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Statistics Final Exam Review Chapters 3, 4, 5 Practice Problems with Solutions Chapter 5 Practice Problems 1) Find the area under the standard normal curve between z = 0 and z = 3. 1 A) 0.9987 B) 0.4987 Normalcdf(0,3.1) = 0.4990 C) 0.0010 D) 0.4641 B 2) Find the area under the standard normal curve to the left of z = 1.25. A) 0.1056 B) 0.8944 C) 0.2318 Normalcdf(-10000,1.25) = 0.8944 D) 0.7682 B Chapter 5 Practice Problems 3) Find the area of the indicated region under the standard normal curve. A) 0.0968 B) 0.0823 C) 0.9032 D) 0.9177 Normalcdf(-1.30,1.0000) = 0.9032 C 4) For the standard normal curve, find the z-score that corresponds to the first quartile. A) 0.67 B) -0.23 C) 0.77 D) -0.67 The first quartile means that P = 0.2500. The z-score that corresponds to this percentage from the Normal Distribution chart is – 0.67. The answer is D. Chapter 5 Practice Problems 5) IQ test scores are normally distributed with a mean of 99 and a standard deviation of 11. An individual's IQ score is found to be 109. Find the z-score corresponding to this value. A) 0.91 B) 1.10 C) -1.10 D) -0.91 Given that the value of = 99, = 11, and x = 109, we can solve for the value of z using the equation z = (x - )/ . Z = 0.91 A 6) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a certain route is approximately normal with μ = 15.5 and = 3.6. What is the probability that during a given week the airline will lose between 10 and 20 suitcases? A) 0.3944 B) 0.8314 C) 0.1056 Normalcdf(10,20,15.5,3.6) = 0.8311 B D) 0.4040 Chapter 5 Practice Problems 7. With 5% of the data not included on either side, we look up the percent value of .0500 on the chart and we get that z = -1.645. The z value for the other side of the chart will be +1.645. The solution is B. 8. Using the equation x = z + , we see that x = (15)(2.33) + 100 = 134.95. The correct choice will be D. Chapter 5 Practice Problems 9. Solve for the value of z for each of the situations and then compare the results; the larger of the two z scores will have performed better. The first z-value will be z = (75 – 65)/8 which equals 1.25. The second z-value will be z = (75 – 70)/4 which equals 1.25. Since both z scores are the same the correct choice will be C. Chapter 4 Practice Problems 10. State whether the variable is discrete or continuous. The number of cups of coffee sold in a cafeteria during lunch A) discrete B) continuous A) discrete 11. State whether the variable is discrete or continuous. The blood pressures of a group of students the day before their final exam A) continuous B) discrete A) continuous Chapter 4 Practice Problems 12) The random variable x represents the number of cars per household in a town of 1000 households. Find the probability of randomly selecting a household that has less than two cars. A) 0.809 B) 0.125 C) 0.553 D) 0.428 Cars Households 0 125 1 428 2 256 3 108 4 83 P(x) = (428 + 125)/1000 = 0.553 C) 0.553 Chapter 4 Practice Problems 13) The random variable x represents the number of credit cards that adults have along with the corresponding probabilities. Find the mean and standard deviation. A) mean: 1.30; standard deviation: 0.44 B) mean: 1.23; standard deviation: 0.66 C) mean: 1.23; standard deviation: 0.44 D) mean: 1.30; standard deviation: 0.32 x 0 1 2 3 4 P(x) 0.07 0.68 0.21 0.03 0.01 x P(x) xP(x) (x - ) (x - )2 (x - )2P(x) 0 0.07 0.00 -1.23 1.513 0.106 1 0.68 0.68 -0.23 .053 0.036 2 0.21 0.42 0.77 .593 0.125 3 0.03 0.09 1.77 3.133 0.094 4 0.01 0.04 2.77 7.673 0.077 Total 1.00 1.23 0.4.38 2 ( x ) P( x) 0.438 0.66 B) mean: 1.23; standard deviation: 0.66 Chapter 4 Practice Problems 14) At a raffle, 10,000 tickets are sold at $5 each for three prizes valued at $4,800, $1,200, and $400. What is the expected value of one ticket? A) $4.36 B) -$4.36 C) $0.64 D) -$0.64 B) -$4.36 1 1 1 xP ( x ) ( $4,800) ( $1, 200) ( $400) 10, 000 9,999 9,998 999 ( $5) $0.48 $0.12 $0.04 ($5.00) $4.36 10000 15) A sports analyst records the winners of NASCAR Winston Cup races for a recent season. The random variable x represents the races won by a driver in one season. Use the frequency distribution to construct a probability distribution. Wins 1234567 Drivers 12 2 0 2 0 0 1 x 1 P(x) 0.71 2 0.12 3 0 4 0.12 5 0 6 0 7 0.06 Chapter 4 Practice Problems 16) Determine whether the distribution represents a probability distribution. If not, identify any requirements that are not satisfied. x 1 2 3 4 5 P(x) 0.2 0.2 0.2 0.2 0.2 It is a probability distribution, because each value is between 0 and 1 and the sum of P(x) = 1. 