Information Pertaining to the Final Exam
Basics
Your exam solutions are due at 10 PM on Tuesday, Dec. 15. Please put your solutions in the proper order and draw boxes around, boldface, or highlight your final answers.
During the period of time that goes from when I first post a link to the exam to when solutions are due, you should not discuss the material of this course
(and particularly the exam problems) with
anyone except for me (and I don't intend to give you any help with solving the problems ... I will only clarify what is being requested and investigate any
suspected typos or glitches that you think you've discovered).
The exam is an open books and open notes exam,
but you cannot share books or notes during the exam.
Description of Exam
The exam will have 5 problems, having a total of 10 parts, with each part being worth 5, 10, or 12.5 points (with most of them being worth 10 points).
- 3 parts, worth 30 points, deal with point estimation. You will be requested to obtain a method of moments estimator, a MLE, and a UMVUE (and for one of the
confidence interval parts you may need to obtain a MLE). You will not be expected to obtain a Bayes estimator or a Pitman estimator.
- 2 parts, worth 20 points, pertain to confidence intervals. You will be requested to obtain one exact confidence interval (using any method use choose, but I
recommend using some sort of pivot having a chi-squared distribution), and one approximate confidence interval (based on the asymptotic normality of a MLE).
- 5 parts, worth 50 points, pertain to hypothesis testing. In the various parts, you may be requested to obtain a test function, a p-value, or a power. You
should know how to use the Neyman-Paerson lemma to do a most powerful test, how to use a statistic having a monotone likelihood ratio to do a UMP test, and you
should know how to do a (general) likelihood ratio test. (3 of the 5 parts dealing with hypothesis testing pertain to the asymptotic results covered during the
last week of lectures.)
Some of the parts of the exam will be similar to some of the easier parts of HW #3, HW #4, and HW #6, and some of the parts will be similar to some of the
Ch. 10 example problems given below. I think that two or three of the 10 parts are a bit more challenging than the rest, but I don't think that any of the parts are
terribly difficult.
I think the best way to prepare for the exam will be to study the homework problems listed below, along with similar examples from the course notes, and to also
study the example problems based on Ch. 10 material that are given below. (It may be best to try to mimic the style of my solutions for these problems when
writing your exam solutions.)
Homework Problems to Review
I recommend trying to understand these homework problems, and not to worry
about other homework problems this semester. Pay particular attention to the style of the justifications (and perhaps try to mimic some of the justifications in your
exam solutions).
- 1(a), 1(b), 1(c) of HW #3
- 1(a) of HW #4
- 2 of HW #4
- 1 of HW #6
- 2 of HW #6
- 3(a), 3(b), 3(c) of HW #6
- 4(a), 4(b) of HW #6
- 5(a) of HW #6
- 6 of HW #6
Problems Based on Ch. 10 Material
(since none of the HW problems covered Ch. 10)
Note: Complete solutions for these problems can be found in the Final Exam period folder of the Blackboard site.
- Problem A:
- Consider 90 iid geometric random variables having parameter θ (and thus mean 1/θ), and a UMP test of the null hypothesis
that θ <= 0.1 against the alternative that θ > 0.1. Give the approximate p-value
(based on large sample results) which results for the case of the sum of the observations equal to 723.
- answer:
- 0.025
- Problem B:
- Consider n iid random variables having the
pdf indicated here.
Give the approximate p-value
(based on large sample results) which results from a LRT of the null hypothesis that θ = 1 against the
general alternative for the case of n = 50 and the sum of the observations equal to 115.5.
- answer:
- 0.14
- Problem C:
- Consider 100 independent random variables:
X1,
X2, ...,
X50
being iid normal random variables having mean 0 and variance θX, and
Y1,
Y2, ...,
Y50
being iid normal random variables having mean 0 and variance θY.
Given that
Σ xi2 = 37.5
& Σ yi2 = 62.5,
give the approximate p-value (based on large sample results) which results from a LRT of the null hypothesis that the two variances are equal against the
general alternative (that the variances are not equal).
- answer:
- 0.072
- Problem D:
- Consider 100 independent random variables:
X1,
X2, ...,
X50
being iid normal random variables having mean 0 and variance θX, and
Y1,
Y2, ...,
Y50
being iid normal random variables having mean 0 and variance θY.
Given that
Σ xi2 = 37.5
& Σ yi2 = 62.5,
give the approximate p-value (based on large sample results) which results from a LRT of the null hypothesis that the variances both equal 1 against the
general alternative (that at least one of the two variances is not equal to 1).
- answer:
- 0.20
- Problem E:
- Consider n iid random variables having the
pdf indicated here.
Give an approximate 95% confidence interval for θ
(based on large sample results)
for the case of n = 100 and the sum of the logs of the observations equal to -50.
- answer:
- (1.67, 2.49) if you do something similar to the example on p. 10.4.2 of the course notes (and (1.61, 2.39)
if you use the standardized sum of the t(Xi) as an approximately standard normal pivot)
- Problem F:
- Consider n iid random variables having the
pdf indicated here.
Give an approximate 95% confidence interval for θ
(based on large sample results)
for the case of n = 100 and
Σ xi4 = 74.6.
- answer:
- (1.12, 1.67) if you do something similar to the example on p. 10.4.2 of the course notes, which is preferred