April 15, 1998

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ID Number:

1. Write you name and id number on the top of this page, and leave your ID on the top of the desk until the proctor examines it.

 

2. You may begin your work on this examination immediately after you have put your name on it. Work each problem in the space provided underneath each question. Record the answer that you wish to be graded on in the box after each problem.

 

3. You may use your calculator, provided that you permit the proctor to inspect it. You may also use your text. You may not share your calculator or your text with any other student, and you may not borrow another student’s calculator or text. If you calculator fails, do the best that you can to estimate the answer.

 

4. You may use your text and must use the tables in your text. You may write notes in your text, and you can staple or tape notes into your text. You may not have notes on paper that is not affixed to your text.

 

5. You may use only paper that I have supplied to you. Do not bring or write on any other paper. Possession of other paper on your desk or having other papers visible from your desk is a violation of academic honesty regulations and will be penalized by a reduction in your grade on this examination of two letters.

 

6. Each problem is worth ten points.

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This is the start of the examination.

 

Background Information for Questions 1-3.

 

A research team will test the null hypothesis that E(Y)=500 at the 0.01 level of significance against the alternative that E(Y)<500. When the null hypothesis is true, Y has a normal distribution with standard deviation σ_Y=200. They will take a random sample of 400 observations and use the statistic  ₄₀₀, the sample average of the random sample of size 400, to test the null hypothesis.

 

1. What is the standard deviation of ₄₀₀ under the null hypothesis?

 

My answer is:

 

2. What is the probability of a Type II error when E(Y)=440, σ_Y=200, n=400, and α=0.01?

 

My answer is:

 

 

 

 

3. What is the smallest value of n, the sample size, so that the probability of a Type II error is no more than 0.01 when E(Y)=440, σ_Y=200, and α=0.01?

 

My answer is:

 

 

 

 

 

 

 

 

 

End of the Group of Problems

 

Common Information for Questions 4 and 5

 

A research team had six litters of animals available for study. They randomly selected two animals from each litter and randomly assigned one to treatment V and the other to treatment W. Their experimental results are given in Table 1.

 

Table 1

Experimental Results by Litter

Treatment V and Treatment W

 

Litter Treatment V Result Treatment W Result

 

1 200 180

2 180 110

3 170 160

4 400 270

5 420 330

6 500 460

 

4. What is the left endpoint of the 99% confidence interval for E(V)-E(W)?

 

My answer is:

 

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

I. H₀: E(V)- E(W)= 0, α=0.01, H₁: E(V)- E(W) ¹ 0.

II. H₀: E(V)- E(W)=100, α=0.01, H₁: E(V)- E(W) ¹100.

 

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.

 

My answer is:

 

End of Group of Questions

 

6. Each patient in a study took a specified medicine, and the patient’s response to that medicine was measured. Fifteen patients were randomly assigned to two groups. Group 1 had eight patients and received 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 had seven patients and received the best 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 observed sample averages were ₈=171.3 and ₇=919.7. The observed standard deviations were s_X=23.3 and s_B=36.5. The resulting pooled estimate of the standard deviation σ was 30.1. What is the right endpoint of the 95% confidence interval for E(X-B)?

 

My answer is:

 

Common Information for Questions 7-8.

 

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 σ_X=σ_B=300. The null hypothesis to be tested is that E(X)-E(B)=0 against the alternative that E(X)-E(B)<0 at the 0.05 level of significance, and the test statistic is  ₂₀₀- ₂₀₀.

 

7. What is the probability of a Type II error for the test specified in the common section when E(X)-E(B)= -80 and σ_X=σ_B=300?

 

My answer is:

 

8. What is the 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 0.05 when E(X)-E(B)= -80 and σ_X=σ_B=300?

 

My answer is:

 

End of Common Group

 

9. Researchers wish to predict the value of Y, a normally distributed random variable with unknown mean and standard deviation. They took a random sample of 3 observations from Y. The observed mean was ₃=5876, and the observed estimate of the standard deviation was s=713. What is the left endpoint of the symmetric 95% two-stage prediction interval for Y?

 

My answer is:

 

Common Information for Questions 10 and 11

 

An animal produces offspring with three types of a characteristic, called A, B, and C. According to a hypothesized genetic theory, the proportion of offspring that are type A should be 15%, the proportion that are type B should be 25%, and the proportion that are type C should be 60%. One thousand offspring are bred and classified. There are 154 of type A, 258 of type B, and 588 of type C.