17. According to government data, the probability that a woman between the ages of 25 and 29 was never married is 40%. In a random survey of 10 women in this age group, what is the probability that at least eight were married? P(8) + P(9) +P(10) = 0.1209 + .0403 + .0060 = .0.1672 B) 0.167 Binompdf(10,.60,8) = 0.1209; Binompdf(10,.60,9) = 0.0403; Binompdf(10,.60,10) = 0.0060 Chapter 4 Practice Problems 17) The probability that a tennis set will go to a tiebreaker is 18%. In 60 randomly selected tennis sets, what is the mean and the standard deviation of the number of tiebreakers? A) mean: 10.8; standard deviation: 2.98 B) mean: 10.2; standard deviation: 2.98 B) C) mean: 10.8; standard deviation: 3.29 D) mean: 10.2; standard deviation: 3.29 Mean: μ = np = 60 x 0.18 = 10.8 Standard Deviation: =npq = 60x.18x.82 = 2.98 A) mean: 10.8; standard deviation: 2.98 Chapter 4 Practice Problems 18) Fifty-seven percent of families say that their children have an influence on their vacation plans. Consider a sample of eight families who are asked if their children influence their vacation plans. Identify the values of n, p, and q, and list the possible values of the random variable x. n = 8; p = 0.57; q = 0.43; x = 0, 1, 2, 3, 4, 5, 6, 7, 8 19) You observe the gender of the next 100 babies born at a local hospital. You count the number of girls born. Identify the values of n, p, and q, and list the possible values of the random variable x n = 100; p = 0.5; q = 0.5; x = 0, 1, 2, . . ., 99, 100 Chapter 4 Practice Problems 20. A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrive. Use the Poisson distribution to find the probability that in a randomly selected office hour in the 10:30 a.m. time slot exactly five students will arrive. A) 0.0137 3) D B) 0.0519 = 3; x = 5; P( x) C) 0.0070 x e D) 0.1008 35 e3 0.1008 x! 5! Poisson(3,5) = 0.1008 Chapter 4 Practice Problems 21. A mail-order company receives an average of five orders per 500 solicitations. If it sends out 100 advertisements, find the probability of receiving at least two orders. Use the Poisson distribution. A) 0.1839 5) D B) 0.9596 C) 0.9048 D) 0.2642 x e 10 e 1 P(0) 0.3679 x! 0! x e 11 e 1 P(1) 0.3679 x! 1! P( x 2) 1 (0.3679 .03679) 0.2642 Poisson(1,0) = 0.3679; Poisson(1,1) = 0.3679; P(0) +P(1)= 2(0.3679)= 0.7358 P( x 2) = 1 – 0.7358 = 0.2642 Chapter 4 Practice Problems 22. Given: The probability that a federal income tax return is filled out incorrectly with an error in favor of the taxpayer is 20%. Question: What is the probability type that of the ten tax returns randomly selected for an audit, three returns will contain only errors favoring the taxpayer? A) Poisson 7) C B) geometric C) binomial 1. The experiment is repeated for a fixed number of trials, where each trial is independent of other trials. 2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success (S) or as a failure (F). 3. The probability of a success P(S) is the same for each trial. 4. The random variable x counts the number of successful trials. Chapter 4 Practice Problems 23 Given: The probability that a federal income tax return is filled out incorrectly with an error in favor of the taxpayer is 20%. Question: What is the probability type that when the ten tax returns are randomly selected for an audit, the sixth return will contain only errors favoring the taxpayer? A) Poisson B) binomial C) geometric 8) C Geometric distribution A discrete probability distribution. Satisfies the following conditions a) A trial is repeated until a success occurs. b) The repeated trials are independent of each other. c) The probability of success p is constant for each trial. Chapter 4 Practice Problems Binomial Geometric repeated for a fix # of trials Random variable x, counts the # of successes out of n trials repeated until a successful outcome occurs Random variable x, is the # of where the first success occurs Poisson Counting the # times