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Math 116 — Practice for Exam 2
Generated October 21, 2018
Name:
Instructor: Section Number:
- This exam has 7 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.
- Do not separate the pages of the exam. If any pages do become separated, write your name on them and point them out to your instructor when you hand in the exam.
- Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam.
- Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate.
- You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3′′^ × 5 ′′^ note card.
- If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use.
- You must use the methods learned in this course to solve all problems.
Semester Exam Problem Name Points Score Winter 2018 2 5 class time 12 Winter 2018 2 10 10 Fall 2013 2 5 pneumonia 10 Fall 2012 2 8 internet cafe 14 Fall 2013 2 9 coffee 7 Fall 2017 3 9 psych experiment 6 Fall 2017 3 3 11 Total 70
Recommended time (based on points): 68 minutes
Math 116 / Exam 2 (March 19, 2018) do not write your name on this exam page 5
- [12 points]
Yennifer’s Introductory Thermodynamics of Muck course is supposed to start at 9: am, but her instructor does not consistently start on time. Let p(x) be the probability density function for the amount of time x, in minutes, between when the instructor is supposed to start the class and when they actually start class.
For parts a.-c., you do not need to justify your answer.
a. [2 points] Yennifer is coming from another class and therefore always arrives at 9:06, exactly 4 minutes before class is supposed to start. Find the probability that class starts before Yennifer arrives.
Answer:
Note that:
- x = 0 represents class starting at 9: am.
- A negative value of x represents starting class early.
- All of the nonzero portion of p(x) is given in the graph below.
- The area of the shaded region is 0.1.
y = p(x)
x
y
b. [3 points] Which of the following statements is best supported by the equation p(12) = 0.02? Circle the one best answer.
i. The probability that the instructor will start class at 9:22 is 2%. ii. The probability that the instructor will start class between 9:21 and 9:23 is about 2%. iii. The probability that the instructor will start class between 9:21 and 9:23 is about 4%. iv. The probability that the instructor has started class by 9:22 is about 2%. v. The probability that the instructor has started class by 9:22 is about 48%.
c. [3 points] Let P (x) be the cumulative distribution function for p(x). Which of the following could be the formula for P (x) on the interval − 2 < x < 8? Circle all answers that could be correct.
i. P (x) = 0 ii. P (x) = 1
iii. P (x) = 0. 06 x iv. P (x) = 0.06(x + 2)
v. P (x) = 0.06(x+2)+0. 3 vi. P (x) = 0. 1 − 0 .06(x−8)
d. [4 points] Find the median value of x. Show your work, and write your answer in exact form.
Answer:
University of Michigan Department of Mathematics Winter, 2018 Math 116 Exam 2 Problem 5 (class time)
Math 116 / Exam 2 (November 13, 2013) page 7
- [10 points] Consider a group of people who have received a new treatment for pneumonia. Let t be the number of days it takes for a person with pneumonia to fully recover. The probability density function giving the distribution of t is
f (t) =
(1 + at)^2 , for t > 0 ,
for some positive constant a.
a. [2 points] Give a practical interpretation of the quantity
3
f (t)dt. You do not need to compute the integral.
b. [5 points] Find a formula for the cumulative distribution function F (t) of f (t) for t > 0. Show all your work. Your answer may include a. Your final answer should not include any integrals.
c. [3 points] Determine the value of a. Show all your work.
University of Michigan Department of Mathematics Fall, 2013 Math 116 Exam 2 Problem 5 (pneumonia)
Math 116 / Exam 2 (November 14 , 2012) page 10
- [14 points] A coffee shop offers only one hour of free internet access to all its customers. The time t in hours a customer uses the internet at the coffee shop has a probability density function p(t) =
at
1 − t^2 0 ≤ t ≤ 1. 0 otherwise. where a is a constant. a. [4 points] For what value of a is p(t) a probability density function? Find its value without using your calculator.
b. [4 points] Find the cumulative distribution function P (t) of p(t). Make sure to indicate the value of P (t) for all values of −∞ < t < ∞. Your final answer should not contain any integrals.
University of Michigan Department of Mathematics Fall, 2012 Math 116 Exam 2 Problem 8 (internet cafe)
Math 116 / Exam 2 (November 13, 2013) page 12
- [7 points] Thanks to the Math Department’s acquisition of a coffee tank in October, there are now 300 cups of coffee available to the graduate students each day. The department wants to assess how much of the coffee is drunk and how much is wasted. Let c be the amount of coffee drunk in one day, measured in hundreds of cups of coffee. The probability density function for c is given by
p(c) =
3 20 c
(^2) for 0 ≤ c ≤ 2
3 5 for 2^ ≤^ c^ ≤^3
0 otherwise.
p H c L
1 2 3 c
a. [4 points] Find the mean of the amount of coffee drunk in one day. Include units. Show all your work.
b. [3 points] Find the median of the amount of coffee drunk in one day. Include units. Show all your work.
University of Michigan Department of Mathematics Fall, 2013 Math 116 Exam 2 Problem 9 (coffee)
Math 116 / Final (December 14, 2017) do not write your name on this exam page 9
- [6 points] Suppose that a psychology experiment is designed so that every participant receives a score that is some (positive or negative) real number. The score that a participant receives is called his or her “experimental score”. The experiment is calibrated so that the probability density function of the distribution of experimental scores is standard normal, call it g(x). a. [3 points] Use a complete sentence to give a practical interpretation of the integral expression (^) ∫ 3
− 4
g(x) dx
in the context of this problem.
b. [3 points] Note that g(1) = √ 21 eπ ≈ 0 .24. Which of the following is the best practical interpretation of the mathematical statement g(1) ≈ 0 .24? Circle the one best option.
i. The fraction of the population having experimental score equal to 1 is approximately 24 percent.
ii. The fraction of the population having experimental score equal to 0.24 is approximately 1 percent.
iii. The fraction of the population having experimental score between 0.9 and 1.1 is approximately 4.8 percent.
iv. The fraction of the population having experimental score between 0.23 and 0.25 is approximately 20 percent.
v. The fraction of the population having experimental score above 1 is approximately 76 percent.
vi. The fraction of the population having experimental score above 0.24 is approximately 1 percent.
University of Michigan Department of Mathematics Fall, 2017 Math 116 Exam 3 Problem 9 (psych experiment)
