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MATH 131 8/11 TEST 2 REVIEW SHEET Links to pencast solutions to selected problems are boxed next to these problems. Written answers to all problems can be found at the end of the review sheet. Important terms and phrases: To locate these terms in the book, please use the index at the back of the book. Sample Space Fundamental Counting Principle Classical Probability Permutation and Combination Empirical Probability Discrete/ Continuous Random Variable Law of Large Numbers Discrete Probability Distribution Empirical Rule Binomial Experiments Chebychev’s Theorem Normal Distribution Conditional Probability Standard Normal Distribution Independent/ Dependent Events Standard Error of the mean Mutually Exclusive Events Central Limit Theorem Multiplication Rule Normal Approximation to Binomial Addition Rule 1. A department store sells sport shirts in three sizes (small, medium and large) and two sleeve lengths (Full and Half). Small Full Sleeve 25 Half Sleeve 28 Total 53 a) b) c) d) e) f) g) Medium 39 52 91 Large 56 30 86 Total 120 110 230 What is the probability that a shirt chosen from this group is a Large? #1 SOLUTION What is the probability that a shirt is Medium and has Full Sleeves? What is the probability that a shirt is Small or has Half Sleeves? What is the probability that a shirt is Large given that it has Half Sleeves? What is the probability that a shirt is Half Sleeves given that it is a size Medium? Are the events “Small size” and “Full Sleeves” mutually exclusive? EXPLAIN. Are the events “Large size” and “Half Sleeves” independent? EXPLAIN. (Hint: You could use 2 of your prior answers.) 2. Of the patients examined at a local clinic, 25% had high blood pressure, 50% were overweight and 15% had both conditions. a) Are the two events "high blood pressure" and "overweight" mutually #2 SOLUTION exclusive? EXPLAIN. b) Are the two events "high blood pressure" and "overweight" independent? EXPLAIN. c) If one of the patients is selected at random, what is the probability that the patient has high blood pressure or is overweight? 3. Rita is playing 3 tennis matches. a) List the sample with the possible win and loss sequences that she can experience for this set of three matches. b) Is this an equally likely sample space? EXPLAIN. 1 #3 SOLUTION MATH 131 8/11 TEST 2 REVIEW SHEET 4. a) Explain clearly the difference between a combination and a permutation. b) New Jersey has 14 four year colleges. Diana wants to apply to 3 of them. In how many different ways can she select #4 SOLUTION three colleges to apply to? 5. It was a well-known rule of the sea that if a ship is sinking, then lifeboats are filled first with women and children. Was this rule followed on Monday, April 5, 1912 when the Titanic sank? Survived Died Total a) b) c) d) e) f) g) h) i) Men 332 1360 1692 Women 318 104 422 Boys 29 35 64 Girls 27 18 45 Total 706 1517 2223 If you select a passenger at random from this group, find the probability that the passenger was a Woman. Survivor. Survivor, given that the passenger was a Woman. Survivor, given that the passenger was a Man. Boy or a Survivor. Man and someone who Died. Do you think Women had better survival rates on the Titanic than Men? (Hint: use your answer in part c and d) Are the events “Men” and “Survivor” independent events? Show the probabilities used to draw your conclusion. Are the events “Women” and “Girls” mutually exclusive events? How do you know? 6. Brookdale offers different summer camp programs for children. Next year they expect 75% of the campers to register for sports camp, 25% to register for arts camp, and 5% to register for both. # 6 SOLUTION a) Are the two events “arts camp” and “sports camp” mutually exclusive? EXPLAIN. b) Find the probability that a camper selected at random goes to arts camp or sports camp. 7. A citizens action committee in NJ is comprised of six Democrats, eight Republicans, and three Conservatives. a) If one member is randomly selected, find the probability of getting a Conservative. b) If one member is selected at random, find the probability of #7 SOLUTION getting a Republican or a Conservative. c) If two different people are selected at random from this committee, find the probability that they are both Democrats. 