STAT 225 Summer 2010 Exam 2 Solution

Transcription

1 STAT 225 Summer 2010 Exam 2 Solution Your Name: Your Instructor: Your class time (circle one): 8:40 9:50 11:00 1:00 Show work for full credit unsupported work will NOT receive full credit All answers should be in decimal form: no fractions, permutation, combination, or exponential form. Round all answers to at least 2 decimal places. You are responsible for upholding the Honor Code of Purdue University. This includes protecting your work from other students. You are allowed 1 page 8.5 x11 handwritten cheat sheet and a calculator. Instructors will not interpret questions, tell you if you re on the right track, or check any answers for you. Only legitimate questions will be answered. You must turn in your Cheat Sheet at the end of the exam and may be asked to show your student ID. Turn off your cell phone before the exam begins. Question Points Possible Points Received Cheat Sheet 1 Total 100

2 1. You and your friends like to play soccer during the weekends. On a given round of penalty kicks, the probability to score a goal is 0.6. For each of the following scenarios, write the correct distribution and its parameter(s). If an approximation can be used, write both the exact and the approximate distributions and parameters to receive full credit. (3 points each) a. A penalty kick is chosen at random. Let X be a success if a goal is scored. X~Bernoulli(0.6) b. The probability to hit the upper bar during a penalty kick is 0.9%. In 1000 trials, let Y be the number of upper-bar hits. Exact: Y~BIN(1000, 0.9%) c. Let Z be the number of goals during 30 penalty kicks. Z~BIN(30, 0.6) d. You and your friends did 2000 penalty kicks during a month out of those 2000 were goals. If we choose 50 penalty kicks without replacement, let M be the number of missed kicks out of that sample. Exact: M~HG(N=2000, n=50, r=400) or HG(N=2000, n=50, p=0.2) Approx: M~BIN(50, 0.2) e. You scored an average of 0.25 goals per minute. Let S be the number of scored goals during 1 hour. S~POI(0.25*60=15) 1

3 2. While spending the evening at the Tippecanoe County Fair you stroll through the carnival area and discover a simple but entertaining game. In order to play you must pay $3.50 and then roll one fair 6-sided die with the payout being equal to the outcome. Let X denote the amount of money you win on any given play of the game. Find the following: (2 points each) a. E[X]=0 b. Var(X)=2.92 On your fifth play of the game you accidently roll the die too hard and it breaks into multiple pieces. In light of the event the game operator offers to switch to a new fair 6-sided die labeled with the numbers {2, 4, 6, 8, 10, 12}. He also informs you that in order to keep the game fair you will now have to pay $7 per game. Let Y denote the amount of money you win on any given play of the game with the new die. Find the following: c. E[Y]=0 d. Var(Y)=Var(2X)=4*Var(X)= The following information about 3 events, A, B and C is given, answer the questions below. (2 points each) P(A)=0.52 P(B)=0.38 P(C)=0.45 P(AB)=0.17 P(A C BC)=0.07 P(AC)=0.19 P(ABC)=0.09. a. P(BC A)=P(ABC)/P(A)=0.09/0.52=0.17 b. P(B C)=P(BC)/P(C)=[P(ABC)+ P(A C BC)]/P(C)=( )/0.45=0.36 c. P(C AC)=1 d. Are B and C independent? Why? No, since P(B)P(C) does not equal P(BC) 2

4 4. One day, a student, Guessy, walked into his Biology class only to discover that he completely forgot about the quiz which was to follow the lecture. The quiz usually has 5 multiple choice questions with 5 possible answers each and the student has to guess at random without preparation. Let X be the number of problems the student answered correctly. (3 points each) a. Name the distribution and parameters of X. X~BIN(5, 0.2) b. What is the probability that Guessy solved more than 3 questions correctly? P(X>3)=P(X=4)+P(X=5)= 5 C C = c. Given that Guessy knows the answer to 2 of the problems, what is the probability that he solved at least 4 of them correctly? P(X>=4 X>=2)=P(X>=4) P(X>=2) = [P(X=4)+P(X=5)] / [ P(X=2) + P(X=3) + P(X=4) + p(x=5)] = / = d. Guessy s instructor gave the quiz in two versions this time and let students choose one to work on. One version includes 5 multiple choice questions with 5 possible answers and the other has 10 true or false questions. Without any preparation, Guessy had to guess ALL the questions on either version but he is 70% likely to choose the one with true or false questions. What is the probability that Guessy got at least 80% of all problems correctly? Let Y={number of True and False question solved correctly} and Y ~BIN(10, 0.5) A={ Guessy got at least 80% of all problems correctly} P(A)=0.3*[P(X=4)+P(X=5)] + 0.7* [P(Y=8)+P(Y=9)+P(Y=10)] = e. If Guessy finally got 80% correct on the quiz, what is the probability that he chose the multiple choice version of the quiz? 0.3*[P(X=4)+P(X=5)] / P(A)=

