MATHSTAT318_HW3_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 3 HW3 is due by 11:59 PM on Thursday, February 1. 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 3, you should upload a single pdf file to Canvas containing your solutions. NOTE: 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 (10 Points) A random sample of 100 PSU first-year students was selected. The results indicated that 28 are taking a Physical Education (PE) course, 31 are taking Biology (B), and 42 are taking English (E). Some of the students were taking more than one of the courses. Specifically, 9 are taking PE and B, 10 are taking PE and E, 6 are B and E, and 4 are taking all three subjects. a. Complete the following Venn diagram. Note that each probability labeled on the plot should represent the probabilities of mutually exclusive regions on the plot. To illustrate the idea, two of the probabilities have been listed. Together, they suggest that ࠵?(࠵?࠵? ∩ ࠵?) = 0.09 . 0.04 0.05 PE B E

3 Problem 1 (Continued) Using the Venn diagram from the previous page, answer the following: b. What proportion of the students took none of the three subjects? c. How many students took physical education, but not biology or English? d. For a student taking biology, what is the probability the student is also taking English? e. Are the events taking biology and taking English independent?

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4 Problem 2 (10 Points) Diagnostic tests of medical conditions (e.g., COVID test, flu test, pregnancy test, etc.) have several results. Regardless of whether a patient has a condition, the test result can be positive or negative. Consider a random sample of 200 patients, some of whom have a medical condition and some of whom do not. Results of a new diagnostic test for the condition are shown. True condition Condition Present Condition Absent Diagnostic Test Test Result Positive 110 20 Test Result Negative 20 50 a. Find the probability that a randomly selected patient with the condition will produce a positive test result. b. Among the patients who produced a negative test results, what is the probability that the patient does not have the condition? c. Find the probability that a randomly selected patient will have the condition present and produce a positive test result. d. What is the probability that the test will be correct? e. Are the events having a negative diagnostic test and truly not having the condition independent? Justify your answer.

5 Problem 3 (5 Points) Consider two events ࠵? and ࠵? . Suppose that you know the following three pieces of information: (1) the probability that ࠵? occurs but ࠵? does not occur is 0.2, (2) the probability that ࠵? occurs but ࠵? does not occur is 0.1, and (3) the probability that neither occurs is 0.6. Find ࠵?(࠵? ∣ ࠵?) . Problem 4 (4 Points) The probability of surviving a transplant surgery is 0.55. If the person survives the surgery, the probability that the person’s body will reject the transplant within a month is 0.20. What is the probability that a person survives the surgery and does not reject the transplant within a month?

6 Problem 5 (5 Points) (Problem 1.3-6 in textbook) A researcher finds that, of 982 men who died in 2002, 221 died from some form of heart disease. Also, of the 982, 334 had at least one parent who had some heart disease. Of the latter 334 men, 111 died from some form of heart disease. A man is selected at random. Given that neither of his parents had some heart disease, find the probability that this man died of heart disease. Problem 6 (5 Points) A major manufacturer of computers introduced a new laptop model. As with most new products, there seems to be some problems. The reliability engineers have found that the probability the hard drive fails is 0.02, the probability the battery drive fails is 0.05, and the probability that the hard drive or battery drive fails is 0.06. Are the events that the battery drive fails and the hard drive fails independent? Justify your answer.

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7 Problem 7 (6 Points) A course you are taking is being taught by three different instructors: Instructor A, Instructor B, and Instructor C. Only one of the instructors writes the weekly quiz. The proportion of times that A, B, and C write the quizzes are 0.20, 0.30, and 0.50 respectively. The probability that you will pass if A, B, and C write the quiz are 0.70, 0.90, and 0.95 respectively. a. What is the probability that you pass this week’s quiz? b. Given that you pass this week’s quiz, what is the conditional probability that instructor B wrote the quiz?

8 Problem 8 (5 Points) Urn I contains two red chips and four white chips. Urn II contains three red and one white. A chip is drawn at random from urn I and transferred to urn II. Then a chip is drawn from urn II. What is the probability that the chip drawn from urn II is red?

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