of the Examination

Solution

• 1. C 2. 1180 3. 2333 4. 0.19 5. 0.1445
• 6. 0.0381 7. 0.3633 8. 0.0026 9. A 10. 1037 or more
• 11. 489.03 12. 0 13. 44 or more 14. 165.6 15. B
• 16. C 17. 100.47 18. D 19. 116.3 20. 0.6277
• 21. 1083 or more 22. 0.7071*621.8 23. 3801.47 24. B 25. 0.005