STAT 200 Week 3 Homework Complete SolutionA+
STAT 200 Week 3 Homework Complete SolutionA+
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Date Created: 11/06/15
Week 3 Instructions Week Text Chapter Exercise page 3 1 Lane et al 5 7 217 3 2 Lane et al 5 25 221 3 3 Lane et al 5 27 222 3 4 Illowsky et al 3 86 207 3 5 Illowsky et al 3 98 209 3 6 Illowsky et al 3 112 210 3 7 Illowsky et al 3 124 214 3 8 Illowsky et al 4 72 268 3 9 Illowsky et al 4 80 270 3 10 Illowsky et al 4 88 271 7 You ip a coin three times The possible outcomes of this random experiment are S HHH HHT HTH THH HTT TTH THT TTT n S 8 a What is the probability of getting heads on only one of your ips The cases favorable to the event E1 of getting heads on only one of your ips are E1 HTT TTH THT 11 E1 3 P E1 11 E1 n S 38 0375 b What is the probability of getting heads on at least one ip The cases favorable to the event E2 of getting heads on only one of your ips are E2 HHH HHT HTH THH HTT TTH THT 11 E2 7 P E2 11 E2 n S 78 0875 25 You are to participate in an exam for which you had no chance to study and for that reason cannot do anything but guess for each question all questions being of the multiple choice type so the chance of guessing the correct answer for each question is 1d d being the number of options distractors per question so in case of a 4choice question your guess chance is 025 Your instructor offers you the opportunity to choose amongst the following exam formats I 6 questions of the 4choice type you pass when 5 or more answers are correct 11 5 questions of the 5choice type you pass when 4 or more answers are correct III 4 questions of the 10choice type you pass when 3 or more answers are correct Rank the three exam formats according to their attractiveness It should be clear that the format with the highest probability to pass is the most attractive format Which would you choose and why Rank Exam Format Probability 6x 0255 075 0256 1 11 00067 4x 020 080 0205 2 1 00046 4x 0103 090 010 3 111 00037 27 A refrigerator contains 6 apples 5 oranges 10 bananas 3 pears 7 peaches 11 plums and 2 mangos a Imagine you stick your hand in this refrigerator and pull out a piece of fruit at random What is the probability that you will pull out a pear Total number of fruits in the refrigerator n 6 5 10 3 7 11 2 44 Number of pears m 3 The required probability p m In 344 00682 b Imagine now that you put your hand in the refrigerator and pull out a piece of fruit You decide you do not want to eat that fruit so you put it back into the refrigerator and pull out another piece of fruit What is the probability that the first piece of fruit you pull out is a banana and the second piece you pull out is an apple Total number of fruits in the refrigerator n 6 5 10 3 7 11 2 44 Number of bananas m1 10 The required probability of selecting banana first p1 m1 In 1044 02273 Number of apples m2 6 The required probability of selecting apple second p2 m2 In 644 01364 Therefore by independence of two draws the required probability of selecting banana first and then apple second by replacing the banana p p1 X p2 02273 01364 00031 c What is the probability that you stick your hand in the refrigerator one time and pull out a mango or an orange Total number of fruits in the refrigerator n 6 5 10 3 7 11 2 44 Number of mangos m1 2 The required probability of selecting banana first p1 m1 In 244 00455 Number of oranges m2 5 The required probability of selecting apple second p2 m2 In 544 01136 Since pulling out either a mango or an orange at one time are mutually exclusive the required probability of pulling out either a mango or an orange p p1 p2 00455 01136 01591 86 R011 two fair dice Each die has six faces a List the sample space The sample space s is 111213141516212223242526 S 313233343 36414243444546 515253545556616263646566 nS36 b Let A be the event that either a three or four is rolled first followed by an even number Find P A The event A is A 323436424446 nA6 The required probability P A n A i e Let B be the event that the sum of the two rolls is at most seven Find P B The event B is 1112131415162122232425 BS 31323334414243515261 nB 21 The required probability PB nB g l 05833 nS 36 12 d In words explain what P A B represents Find P A B The event P A B indicates that the probability of obtaining either a three or four is rolled first followed by an even number when the sum of the two rolls is at most seven The event A is A 323436424446 gt nA 6 11121314151621222324 25 The event B is B 31323334414243515261 nB21 AnB 323442 gt nAnB 3 nArB i 1 Th 39 d b b39l39 9P AIB e require pro a 1 y quot3 21 7 01429 e Are A and B mutually exclusive events Explain your answer in one to three complete sentences including numerical justification m 1 i 200833 12 Th bb39l39 P A B eproa11ty m nS 36 Since PA m B 00833 7t 0 the events A and B are not mutually exclusive f