BIOSTATS 540 Fall 2018 Exam 2 Page 1 of 12

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1 BIOSTATS 540 Fall 2018 Exam 2 Page 1 of 12 BIOSTATS Introductory Biostatistics Fall 2018 Examination 2 Units 4, 5, 6, & 7 Probabilities in Epidemiology, Sampling, Binomial, Normal Due: Wednesday November 14, 2018 Last Date for Submission with Credit: Wednesday November 21, 2018 (-20 points) Rules: This is an open book take-home exam. You are welcome to use any reference materials you wish. You are welcome to use the computer as you wish, too. However, you MUST work this exam by yourself and you may not consult with anyone. Instructions and Checklist 1. Start each problem on a new page. 2. Write your name on every page 3. Make a copy of your exam for safekeeping (sometimes a mailed exam is lost!) 4. Submit a completed signature page (See next page). How to submit your exam: Worcester in-class Section: 1. Please a pdf of your completed exam to class on Wednesday November 14, 2018 being sure that you have chosen a filename that includes your name! (note exam2.pdf is not a good choice); OR 2. Mail your completed exam to me with post-mark November 14, 2018 to my address below. Blackboard Learn Online Section: 1. Upload your completed exam to the ASSIGNMENT tab no later than 11:59 pm on Wednesday November 14, This must be a single pdf ; OR 2. Mail your completed exam to me with post-mark November 14, 2018 to my address below. Address and Telephone Number for Mailing Carol Bigelow Biostatistics & Epidemiology/402 Arnold House University of Massachusetts/Amherst 715 North Pleasant Street Amherst, MA Tel

2 BIOSTATS 540 Fall 2018 Exam 2 Page 2 of 12 BIOSTATS Introductory Biostatistics Fall 2018 Examination 2 Units 4, 5, 6, & 7 Probabilities in Epidemiology, Sampling, Binomial, Normal Due: Wednesday November 14, 2018 Last Date for Submission with Credit: Wednesday November 21, 2018 (-20 points) Signature This is to confirm that in completing this exam, I worked independently and did not consult with anyone. Signature: Printed Name: Date:

3 BIOSTATS 540 Fall 2018 Exam 2 Page 3 of (10 points total) An article about using a diagnostic test (helical computed tomography) to screen adjust smokers for lung cancer warned that a negative test may cause harm by providing smokers with false reassurance, and a false positive test results in unnecessary operation opening the smoker s chest. 1a. (5 points) Explain what false negatives mean in the context of this diagnostic test. 1b. (5 points) Explain what false positives mean in the context of this diagnostic test.

4 BIOSTATS 540 Fall 2018 Exam 2 Page 4 of (10 points total) For combined ELISA-Western blot blood test for HIV positive status, the sensitivity is about and the specificity is about a. (3 points) Consider a high-risk group in which 10% are truly HIV positive. Construct a tree diagram to summarize this screening test. 2b. (3 points) A person from this high-risk group has a positive test result. What is the probability that this person is truly HIV positive? 2c. (4 points) Explain why a positive test result is more likely to be in error when the prevalence is lower. Hint Play around with your tree diagram.

5 BIOSTATS 540 Fall 2018 Exam 2 Page 5 of (10 points total) The following pertains to questions #3a, #3b, and #3c. After a recent flight, I received an from the airlines asking me to fill out a survey regarding my satisfaction with my travel experience. Suppose the airline analyzes the data from all responses to such s. 3a. (2 points) What is the sample? 3b. (2 points) What is the population that is of interest to the airline? 3c. (2 points) In at most 1-2 sentences, do you expect the survey results to accurately portray customer satisfaction? Why or why not? The following pertains to question #3d. Today s poll for a certain magazine asked, Have you ever hired a personal trainer? Visitors to the website had the option to select yes or no. Suppose that 27% of respondents answered yes. 3d. (2 points) In at most 1-2 sentences, can we infer that 27% of people have hired a personal trainer? Why or why not? The following pertains to question #3e. Right before the 2008 presidential election, the Gallup Poll randomly sampled and collected data on n=2847 Americans. Of those sampled, 52% supported Barack Obama and 42% supported John McCain. 3e. (2 points) In at most 1-2 sentences, can we generalize these results to the entire population of 129 million voters in order to estimate the popular vote in the election? Why or why not?

