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

This is the start of the examination.

- 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?
- 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}? - 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*? - 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?
- What is α, the probability of a Type I error, for this test of the null hypothesis?
- 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?
- What is the p-value for a student with
*e*_{8}=3? - What is the observed significance level (p-value)?
- Which of the following is a correct decision for this result?
- A random variable
*Y*has unknown expected value E(*Y*) and standard deviation σ=50. What is the least sample size_{Y}*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±4?_{n} - What is the critical value for this test?
- What is the probability of a Type II error when E(
*Y*)=450, σ_{Y}=100,*n*=225, and α=.05? - 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, σ_{Y}=100, and α=.05? - What is the right endpoint of the symmetric 95% confidence interval for E(
*Y*)? - Based on the information of the common section, which actions are correct for testing situation I and testing situation II below:
- 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 σ. Their problem was to test the null hypothesis H_{Y}_{0}: E(*Y*)=100 against the alternative hypothesis H_{1}: 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? - What is the left endpoint of the symmetric 99% confidence interval for E(
*Y*)? - Based on the information of the common section, which actions are correct for testing situation I and testing situation II below:
- What is the critical value for this test?
- What is the probability of a Type II error for the test specified in the common section when E(
*X*)-E(*B*)=100 and σ=σ_{X}=500?_{B} - 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 σ=σ_{X}=500?_{B} - What is the standard deviation of
_{4}-_{4}? - What is the right endpoint of the 99% confidence interval for E(
*X-B*)? - Based on the information of the common section, which actions are correct for testing situation I and testing situation II below:
- A research team collected information on ten null hypotheses in a study. They will test the ten null hypotheses: H
_{0,1}: E(*Y*_{1})=0 at level of significance α, ¼, H_{0,10}: E(*Y*_{10})=0 at level of significance α. Let α_{O}be the probability that the researcher rejects at least one of H_{0,1}, ¼, H_{0,10}. What is the largest value of α that guarantees that α_{O}£ 0.05?

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.

** 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 *E*_{8}, the number of errors that the student makes, as the basis for accepting or rejecting the null hypothesis H_{0} that their student is a random guesser. They will reject H_{0} when they observe *E*_{8} £2 and accept H_{0} otherwise.

They computed *F*_{0}, the cumulative distribution function of *E*_{8} under the null hypothesis. They also computed *F*_{1}, the cumulative distribution function of *E*_{8} 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 *E*_{8}

under H_{0} and for Knowledgeable Student

s F_{0}(s) F_{1}(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

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.

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

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.

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.

I. H_{0}: E(Y)=50, α=0.05, H_{1}: E(Y) ¹ 50.

II. H_{0}: E(Y)=250, α=0.05, H_{1}: E(Y) ¹ 250.

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.

**End of a Common Problem Section**

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.

I. H_{0}: E(*Y*)=0, α=0.01, H_{1}: E(*Y*) ¹ 0.

II. H_{0}: E(*Y*)=1000, α=0.01, H_{1}: E(*Y*) ¹ 1000.

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.

**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 σ* _{X}*=σ

**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 s* _{X}*=666.8 and s

I. H_{0}: E(*X-B*)=1000, α=0.01, H_{1}: E(*X-B*) ¹ 1000.

II. H_{0}: E(*X-B*)=4000, α=0.01, H_{1}: E(*X-B*) ¹ 4000.

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.

End of Group of Questions

End of the Examination

Solution