# AMS 315, Examination 1

Question

AMS 315, Examination 1

Name: ID:

Directions: Write your name in the space provided. Work each problem in the space underneath the problem and on the back side of the page. You may use a calculator but not a computer or cell-phone. You may also use a single sheet of notes in your handwriting that is the size of the paper in this examination. You may use only the paper in this form. You are on your honor not to use any other assistance during this examination. Do not make marks on the tables given to you to work this examination. Turn in both your paper and your tables at the end of the examination. There will be no partial credit given for a problem unless you show your work. The total points for this examination is 240. There are five problems, and each is worth 40 points, except problem 4, which is worth 80 points. In the event of a fire alarm, please take your papers, exit the room, find a private place to work, and turn in your examination to me in my office (Math Tower 1-113) by 9:00 pm today. In this event, you are still on your honor not to give or receive assistance.

Since the course satisfies requirements for actuarial credentials, academic integrity standards will be enforced strictly.

1. A research team took a random sample of 5 observations from a normally distributed random variable Y and observed that .gif”> and .gif”>, where .gif”> was the average of the five observations sampled from Y and .gif”> was the unbiased estimate of .gif”>. A second research team took a random sample of 3 observations from a normally distributed random variable X and observed that .gif”> and .gif”>, where .gif”> was the average of the three observations sampled from X and .gif”> was the unbiased estimate of .gif”>.Calculate the 99% confidence interval for .gif”> using the pooled variance estimator.

2. In a clinical trial, 2J patients suffering from an illness will be randomly assigned to one of two groups so that J will receive an experimental treatment and J will receive the best available treatment. The random variable X is the response of a patient to the experimental medicine, and the random variable Bis the response of a patient to the best currently available treatment. Both Xand B are normally distributed with .gif”>. The null hypothesis to be tested is that .gif”>against the alternative that.gif”>at the 0.005 level of significance. What is the number J 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 .gif”>?

3. A research team took a random sample of 4 observations from a normally distributed random variable Y and observed that .gif”> and .gif”>, where .gif”> was the average of the four observations sampled from Y and .gif”> was the unbiased estimate of .gif”>. A second research team took a random sample of 3 observations from a normally distributed random variable X and observed that .gif”> and .gif”>, where .gif”> was the average of the three observations sampled from X and .gif”> was the unbiased estimate of .gif”>. Test the null hypothesis .gif”> against the alternative .gif”> at the 0.10, 0.05, and 0.01 levels of significance.

4. A research team studied Y, the blemished area on the skin of a laboratory animal, and how Y was affected by the dose of medicine. They used four doses of medicine: 0, 1, 2, and 3 units respectively. They randomly assigned 15 animals to each dosage and observed that the average values of Y at each dosage were .gif”> where .gif”> was the average of the observations taken with dosage .gif”>, respectively. They also observed that .gif”> where .gif”> was the unbiased estimate of the variance for the observations taken with dosage .gif”>, respectively. The research team seeks to minimize .gif”>.

Complete the balanced one way analysis of variance table for these results; that is, be sure to specify the degrees of freedom, sum of squares, mean square, F-test, and your conclusion. Use the 0.10, 0.05, and 0.01 levels of significance. (30 points)

Find the sum of squares due to the linear, quadratic, and cubic contrasts. The coefficients of the linear contrast are .gif”> ; the coefficients of the quadratic contrast are .gif”>; and the coefficients of the cubic contrast are .gif”>. (20 points)

Find the 99% Scheffe confidence intervals for the linear, quadratic, and cubic contrasts. (20 points)

Which setting of dosage is optimal? (10 points) The four parts of this problem are worth 80 points.

Analysis of Variance Table, Problem 4A

SS

DF

MS

F

Dosages

151.35

3

50.45

0.679

Error

4159.96

56

74.285

Total

4311.31

59

5. The random variable X has expected value .gif”>and variance .gif”>. The random variable Y has expected value .gif”>and variance .gif”>. The random variable W has expected value .gif”>and variance .gif”>.These random variables are not necessarily independent. That is, the covariances of the pairs of random variables are given by.gif”>, .gif”>, and .gif”>. Let .gif”>. Find .gif”>and .gif”>.

End of Examination

AMS 315, Examination 1

1. A research team took a random sample of 2 observations from a normally distributed random variable Y and observed that .gif”> and .gif”>, where .gif”> was the average of the two observations sampled from Y and .gif”> was the unbiased estimate of .gif”>. A second research team took a random sample of 3 observations from a normally distributed random variable X and observed that .gif”> and .gif”>, where .gif”> was the average of the three observations sampled from X and .gif”> was the unbiased estimate of .gif”>.Test the null hypothesis .gif”> against the alternative .gif”> at the 0.10, 0.05, and 0.01 levels of significance using the pooled variance t-test.

2. In a clinical trial, 2J patients suffering from an illness will be randomly assigned to one of two groups so that J will receive an experimental treatment and J will 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 Xand B are normally distributed and have .gif”>. The null hypothesis to be tested is that .gif”>against the alternative that.gif”>at the 0.025 level of significance. What is the number J 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 .gif”>?

3. A research team took a sample of 4 observations from the random variable Y, which had a normal distribution .gif”>. They observed .gif”>, where .gif”> was the average of the four sampled observations and .gif”> was the observed value of the unbiased estimate of .gif”>, based on the sample values. Test the null hypothesis that .gif”> against the alternative .gif”> at the 0.10, 0.05, and 0.01 levels of significance.

4. A research team studied Y, the blemished area on the skin of a laboratory animal, and how Y was affected by the dose of medicine. They used four doses of medicine: 0, 1, 2, and 3 units respectively. They randomly assigned 25 animals to each dosage and observed that the average values of Y at each dosage were .gif”> where .gif”> was the average of the observations taken with dosage .gif”>, respectively. They also observed that .gif”> where .gif”> was the unbiased estimate of the variance for the observations taken with dosage .gif”>, respectively. The research team seeks to minimize .gif”>.

a. Complete the balanced one way analysis of variance table for these results; that is, be sure to specify the degrees of freedom, sum of squares, mean square, F-test, and your conclusion. Use the 0.10, 0.05, and 0.01 levels of significance. (30 points)

b. Find the sum of squares due to the linear, quadratic, and cubic contrasts. The coefficients of the linear contrast are .gif”> ; the coefficients of the quadratic contrast are .gif”>; and the coefficients of the cubic contrast are .gif”>. (20 points)

c. Find the 95% Scheffe confidence intervals for the linear, quadratic, and cubic contrasts. (20 points)

d. What is the optimal setting of the dosage? (10 points) The four parts of this problem are worth 80 points.

5. The random variable X has expected value .gif”>and variance .gif”>. The random variable Y has expected value .gif”>and variance .gif”>. The random variable W has expected value .gif”>and variance .gif”>.These random variables are not necessarily independent. That is, the covariances of the pairs of random variables are given by.gif”>, .gif”>, and .gif”>. Let .gif”>. Find .gif”>and .gif”>.

End of Examination

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