8. Ten equally qualified people apply for a job and three will be interviewed by the Manager. How many different groups of three people can be selected for the interview? 2 MATH 131 8/11 TEST 2 REVIEW SHEET 9. A state lottery involves the random selection of six numbers between 1 and 44. To win the lottery, you must select the six winning numbers but not necessarily in any specific order. If you select one set of six numbers, what is the probability it will be the winning set of numbers? 10. How many ways can seven people be seated around a table? 11. According to the Census Bureau’s Current Population Report, 1/3 of American children (under 18 years old) are not living with both parents. If 5 American children are selected at random, find the probability that exactly two of them are not living with both parents. (Use the appropriate formula to find the probability. Leave answer in fraction form.) 12. Seventy-five percent of households in the United States subscribe to cable TV. If you randomly select 5 households and ask each if they subscribe to cable TV, find the probability that a) at least four houses have cable. #12 SOLUTION b) at most 2 houses have cable. c) all of the houses have cable. 13. The rates of on-time flights for commercial jets are continuously tracked by the U. S. Department of Transportation. Recently, Southwest Air had the best rate with 80% of its flights arriving on time. If 15 Southwest flights are randomly observed, find the probability that a) none are on time b) all are on time #13 SOLUTION c) at least 10 are on time d) no more than 4 are on time 14. A publisher introduces a new weekly magazine for teenagers. The company’s marketers estimate that the sales x (in thousands) will be approximated by the distribution on the right. a) Check whether the given distribution is a probability distribution. b) If it is a probability distribution, find the mean for this distribution. c) If it is a probability distribution, find the standard deviation. x 10 15 20 25 P(x) .200 .300 .150 .350 #14 SOLUTION 15. A package of frozen vegetables has a label that says “contents 32 oz.” However, the company that produces these packages knows that the weights are normally distributed with mean 32 oz. and standard deviation 2 oz. If a package is chosen at random, what is the probability that it will weigh? a) more than 35 oz.? #15 SOLUTION b) less than 30 oz.? c) between 30 and 35 oz.? 16. Suppose that the test scores for a college entrance exam are normally distributed with a mean of 450 and a standard deviation of 100. 3 MATH 131 8/11 TEST 2 REVIEW SHEET a) What percentage of those taking the exam score between 350 and 450? b) A student scoring above 400 is automatically admitted. What percent score above 400? c) The upper 5% receive scholarships. What score must a student make on the exam to receive a scholarship? 17. The heights of adult males are normally distributed with a mean height of 70 inches and a standard deviation of 2.6 inches. How high should the doorway be #17 SOLUTION constructed so that 90% of the men can pass through it without having to bend? 18. An advertising agency was hired to introduce a new product. It is claimed that after its campaign 40% of all consumers were familiar with the product. a) Is this a binomial experiment? EXPLAIN. #18 SOLUTION b) Find that the probability that out of 500 consumers, 215 or more of them will become familiar with the product because of the campaign. 