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AMS315/576, Second Examination, Spring Semester, 1998
April 15, 1998
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This is the start of the examination.
Background Information for Questions 13.
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}=50. They will take a random sample of 100 observations and use the statistic _{100}, the sample average of the random sample of size 100, to test the null hypothesis.
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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
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I. H_{0}: E(V) E(W)= 0, α=0.01, H_{1}: E(V) E(W) ¹ 0.
II. H_{0}: E(V) E(W)=150, α=0.01, H_{1}: E(V) E(W) ¹150.
a. Accept H_{0} in both situations I and II.
b. Accept H_{0} in situation I, and reject H_{0} in situation II.
c. Reject H_{0} in situation I, and accept H_{0} in situation II.
d. Reject H_{0} in both situations I and II.
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End of Group of Questions
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Common Information for Questions 78.
In a clinical trial, 100 patients suffering from an illness will be randomly assigned to one of two groups so that 50 receive an experimental treatment and 50 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}=200. 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 _{50} _{50}.
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End of Common Group
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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 142 of type A, 234 of type B, and 624 of type C.
My answer is: 

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.
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End of Group
Common Information for Questions 1213
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. The second blood factor L was present in 1000 of the subjects and absent in 1000 subjects. There were 420 subjects who had both blood factor H and blood factor L.
My answer is: 

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.
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End of Group of Questions
Information for Problems 1416.
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_{1.0}x_{i}+σZ_{i}.
Here β_{0.1} and β_{1.0} are fixed but unknown parameters, σ=400, 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 H_{0}: β_{1.0}=0, α=0.01, and the alternative is H_{1}: β_{1.0}>0. The random variable B_{1.0} is the ordinary least squares estimator of β_{1.0}.
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End of Group of Problems.
Information for Questions 1725.
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_{1×0}x_{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 0.417, _{40}=579, and S(y_{i} _{40})^{2}=186,943.
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a. Reject at the .01 level.
b. Reject at the .05 level and accept at the .01 level.
c. Accept at the .05 level.
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End of Group of Questions
End of the Examination