STAT 230 First Midterm Exam Solution

STAT 230 First Midterm Exam University of British Columbia, Okanagan You have 75 minutes to complete the test. Please round all answers and probabilities to 3 decimal places. Question | Point Score 1 7 2 3 3 2 4 3 5 4 s | 3 7 5 8 8 9 5 Total: 35

Consider the following contingency table summarizing the fate of 1207 passengers on the fatal maiden voyage of the ocean liner Titanic according to economic status (i.e. class). For instance, of the 765 passengers that did not survive, 122 of them were from first class. 1. Survived Class No Yes Total I 122 197 319 o 167 94 261 3% 476 151 627 Total 765 442 1207 (a) (1 point) Are A and D mutually exclusive? N = \ l\(,)‘ /i\ ““~\v'\,‘\~\x\t D A coun e \\\ \\\ \ GO Let A be the event that the passenger was traveling first class Let B be the event that the passenger was traveling second class Let C be the event that the passenger was traveling third class Let D be the event that the passenger survived SN \VEL (b) (1 point) In symbols, what is the probability that a randomly selected third-class passenger survived? ¥ (D\C) (c) (1 point) What does P(A N D) present in words? The probability that a randomly selected passenger... A. was not traveling first class given that they survived B. survived given that they were not traveling first class C. was traveling in second or third class or survived @) was traveling in second or third class and survived E. None of the above (d) (4 points) What is the probability that a randomly sampled passenger survived or was in first class? (’ - — f b

5. (4 points) The probability mass function for X = the number of major defects on a randomly selected appliance of a certain type is p(x) = 0.75(0.25)*, forx=0,1,2,3,.. What is the probability of 1 or fewer major defects on such an appliance? '\i;g'.)‘) £ 0. 1o (8-43) » Pl <\\'/"\ )\ ) P (0) « \\" Ln) I O A - AT D @ @ @ : (\'};fi'\ "\ f(‘*/ @ 6. (3 points) 1% of a population has a certain genetic defect. 90% of tests for the genetic defect detect the defect (true positives). 9.6% of the tests are false positives. If a person gets a positive test result, what is the probability they actually have the genetic defect? ~ \\\\ { >\< ‘\(i\ \)\ \\)\\\‘\“‘ \\&\\{ (1o Lol ‘\\ \ ,\; ¥ hwve \ - —~ \ ® ® O 7. An animal grooming service schedules 25 cats to bathed in day. Assuming the probability that a cat scratches the pet stylist is 45%, and cats are independent of one another, answer the following questions. (a) (1 point) What is the probability that exactly 14 of these cats will be scratched by the pet stylist? o ‘ N % aoe [ 23 s 2514 y -~ o\ | ~ \ [ A€ (\_ 0.1 P (X i (N} =1 . gy O ) L S)

S(a-H o, b M P L2 (= \) 8. Suppose the random variable X follows a normal distribution with a mean of u =120 and avariance 02 = 400. -~ G = 20 (a) (3 points) What is the probability P(106 < X < 142). | o Q- PL06Z ) - P (18e=l20 (7 f 142-00) Pr g {20 4 P (Z(\~\> - (‘—O?{Z> - 0.9643% _ p2y20 =0.6223 = 0.622 () (b) (3 points) Find the value of ¢ such that P(X < q) = From Z telde @ P (Z SC),(A,\ = U243 @ Y- = (-062%)-20 y\20 = Y 3 =106.4Q0) 9. A random variable has a cumulative distribution function given by 0, forx <1 F(x) =<(x—-1)3? forl<x<2 1, forx =2 (a) (1 point) What is P(X < 1.4) PxLnu)y = F (rd)= 0:\6 (b) (2 points) Whatis P(1.4 < X < 1.55) PO '\4)\ “?? - F(s3)-F(\2)=0.3025 ~0-\6 = 04U2S c 0143 (¢) (2 points) Write down the pdf and calculate its value when x = 1.4. / Fooy=You = 2% -2 - A=k s ©O8