19. Thirty-three percent of adults consider public schools good at preparing students for college. You randomly selected 40 adults and ask them if they think public schools are good at preparing students for college. Decide whether you can use the normal distribution to approximate the binomial distribution. Find the probability that between 15 to 17 inclusive adults will say yes. 20. Currently, most of U.S. companies test new employees for drugs, and 3.8% of those prospective employees test positive (data based on American #20 SOLUTION Management Association). The Alpha Electronics company tests 150 prospective employees. Estimate the probability of at most 10 of them testing positive for drugs. 21. The heights of college age men are known to be normally distributed with a mean of 68 in. and a standard deviation of 3 in. a) What is the probability that the mean height of a random sample of nine college men is between 67 and 69 inches? b) What is the probability that the mean height of a random sample of 100 college men is between 67 and 69 inches? c) Compare your answers to parts a and b. Was the probability in part b much higher? Why would you expect this? d) Name the theorem you used to solve this problem. e) List the criteria to apply the theorem. 22. According to the book "Standing up to the SAT", the average SAT score for Native American males is 852. Assume that the standard deviation is 120 points. a) Do you think that the distribution of the SAT scores is normal for #22 SOLUTION any group? Sketch what the distribution would look like if it was normal. Use this in your explanation. 4 MATH 131 TEST 2 REVIEW SHEET 8/11 b) If a random sample of 50 Native Americans is surveyed, what is the probability that their mean SAT score is more than 900? c) Could you have answered the question if the sample size was 25 instead of 50? EXPLAIN your answer, mentioning any appropriate theorem. 23. The amount of coffee dispensed by a vending machine is normally distributed with a mean of 7.8 oz. and a standard deviation of 0.1 oz. The size cup used in this machine is an 8 oz. cup. a) What percentage of cups will overflow with coffee? b) When coffee overflows a cup, the company loses money. #23 SOLUTION What could the company do to further limit the amount of overflow? 24. The monthly utility bills in a certain city are normally distributed, with a mean of $100 and a standard deviation of $12. A utility bill is randomly selected. a) Find the probability that the utility bill is less than $80. #24 SOLUTION b) Find the probability that the utility bill is between $80 and $115. c) Find the probability that the utility bill is more than $115. 25. a) What is the probability of an event that is certain to occur? b) What is the probability of an impossible event? c) On a true/false problem, what is the probability of answering a question correctly if you make a random guess? #25 SOLUTION d) On a multiple-choice test with five possible answers for each question, what is the probability of answering a question correctly if you make a random guess? 26. In a research poll, respondents were asked how a fruitcake should be used. Use Doorstop Birdfeed Landfill Gift Other Number of respondents, f 50 25 35 10 5 #26 SOLUTION a) If a respondent is picked at random, what is the probability that they would use the fruitcake as a doorstop? b) If a respondent is picked at random, what is the probability that they would not use it as birdfeed? 27. Explain, clearly and in your own words, the Law of Large Numbers. 