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PoNQM6uwFe, Exercises of Probability and Statistics

Kent State University (KSU) - Ashtabula CampusProbability and Statistics

Probability & Statistics: Practice questions for the final exam | PSTAT 120C

Typology: Exercises

2019/2020

Uploaded on 06/08/2020

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PSTAT 120C: Final Exam Information Tuesday, June 9 8:00–11:00
The policies for the final will be essentially the same as they were for the midterm except you will
be allowed an additional page of notes.
•Please arrive early so that we can get started on time.
•You will have 3hours to complete the exam.
•You will be allowed to consult your textbook and 2sides of an 8 1
2×11 page for whatever notes
you wish. No other books or resources will be allowed. Make sure that the tables are complete
and legible in your text.
•You will need a calculator, and you will need bluebooks or paper to write out your answers.
•There will be 4–6 questions. The midterm and the shorter homework questions should serve
as models for the type of questions I will ask.
Practice Questions
These questions cover the material from the second half of the course. You should also review the
midterm and the Bayesian inference questions from assignment 7 for additional relevant material.
1. My friend Ron is convinced that his computer solitaire game is cheating him. He records for
16 games whether he Won or Lost.
Lost Won Lost Won Lost Lost Won Lost
Won Lost Lost Lost Won Lost Won Lost
(a) Calculate a Pvalue for testing the null hypothesis that the outcome of each successive
game is independent of the others.
(b) How would you interpret your result? (Use α= 0.05)
2. The grades in an upp er division class are tabled to compare the performance of juniors and
seniors:
A B C F
Juniors 16 32 23 2
Seniors 8 24 16 3
We want to test whether or not the distribution of grades was the same for the two classes of
students.
(a) Is the normal approximation appropriate for this table? If necessary, redraw the table in
such a way that it corrects the problem.
(b) Calculate the appropriate test statistic for the χ2test of independence.
(c) What is the critical value for this test with size α= 0.01?
(d) What do you conclude?
1
pf3
pf4

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PSTAT 120C: Final Exam Information Tuesday, June 9 8:00–11:

The policies for the final will be essentially the same as they were for the midterm except you will be allowed an additional page of notes.

  • Please arrive early so that we can get started on time.
  • You will have 3 hours to complete the exam.
  • You will be allowed to consult your textbook and 2 sides of an 8 12 × 11 page for whatever notes you wish. No other books or resources will be allowed. Make sure that the tables are complete and legible in your text.
  • You will need a calculator, and you will need bluebooks or paper to write out your answers.
  • There will be 4–6 questions. The midterm and the shorter homework questions should serve as models for the type of questions I will ask.

Practice Questions

These questions cover the material from the second half of the course. You should also review the midterm and the Bayesian inference questions from assignment 7 for additional relevant material.

  1. My friend Ron is convinced that his computer solitaire game is cheating him. He records for 16 games whether he Won or Lost.

Lost Won Lost Won Lost Lost Won Lost Won Lost Lost Lost Won Lost Won Lost

(a) Calculate a P value for testing the null hypothesis that the outcome of each successive game is independent of the others. (b) How would you interpret your result? (Use α = 0.05)

  1. The grades in an upper division class are tabled to compare the performance of juniors and seniors:

A B C F Juniors 16 32 23 2 Seniors 8 24 16 3

We want to test whether or not the distribution of grades was the same for the two classes of students.

(a) Is the normal approximation appropriate for this table? If necessary, redraw the table in such a way that it corrects the problem. (b) Calculate the appropriate test statistic for the χ^2 test of independence. (c) What is the critical value for this test with size α = 0.01? (d) What do you conclude?

  1. The following data are independent observations from some unknown distribution

2.1 2.3 3.8 3.9 3.9 4.2 4.2 4.7 5.1 5. 5.4 5.5 5.7 5.8 6.0 6.2 6.8 7.9 9.3 9. 10.3 10.4 10.5 10.8 11.2 11.2 12.3 14.9 17.3 17.

We can generate a table of the values

0 < X ≤ 3 3 < X ≤ 6 6 < X ≤ 9 9 < X ≤ 14 X > 14 Counts 2 13 3 9 3

Use a χ^2 goodness-of-fit test to test the null hypothesis that this data comes from an expo- nential distribution with some unknown parameter λ. (Use α = 0.05)

  1. The following table counts the number of outcomes in 100 trials for each of four events: Event A B C D Outcomes 34 30 12 24 Perform a goodness-of-fit test of size α = 0.05 to test the hypothesis that P(A) = P(B) and P(C) = P(D).
  2. The chair of the statistic department is interested in the majors of students in the 120 courses. The registrar tabulated the number of engineers, economist, and statisticians in each semester of 120.

Engineering Economics Statistics 120A 112 63 35 120B 80 53 27 120C 40 23 12

Test whether or not the distribution of majors is the same for the three semesters.(Use size α = 0.05 What interpretation would you give to the results?

  1. (WMS 15.23) An industrial wants to test the strengths of two materials.

Material A Material B 15.3 21. 18.7 22. 22.3 18. 17.6 19. 19.1 17. 14.8 27.

We want to use a Mann-Whitney U test to test whether the strengths of the two materials is the same.

(a) What assumptions do we make about the data to be sure that the Mann-Whitney test is appropriate?

Men No Show Deferred Served Total White 225 133 20 378 Black 47 52 7 106 Hispanic 87 52 2 141 Women No Show Deferred Served Total White 213 117 19 349 Black 56 68 5 129 Hispanic 100 63 5 168

We want to test the hypothesis that their jury service is marginally independent of gender and race.

(a) Calculate the expected number Hispanic women that served on a jury and the expected number of Hispanic men that served on a jury. (b) Is the normal approximation appropriate for this table? If not, then how can it be fixed? (c) If the test statistic is X^2 = 28.75, then what do we conclude at an α = 0.01 level?

  1. A model for incomes in a heterogeneous population takes n independent observations X 1 ,... , Xn and transforms them to generate Yi = log Xi. Assume that Yi have independent normal distributions with unknown mean μ and variance σ^2 = 1. We will impose a prior distribution on this mean parameter that is normal with mean 10 and variance 5.

(a) For 38 observations with ¯x = 11.129, what is our Bayes estimator of μ? (b) Give a 95% Bayesian Credible Interval for μ. (c) To generate an estimate on the original scale we need eμ. Calculate the Bayes estimator for eμ.