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Stats 07 - Lakehead UNIVERSITY Department of Mathematical...

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Unformatted text preview: Lakehead UNIVERSITY Department of Mathematical Sciences Examination Cover Sheet Name: Student No.: Mathematics Math 4030 FAIFB SUBJECT COURSE NO. SECTION Probability & Statistics Dr. W. Huagg COURSE TITLE INSTRUCTOR December 14, 2007 1:00 pm. 3 Hours EXAM DATE EXAM TIME DURATION TYPE OF EXAMINATION: I FINAL I:I MIDTERM I:I SPECIAL I:I DEFERRED Important: Be sure to fill in Section above. (FA or F8) Closed Book Exam 4 - 8% x 11 pages of Study Notes are allowed Non Programmable Calculator Allowed This examination question paper [:I MAY - MAY NOT be taken from the examination room. The Department of Mathematical Sciences regards cheating as a serious matter, one requiring strong counteraction. The Department requires its instructors to report all instances of cheating to the Chair, who shall seek a severe penalty consistent with the oflrense and the evidence, up to and including expulsion from the University, under the terms and procedures of the Lakehead University Code of Student Behaviour and Disciplinary Procedures, I985. STUDENTS PLEASE NOTE: This question paper has Pages. YQU MUST count the number of pages in this examination question paper BEFORE beginning to write, and report any discrepancy Immediater to a proctor. This is PAGE 1 of _14__ Math 4030 Final Exam Name: 1. (5 points.) Given the following data set 55,67,89,33,31,22, find mean, median, the 75th percentile, and the first quartile. Construct the boxplot. This is page 2 of 14 Math 4030 Final Exam Name: 2. (5 point_s.) Given P(A) = 0.3, P(B) = 0.5, and P(A 03) = 0.24, find P(A UB), P(Z n B), and P(A n F). Are A and B mutually exclusive? independent? 3. (3 points.) Xis a random variable with mean p. Prove that E[(X— M] = 0, where E['] is the mathematical expectation. This is page 3 of 14 Math 4030 Final Exam Name: 4. (4 points.) X is a random variable with mean 3 and variance 2, Y is a random variable with mean 1 and variance 2. Assume thatXand Yare independent. Find a. E[X—3Y]; b. Var(X—3Y); 6- E[XY]; d. C0v(X,Y). 5. (4 points.) Estimate the mean and the median from the following grouped frequency table: Grade Frequency r 40-49 . 5 '7 736169 TV 27 a 70-79 15 80-89 6 90—99 2 This is page 4 of 14 Math 4030 Final Exam Name: 6. (4 points) Randomly order five letters A, E, M, S. and X. What is the probability that the result is the word “EXAMS”? 7. (4 points) Amy commutes to work by two different routes A and B. If she comes home by route A. then she will be home no later than 6 PM with probability 0.8. If she comes home by route B. then she will be home no later than 6 PM with probability 0.7. In the past, the proportion of times that Amy chose route A is 0.4. a. What proportion of times is Amy home no later than 6 PM? b. If Amy is home after 6 PM today, what is the probability that she took route B? This is page 5 of 14 Math 4030 Final Exam Name: 8. (4 points) A basketball player makes 90% of her free throws. What is the probability she will miss for the first time on the fifth shot? 9. (4 points) Investing is a game of chance. Suppose there is a 30% chance that a risky stock investment will end up in a total loss of your investment. Because the rewards are so high, you decide to invest in four independent such risky stocks. a. Find the probability that at least one of your investments becomes a total loss. b. Find the probability that all four stocks become total loss. This is page 6 of 14 Math 4030 Final Exam Name: 10. (8 points) Discrete random variables X and Y has the joint distribution given by the following table. Find a. P(X=2,Y=1) b. P(X=Y) c. P(X>O,2.<_YS4) d. P(X+YZ4) This is page 7 of 14 Math 4030 Final Exam Name: 11. (4 points.) In a city. there are averager 7 fire incidents per year (52 weeks). Find the probability that there are at least 2 fire incidents in a randomly selected week. 12. (4 points.) Random variable X has the following density function Find the mean E[X]. This is page 8 of 14 Math 4030 Final Exam Name: 13. (6 points.) The burning time of an experimental rocket is a random variable having the normal distribution with [,1 = 4.76 seconds and o = 0.04 seconds. What is the probability that this kind of rocket will burn a. less than 4.66 seconds; b. more than 4. 80 seconds; c. anywhere from 4.70 to 4.82 seconds. This is page 9 of 14 Math 4030 Final Exam Name: 14. (8 points.) If two random variables X and Y have the joint density x—g’i, 0<x<l,0<y<2, flw) = _ 0, otherwzse. Find the probability ofP[0 < X < 1, 0 < Y < 1]. Find the probability of P[0 < X < 1, Y = 1]. Find the marginal densityf; (x). Find the conditional densityf2<y | x). 9-957.” This is page 10 of 14 Math 4030 Final Exam Name: 15. (5 points.) A NAPA Auto Parts supplier wants information about how long car owners plan to keep their cars. A random sample of 15 car owners results in 7 = 7.5 years and s = 3.1 years, respectively. Assume that the sample is drawn from a normally distributed population. Find a 95% confidence interval in estimating the population mean. 16. (5 points.) A medical researcher wishes to estimate the serum cholesterol level (in mg/1OO mL) of all women aged 18 and 24. There is strong evidence suggesting that o = 41.0 mg/1OO mL. If the researcher wants to be 95% confident of obtaining a sample mean that is off by no more than 4 units (mg/100 mL)from the true population mean, how large must the sample be? This is page 11 of 14 17. Math 4030 Final Exam Name: (5 points.) The force vital capacity (FVC) is often used by physicians to assess a person’s ability to move air in and out of their lungs. It is the maximum amount of air that can be exhaled after a deep breath. For adult males, the average FVC is 5.0 liters. A researcher wants to perform a hypothesis test to determine whether the average force vital capacity for women differs from this value. The mean force vital capacity for a random sample of 50 women was 4. 8 liters. Do the data provide sufficient evidence to conclude that the mean force vital capacity for women is less than the mean value for men 5.0 L? Perform the appropriate hypothesis test using a significance level of 0.05. Assume that a = 0.9 liters. This is page 12 of 14 Math 4030 Final Exam Name: 18. (5 points.) If 26 measurements of the boiling point of sulfur have a standard deviation of 0.83 degree Celsius, construct a 98% confidence interval for the true standard deviation of such measurements. Assume normality. This is page 13 of 14 Math 4030 Final Exam Name: 19. (8 points.) Following table gives the data of (X, Y) pairs. rx’l 367 y 1 i4 40 85 8] a. Find the value of the correlation coefficient. b. Find the equation of the best fitting line (the least square regression line). c. Use the best fitting line to predict the Y value when X = 4. d. When using the best fitting line to explain the data variation, what percentage of the variation can be explained by the regression line? This is page 14 of 14 ...
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