5 #27 SOLUTION MATH 131 TEST 2 REVIEW SHEET 8/11 2 3 MATH 131 11. 5 C2 (1/3) (2/3) = 80/243 Test 2 Review Sheet/Answer Key 12. a) P(4) + P(5) = .633 1. a) b) c) d) e) f) g) 2. a) b) c) 3. a) b) P(0) = 0.000 P(15) = 0.0352 P(10) + P(11) +...+ P(15) = 0.939 P(0) + P(1)...+ P(4) = 0.000 14. a) Yes it is. Each probability is between 0 and 1 sum of the probabilities equals 1. b) 18.25 c) 5.7608 15. a) z 35 32 1.5 2 P( x 35) P( z 1.5) 1 0.9332 0.0668 WWW, WWL, WLW, WLL LWW, LWL, LLW, LLL Maybe. Is she a 50/50 tennis player? Is she as likely to win as to lose? See Section 3.4. 5. a) b) c) d) e) f) g) 422/2223 = 0.1898 706/2223 = 0.3176 318/422 = 0.7536 332/1692 = 0.1962 64/2223 + 706/2223 – 29/2223 = 0.3333 1360/2223 = 0.6118 Yes. 75.4% of the women survived as compared to only 19.6% of the men. Not independent. P(Man) = 1692/2223 = 0.7611 and does not equal P(Man given Survivor) = 332/706 = 0.4702 Yes, because no one is both a Woman and a Girl. P(Woman and Girl) = 0 i) 13. a) b) c) d) No, 15% are both. No, P(H and O) = 0.15 0.125 = P(H) P(O). It has been shown that patients who are overweight are more likely to have high blood pressure. 0.60 4. a) b) h) b) P(0) + P(1) + P(2) = .104 c) P(5) = .237 86/230 = 0.3739 39/230 = 0.1696 53/230 + 110/230 – 28/230 = .5870 30/110 = 0.2727 52/91=0.5714 No, 25 shirts are both. Comparing answers (a) and (d), you can see P(Large) = 0.3739 0.2727 P(Large, given Half Sleeves). Not independent. 14 C3 364 6. a) b) No, 5% go for both. 95% 7. a) b) c) 3/17=0.1765 11/17=0.6471 (6/17)(5/16) =0.1103 8. 10 9. 10. 1/7,059,052= 0.000001416 7 = 5040 = 7 P7 C3 120 6 b) z c) z 30 32 2 1 P( x 30) P( z 0.1587 35 32 30 32 1.5 z 1 2 2 P(30 x 35) P( 1 z 1.5) 0.9332 0.1587 0.7745 1) MATH 131 8/11 TEST 2 REVIEW SHEET c) x 450 350 450 16. a) z 1 100 P(350 x 450) P( 1 z 0) 0.5 0.1587 0.3413 34.13% b) z 1.645 100 614.5 17. x x x z 70 (2.6)(1.28) 73.3 inches 18. a) Yes, it is a binomial experiment. See Section 4.2. b) np 500(0.4) 200 5 nq 500(0.6) 300 5 Use the normal distribution to approximate the binomial distribution. np 200 400 450 0.5 100 P( x 400) 1 P( z 0.5) 1 0.3085 0.6915 69.15% z 500(0.4)(0.6) P( x 215) P( x 214.5) P( z 1.32) 1 0.9066 0.0934 19. np 40(0.33) 13.2 5 nq 40(0.67) 26.8 5 We can use the normal distribution to approximate the binomial distribution. np 13.2 Since npq 7 40(0.33)(0.67) MATH 131 8/11 TEST 2 REVIEW SHEET P(15 y 17) P(14.5 x 17.5) P(0.44 z 1.45) 0.9265 0.67 0.2565 b) z z 69 68 3/ 100 67 68 3.33 3.33 3 / 100 20. Let x = prospective employees test positive. n = 150, p(x) = 3.8% = 0.038 np 150(0.038) 5.7 5 nq 150(0.962) 144.3 5 P(67 x 69) P( 3.33 z 3.33) 0.9996 0.0004 0.9992 c) Yes, explain. d) Central Limit Theorem e) See Section 5.4. Use normal distribution to approximate the binomial distribution. np 5.7 22. a) b) npq (150)(0.038)(0.962) P( x 10) P( x 10.5) P( z 2.05) 0.9798 10.5 5.7 z 2.05 150 0.038 0.962 21. a) Use CLT because is finding the mean height P(67 x 69) P( 1 z 1) 68 x 0.8413 0.1587 0.6826 z 69 68 3/ 9 1 z 67 68 1 3/ 9 8 Explain n = 50 >30 use CLT 852 x 120 x n 50 x 900 852 z / n 120 / 50 2.83 MATH 131 8/11 P( x TEST 2 REVIEW SHEET 900) P( z 2.83) 1 0.9977 0.0023 c) With a sample size of 25 the CLT does not apply. Methods of nonparametric statistics would be needed. 23. a) z P( x 8) 8 7.8 0.1 P( z 2) 1 0.9772 0.0228 2.28% c) b) Lower the standard deviation. Set the mean lower. Use a bigger cup. 24a) 80 100 1.67 12 P( x 80) P( z 1.67) 0.0475 z P( x 115) P ( x 1.25) 1 P( z 1.25) 1 0.8944 0.1056 25. a) 100% b) 0% 1 2 1 d) 20% or .2 or 5 c) b) P(80 x 115) ? 115 100 z 1.25 12 P(80 x 115) P( 1.67 0.8944 0.0475 0.8469 50% or .5 or 2 50 = or 40% or .4 125 5 4 25 b) 1 = or .8 or 80% 5 125 26. a) x 1.25) 27. See Section 3.1. 9