 

10. What is the value of the chi-squared test of goodness of fit of the hypothesized genetic theory?

 

My answer is:

 

11. Which of the following is the correct decision for accepting or rejecting the null hypothesis that the proportions predicted by the genetic theory are consistent with the observed counts?

 

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.

 

My answer is:

 

End of Group

 

 

 

 

 

 

Common Information for Questions 12-13

 

A research team classified 2000 subjects with regard to whether or not the subject’s blood had factor H and whether or not the subject’s blood had factor L. Of these, 800 did not have blood factor H, and 1200 had blood factor H. There was a second blood factor L that was present in 1000 of the subjects and absent in 1000 subjects. There were 387 subjects who had both blood factor H and blood factor L.

 

12. What is the value of the X² test for independence of having blood factor H and having blood factor L?

 

 

My answer is:

 

 

13. Which of the following is the correct decision for accepting or rejecting the null hypothesis of independence of having blood factor H and having blood factor L?

 

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.

 

My answer is:

 

 

End of Group of Questions

 

 

 

Information for Problems 14-16.

 

One hundred values of an independent variable x_i are chosen such that S(x_i- ₁₀₀)²=4,000,000. For each setting of x_i, the random variable Y_i will be observed where

 

Y_i=β_0.1+β_1.0x_i+σZ_i.

 

Here β_0.1 and β_1.0 are fixed but unknown parameters, σ=800, and the Z_i are normally and independently distributed with E(Z_i)=0 and var(Z_i)=1. The null hypothesis to be tested is H₀: β_1.0=0, α=0.05, and the alternative is H₁: β_1.0>0. The random variable B_1.0 is the ordinary least squares estimator of β_1.0.

14. What is the probability of a Type II error in the test specified in the common section using B_1.0, the ordinary least squares estimator of the slope, as the test statistic when α=.05 and Σ(x_i-₁₀₀)²=4,000,000 for the alternative with β_1.0=1.9 and σ=800?

 

My answer is:

 

 

15. How many observations n are necessary so that the probability of a Type II error in the test specified in the common section using as the test statistic B_1.0, the ordinary least squares estimator of the slope β_1.0, is 0.05 when α=.05, Σ( x_i-_n)²=40,000n, β_1.0=1.9, and σ=800?

 

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16. The probability of a Type II error in a linear regression analysis can also be reduced by increasing Σ( x_i-₁₀₀)² to 4,000,000 T², or by a combination of increasing sample size and increasing the variability of the x_i values. What value of T is necessary so that the probability of a Type II error is 0.05 when α=0.05, Σ( x_i-₁₀₀)²=4,000,000T², β_1.0=1.9, and σ=800?

 

My answer is:

 

 

 

End of Group of Problems.

Information for Questions 17-25.

 

Forty values of an independent variable x_i were chosen such that S(x_i- ₄₀)²= 100,000. For each setting of x_i, the random variable Y_i=β_0×1+β_1×0x_i+σZ_i was observed. Here β_0×1 , β_1×0, and σ were fixed but unknown parameters, and the Z_i were independent standard normal random variables (that is, E(Z_i)=0 and var(Z_i)=1). The results observed were that  ₄₀ =2500, the correlation coefficient of x and y was 0.248,  ₄₀=1579, and S(y_i- ₄₀)²=816,943.

 

17. What is the value of the sum of squares due to the regression?

 

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18. What is the value of the sum of squares for error?

 

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19. What is the value of the F test of the null hypothesis H₀: β_1.0=0 against the alternative hypothesis H₁: β_1.0 ¹0?

 

My answer is:

 

 

20. Which of the following is a correct decision about the null hypothesis H₀: β_1×0=0, against the alternative hypothesis H₁: β_1.0 ¹ 0?

 

a. Reject at the .01 level.

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

c. Accept at the .05 level.

 

 

My answer is:

 

 

 

21. What is the estimated standard deviation of B₁×0, the ordinary least squares estimate of the slope, based on the statistics reported in the common section?

 

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22. What is the left endpoint of the 99% confidence interval for β_1×0?

 

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23. What is the fitted value for Y when the value of x has been set to 2650?

 

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24. What is the left endpoint of the 99% confidence interval for E(Y|x=2650), the value of the regression function when X=2650?

 

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25. What is the left endpoint of the symmetric two stage 99% prediction interval for a future value of Y when X=2650?