MATHSTAT318_HW2_Solutions

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Pennsylvania State University *

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318

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Statistics

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Feb 20, 2024

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1 MATH/STAT 318 – Homework 2 HW2 is due by 11:59 PM on Thursday, January 25. Instructions Some details: 1. When submitting homework in Canvas, multiple pages must be scanned and saved as a single PDF document. If you submit other file types, such as .jpeg, .png, .heic, etc., instead of a .pdf, will result in points being deducted or a score of 0. 2. Not knowing how to scan your homework assignment into a PDF document is not a valid excuse for late submissions. If you have never scanned your homework previously, try scanning a few sheets of paper just to check whether you are able to do it. 3. Failure to upload your homework assignment OR uploading the incorrect assignment will result in a grade of 0 for the assignment. It is each student's responsibility to check that the correct homework assignment is uploaded before the due date! 4. It is each student’s responsibility to make sure that the file you submit is complete, the problems are in the correct order, and the scan is readable. 5. The Instructor and TA must be able to easily read all uploaded homework. Messy and hard-to-read Canvas submissions will have points deducted or receive a score of 0. 6. While completing the homework problems, show all relevant work. Credit is given mostly for the method/work, not the answer. 7. You are encouraged to work with other students, but each student must write and submit their own assignment. Even if you have worked with others, you should not be submitting identical assignments. 8. For the policy of late assignments, see the syllabus. The rest of this document is formatted like an exam, roughly one page per problem. You might want to consider printing this document. I would suggest doing the problems on scratch paper first. Once you have found a satisfactory solution, rewrite the solution neatly on this document. When appropriate, please circle your final answer. At the conclusion of the Homework 2, you should upload a single pdf file to Canvas containing your solutions. NOTE 1: The first two problems on this assignment are similar several of the problems on HW1. These are problems are meant to be at a slightly higher level of difficulty than the problems on HW1. NOTE 2: Your work matters as much, if not more, than just writing the final answer. Rather than simply writing a final numerical answer, be sure to include sufficient work to communicate how you are arriving at the answer.

2 Problem 1 (Problem 1.1-1 in textbook.) Of a group of patients having injuries, 28% visit both a physical therapist and a chiropractor, while 8% visit neither. Suppose that the probability of visiting a physical therapist exceeds the probability of visiting a chiropractor by 16 percentage points. What is the probability that a randomly selected person from this group visits a physical therapist?

3 Problem 2 During a visit to a primary care physician’s office, the probability of having neither lab work nor referral to a specialist is 0.21. Of those coming to that office, the probability of having lab work is 0.41 and the probability of having a referral is 0.53. What is the probability of having both lab work and a referral? HINT: Although I would not provide this on an exam, you are still learning tricks you can use to solve these problems. When solving this question, I found the complement rule may be useful. 0.15

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4 Problem 3 Five awards (highest overall grade, best exam taker, best project, etc.) are to be presented to selected students from a class of 25 students. How many different outcomes are possible if: a. a student can receive more than one award? b. each student can receive at most one award? Problem 4 In the 2016 Olympics, 80 women competed in the 100-meter race as part of the track-and-field events. The top 3 contestants are awarded a gold, silver, and bronze medal, respectively. In how many ways could the medals be awarded? Problem 5 (Problem 1.2-8 in textbook). How many different varieties of pizza can be made if you have the following choices: size (small, medium, large), crust (thin, hand-tossed, or pan crust), and toppings (there are 12 toppings from which you may select from 0 to 12).

5 Problem 6 How many distinguishable ordered arrangements are there of the letters in “STATISTICS”? Problem 7 Suppose that in a class of 30 graduate students and 15 undergraduate students, 5 students are to be selected at random to work together for a special project. What is the probability that exactly 3 undergraduate students are chosen for the project? Problem 8 Fifteen people are available to use in a study to compare the effectiveness of three different exercise programs. Each of the exercise programs is to be used on five randomly selected people. In how many ways can the exercise programs be assigned to the people?

6 Problem 9 Suppose we wish to create a 4-character password using letters (uppercase A-Z and lowercase a-z), numbers (0-9), and symbols (eight possible symbols such as *, $, #, etc.). The only restriction is that the first character in the password must better a letter. Do not impose any other restrictions (such as the person must use a symbol or that characters cannot repeat). a. How many 4-character passwords are possible, assuming the first character must be a letter? b. If a hacker can guess and check one password per second, what is the probability that the hacker can guess the password correctly during a 24-hour period? You may assume the hacker knows that the first character must be a letter so that all of the hacker’s guesses are valid potential passwords.

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7 Problem 10 In a shipment of 100 car batteries, a small number of batteries will be defective while the remaining batteries will be functioning properly. The following procedure is used to inspect the shipment. Five of the batteries are randomly selected and tested. If one or more of the selected batteries is defective, the shipment is rejected. Otherwise, the shipment is accepted. Suppose that of the 100 batteries in the shipment, 10 of the batteries are defective. What is the probability that the shipment is accepted?

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