6 BIOSTATS 540 Fall 2018 Exam 2 Page 6 of (10 points total) For each of the following settings, determine whether the setting describes a binomial random variable. If yes, give the values of n and π. If the setting is not described by a binomial random variable, explain why not. Question (points) 4a (2 points) Count the number of sixes in 10 dice rolls 4b (2 points) Roll a die until you get 5 sixes and count the number of rolls required 4c (2 points) Sample 50 students who have taken BIOSTAT 540 and record the final grade for each 4d (2 points) Suppose 30% of students at UMass take an introductory statistics course. Randomly sample 75 students and count the number who have taken an introductory statistics course 4e (2 points) Worldwide the proportion of babies who are born male is about We randomly sample 100 newborns and count the number of males. Binomial or not? Yes or No If binomial, n = π = If not binomial, why not

7 BIOSTATS 540 Fall 2018 Exam 2 Page 7 of (10 points total) Let X = the number of living grandparents that a randomly selected adult has. Suppose further that, according to a recent survey, its probability distribution is approximately P(0)=.71, P(1)=.15, P(2)=.09, P(3)=.03, and P(4)=.02. 5a. (3 points) In 1 sentence, does this setting refer to a discrete or a continuous random variable? Explain. 5b. (3 points) Show that the probabilities satisfy the two conditions for a well-defined probability distribution. 5c. (4 points) Calculate the mean (µ) of this probability distribution.

8 BIOSTATS 540 Fall 2018 Exam 2 Page 8 of (10 points total) Suppose the state of Massachusetts has a lottery game in which you pick a 3-digit number between 000 and 999. Note this means there are actually 1000 such 3-digit numbers. Suppose that a player wins $275 if his/her 3-digit number is the number for the day and nothing otherwise. Suppose further that it costs $1 to play. 6a. (3 points) Construct the probability distribution of the random variable X = winnings of a single play. 6b. (3 points) Calculate the mean of this probability distribution. 6c. (4 points) Based on the mean you obtained in #6b and the $1 cost to play, how much can you expect to lose each time you play?

9 BIOSTATS 540 Fall 2018 Exam 2 Page 9 of (10 points total) Abraham Lincoln once said you can t please all the people all the time. Suppose that you can please each individual 9 times out of 10 and that there are 8 people you want to please, all independent of one another. 7a. (2 points) What is the probability that you will please all of them? 7b. (2 points) What is the probability that you will please at least 6 of them? 7c. (2 points) What is the probability that you will please fewer than 4 of them? 7d. (2 points) What is the expected number of people you will please? 7e. (2 points) What is the probability that you will please exactly the expected number of people you will please?

10 BIOSTATS 540 Fall 2018 Exam 2 Page 10 of (10 points total) Suppose it is known that the length of time (minutes) needed to service a car at a particular gas station is distributed Normal with µ = 4.5 minutes and standard deviation σ = 1.1 minutes. 8a. (2 points) What is the probability that a randomly selected car at this gas station will require more than 6 minutes of service or less than 5 minutes of service? 8b. (2 points) What is the probability that a randomly selected car at this gas station will require between 3.5 and 5.6 minutes of service? 8c. (2 points) What is the probability that a randomly selected car at this gas station will require at most 3.5 minutes of service? 8d. (2 points) What is the servicing time such that only 5% of all cars at this gas station require more than this amount of time? 8e. (2 points) What are the two servicing times, symmetric about the mean, within which 50% of all the cars at this gas station require this range of servicing time?

11 BIOSTATS 540 Fall 2018 Exam 2 Page 11 of (10 points total) 9a. (5 points) Suppose it is known that the serum cholesterol levels of Wisconsin children are distributed normal with µ = 175 mg/dl and standard deviation σ = 30 mg/dl. Suppose further that a normal value is defined as any value between two standard deviations of the mean. What are the values of these normal limits? 9b. (5 points) Suppose it is known that IQ is distributed normal with µ = 100 and standard deviation σ = 15. Now consider the average IQ of classes of 25 students. What are the population mean and variance of these class averages?

12 BIOSTATS 540 Fall 2018 Exam 2 Page 12 of (10 points total) The following pertains to questions #10a and #10b. Suppose that the distribution of exam grades for all sections of an introductory biostatistics course is distributed normal with µ = 75 and standard deviation σ = a. (3 points) If the lowest 5% will be encouraged to attend additional tutoring sessions, what grade is the cut-off for this suggestion?? 10b. (3 points) If the highest 10% will be given a grade of A, what is the cut-off score for an A? The following pertains to questions #10c and #10d. Next suppose that in one administration of this exam, the fire alarm went off with 10 minutes to go before the class period ended. Finally, suppose that in grading these exams, the instructor found that mean grade was 62 and the standard deviation of 18. Thus, consider a new fire-alarm related normal distribution with µ = 62 and standard deviation σ = 18. To be fair, the instructor decides to curve the firealarm class exam scores to match the normal distribution with µ = 75 and standard deviation σ = c. (2 points) For a student in the fire-alarm class who scores a 47, what is his curved score? That is, what is the conversion of their score of 47 to its equivalent score with respect to the normal distribution for the rest of the university? 10d. (2 points) How about a fire-alarm class student who scores a 90? That is, what is the conversion of their score of 90 to its equivalent score with respect to the normal distribution for the rest of the university?