Are A and B independent events Explain your answer in one to three complete sentences including numerical justification m 3 2i 200833 6 12 PAPB 00972 since PA n B at PA P B the events A and B are not independent The probability P A n B 98 At a college 72 of courses have final exams and 46 of courses require research papers Suppose that 32 of courses have a research paper and a final exam Let F be the event that a course has a final exam Let R be the event that a course requires a research paper a Find the probability that a course has a final exam or a research project PF orR PF PR PF and R 072 046 032 086 b Find the probability that a course has NEITHER of these two requirements P FC and RC P F or RC 1 PF 0139 R 1 086 0l4 112 Table 322 identifies a group of children by one of four hair colors and by type of hair Hair Type Brown Blond Black Red Totals Wavy 20 15 3 43 Straight 80 15 12 Totals 20 215 a Complete the table Hair Type Brown Blond Black Red Totals Wavy 20 5 15 3 43 Straight 80 15 65 12 172 Totals 100 20 80 15 215 b What is the probability that a randomly selected child Will have wavy hair The required probability 43215 020 c What is the probability that a randomly selected child Will have either brown or blond hair The required probability 100 20215 120215 05581 d What is the probability that a randomly selected child Will have wavy brown hair The required probability 20215 0093 e What is the probability that a randomly selected child Will have red hair given that he or she has straight hair The required probability 12172 00698 f If B is the event of a child having brown hair find the probability of the complement of B The required probability P BC 1 P B 1 100215 115215 05349 g In words What does the complement of B represent The complement of B represents the event of a child does not having brown hair 72 You buy a lottery ticket to a lottery that costs 10 per ticket There are only 100 tickets available to be sold in this lottery In this lottery there are one 500 prize two 100 prizes and four 25 prizes Find your expected gain or loss Expected gain or 1088 1 100 X 500 2100 X 100 4100 X 25 93100 X 10 13 Since the expected value is negative it can be treated as loss 124 Suppose that 10000 US licensed drivers are randomly selected The percent of licensed US drivers from a recent year that are female is 4860 Of the females 503 are age 19 and under 8136 are age 20 64 1361 are age 65 or over Of the licensed US male drivers 504 are age 19 and under 8143 are age 20 64 1353 are age 65 or over a How many would you expect to be male Expected number of males 10000 X 1 04860 5140 b Using the table or tree diagram construct a contingency table of gender versus age group Gender Age 19 and under 2064 65 or over Total Male 00259 04182 00700 05140 Female 00244 03954 00661 04860 Total 00503 08136 01361 10000 c Using the contingency table find the probability that out of the age 20 64 group a randomly selected driver is female The required probability 03954 80 Florida State University has 14 statistics classes scheduled for its Summer 2013 term One class has space available for 30 students eight classes have space for 60 students one class has space for 70 students and four classes have space for 100 students a What is the average class size assuming each class is filled to capacity The average class size assuming each class is lled to capacity 1gtlt308gtlt601gtlt704gtlt100 1814 3048070400 14 14 b Space is available for 980 students Suppose that each class is filled to capacity and select a statistics student at random Let the random variable X equal the size of the student s class Define the PDF for X Variable X Frequency f P X X fN 30 1 114 00714 60 8 814 05714 70 1 114 00714 100 4 414 02857 Total N 14 1 88 A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet Vietnamese New Year festivities this year Based on past years she knows that 18 of students attend Tet festivities We are interested in the number of students who will attend the festivities a In words define the random variable X The random variable X is the number of students who will attend the festivities b List the values that X may take on The values that X may take on are 0 1 2 3 4 5 6 7 8 9 10 11 and 12 c Give the distribution of X X Binomial n 12 p 18 018 d How many of the 12 students do we eXpect to attend the festivities E X up 12 X 018 216 e Find the probability that at most four students will attend Here X Binomialn 21219 2 018 gtPX x12 018quot1 018 quot x 0112 x The probability that at most four students will attend is PX g 4 i1018x1 01812x x0 f Find the probability that more than two students will attend Here X Binomia1n 12 p 018 gtPX x12 018x1 01812x x 0112 X The probability that more than two students will attend is 2 PX gt 2 1 PX g 2 1 Z12018x1 O1812x 1 O6298 x0 x
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