5 5. A researcher is conducting a pilot study. He wants to select 6 rats from a group of 15 to conduct an experiment. Currently, he has 6 male and 9 female rats. (3 points each) a. What is the probability that he gets 3 rats of each gender. 6C 3 * 9 C 3 / 15 C 6 = b. The results of the pilot study are very good and the researcher finally received a grant which allows him to conduct a large scale experiment. In the new experiment, he will have to select 150 rats from a total of 5000 males and 7500 females. Let MR represents the number of male rats selected into the experiment, find the approximate probability that MR=75. MR~BIN(150, 5000/( )=0.4) P(MR=75)= 150 C * = The U.S. Postal Service uses machines to help sort outgoing mail according to the zip code of the addressee. Suppose the machine at the main Lafayette, IN post office correctly sorts letters independently but makes a mistake on any given sort with probability.001. On a certain day the machine sorts 4,500 parcels. Let P be the number of incorrectly sorted parcels. (3 points each) a. Find the exact probability the machine will correctly sort all 5,500 parcels. P~BIN(4500, 0.001) P(P=0)=0.999^4500= b. State the name and parameter(s) of a good approximating distribution to P. P~POI(4500*0.001=4.5) c. Find the approximate probability the machine will make at least 2 mistakes. P(P>=2)=1-P(P<2)=1-P(P=0)-P(P=1)=

6 7. Once a month Gertrud likes to splurge on a new pair of shoes. However, before making such a decision she always sizes up her boyfriend Jamal s attitude at the time to determine what repercussions she may experience later upon making the purchase. A look of disapproval occurs 70% of the time and in such an instance Gertrud still decides to buy a new pair of shoes 10% of the time. However, if no look is given Gertrud will buy a new pair of shoes 80% of the time. a. Draw a tree diagram of the situation. (4 points) ANSWER: Gertrud 0.7 Look 0.3 NO look 0.1 Purchase NO purchase Purchase NO purchase 0.06 b. What are the chances Gertrud purchases a new pair of shoes in any given month? (3 points) Let A={Gertrud purchases a new pair of shoes in any given month} and B={Jamal gave her a look of disapproval} P(A)=P(A B)P(B)+P(A B C )P(B C )=0.7* *0.8=0.31 c. Knowing Gertrud purchased a pair of shoes this month what are the chances Jamal gave her a look of disapproval? (3 points) P(B A)= P(A B)P(B) / P(A) = d. Knowing Gertrud purchased a pair of shoes on twelve different months, what are the chances Jamal gave her a look of disapproval in a total of seven of those months? (3 points) Let X be the number of months when Jamal gave the look knowing Gertrud made the purchase. 7 X ~ Binn 12, p PLook Purchase ! *11*10*9*8*7 * 24 7 P X 7 p 1 p ! 12 7! !*31 5

7 8. One evening during a meteor shower Sarah walks to the top of the hill behind her house to do some stargazing. She sees shooting stars at the rate of 5 per hour and gazes for 2 hours. Let S be the number of stars Sarah sees during the 2 hour period. a. Name the distribution of S and its parameters. (3 points) S~POI(5*2=10) b. Find the probability Sarah sees at least 3 shooting stars during the first hour and at least 2 shooting stars during the second hour. (4 points) P(S 1 >=3)*P(S 2 >=2) =[1-P(S1<3)]*[1-P(S2<2)]= [1-P(S 1 =0)-P(S 1 =1)-P(S 1 =2)]*[1-P(S 1 =0)-P(S 1 =1)] = [1-e -5 *(5 0 /0!+5 1 /1!+5 2 /2!)]*[ 1-e -5 *(5 0 /0!+5 1 /1!)] = * = c. Find the probability she sees at least 3 stars in at least one of the two 1-hour periods.(4 points) 1-P(S 1 <3)P(S 2 <3)=1-( ) 2 = a. Given that Sarah has already seen 9 stars during the 2 hour period, what is the probability that she will see less than 12 stars by the end of this period? Please only show the steps necessary for the answer, no numeric answer is needed. (4 points) P(S<12 S >=9) = P(9<= S < 12) / P( S> =9 ) = [ P(S=9) + P(S=10) + P(S=11) ] / [1 - P(S=0) - P(S=1) - P(S=2) - P(S=3) - P(S=4) - P(S=6) - P(S=7) - P(S=8) ] = e -10 ( 10 9 /9! /10! /11!) / [ 1- e -10 (10 0 /0! /1! /2! /3! /4! /5! /6! /7! /8! )] 6

8 9. A tricked B into playing an unfair game for money. A put 20 balls (12 white, 4 red and 4 green) into a bag and asked B to pick 3 balls from the bag without replacement. B wins if he gets more colored balls than the white balls or three colored balls. a. What is the probability that B wins a game? (3 points) Let C={number of colored balls B picks from the bag}, B wins if C=2 or 3 P(B wins)=p(c=2)+p(c=3)= 8 C 2 * 12 C 1 / 20 C C 3 * 12 C 0 / 20 C 3 = b. A and B played 10 games. If we use k to represent the number of games won by B, which random variable can be used to describe k? Also find the parameter(s) of this random variable. (3 points) K~ BIN (10, ) c. A and B both started the game with $100 and the winner of each game got $10 from the loser. What is the probability that A ended up with $160 after 10 games? (4 points) A needs to win 8 games to end up with $160, then B wins 2 games. P(k=2)= 10 C 2 * ( ) 8 =