an event occurs for a specified interval The number of occurrences of an event in a specified interval is independent of the # of occurrences of the event in other specified intervals Chapter 4 Practice Problems np Mean Binomial 2 npq Variance npq Standard Deviation Geometric Mean Variance Standard Deviation Poisson 1 p 2 q p2 q p2 Mean mean # occurrences Variance 2 Standard Deviation Chapter 4 Practice Problems 24. Assume the probability that you will make a sale on any given telephone call is 0.19. Find the probability that you (a) make your first sale on the fifth call, (b) make your first sale on the first, second, or third call, and (c) do not make a sale on the first three calls. x = # of trials for 1st success p = probability of success q = probability of failure (a) geometpdf(.19,5) = 0.082 (b) geometpdf(.19,1) + geometpdf(.19,2) + geometpdf(.19,3) = 0.19 + 0.154 + 0.125 = 0.469 P(x) = p(q) x-1 (c) No sales on the first 3 calls = Q(1) + Q(2) + Q(3) = 1 – 0.469 = 0.531 Chapter 4 Practice Problems 24. During a 36-year period, lightning killed 2457 people in the United States. Assume that this rate holds true today and is constant throughout the year. Find the probability that tomorrow (a) no one in the United States will be struck and killed by lightning, (b) one person will be struck and killed, (c) more than one person will be struck and killed. x = # occurrences in a = 2457/13140 = 0.1870 given time or space = average # of occurrences in a given time or space Poissoinpdf(,x) P( x) (a) x = 0; poissonpdf(0.1870,0) = 0.829 (b) x = 1; poissonpdf(0.1870,1) = 0.155 (c) x > 1 = 1 – (P(0) + P(1)) = 0.016 x e x! 36 yrs = 13140 days Chapter 4 Practice Problems 25. A newspaper finds that the mean number of typographical errors per page is four. Find the probability that (a) exactly three typographical errors will be found on a page, (b) at most three typographical errors will be found on a page, and (c) more than three typographical errors will be found on a page. x = # occurrences in a given time or space = average # of occurrences in a given time or space P( x) x e x! Poissoinpdf(,x) (a) = 4, x = 3; P(3) poissonpdf(4,3) = 0.195 (b) P(x 3) = P(0) + P(1) + P(2) + P(3) = .0183 + .0733 + 0.1465 + 0.1954 = 0.433 (c) P(x > 3) = 1 – P( x 3) = 1 – 0.433 = 0.567 Chapter 3 Practice Problems 26. A single six-sided die is rolled. Find the probability of rolling a number less than 3. A) 0.25 B) 0.333 1) B C) 0.1 D) 0.5 2 correct answers/6 possible answers =0.33 27. In a survey of college students, 880 said that they have cheated on an exam and 1721 said that they have not. If one college student is selected at random, find the probability that the student has cheated on an exam. A) 1721/2601 4) D B) 2601/1721 C) 2601/880 D) 880/2601 880 correct answers/(1721+880) possible answers = 880/2601 Chapter 3 Practice Problems 28. The distribution of blood types for 100 Americans is listed in the table. If one donor is selected at random, find the probability of selecting a person with blood type A+ or A-. Blood Type O+ O- A+ A- B+ B- AB+ ABNumber 37 6 34 6 10 2 4 1 A) 0.34 5) C B) 0.06 C) 0.4 D) 0.02 (34 +6) correct answers/100 possible answers = 40/100 = 0.4 29. A card is picked at random from a standard deck of 52 playing cards. Find the odds that it is not a heart. A) 1:3 10) D B) 1:4 C) 4:1 D) 3:1 1 suit of correct answers/4 suits of possible answers = ¼. To not be a heart means that we have 4/4 – ¼ = ¾ . There are a 3:1 odds that a heart will not be chosen randomly. Chapter 3 Practice Problems 29. Classify the events as dependent or independent. Events A and B where P(A) = 0.6, P(B) = 0.3, and P(A and B) = 0.17 A) independent 12) B B) dependent The results of P(A and B) = 0.17 indicates that the event A and the event B are interrelated and so they are dependent. 30. Find the probability of answering two true or false questions correctly if random guesses are made. Only one of the choices is correct. A) 0.25 15) A B) 0.75 C) 0.1 D) 0.5 1 correct answer/4 possible answers =0.25 Chapter 3 Practice Problems 30. Find the probability of getting four consecutive aces when four cards are drawn without replacement from a standard deck of 52 playing cards. 30. P(4-Aces) = (4/52)(3/51)(2/50)(1/49 )= 0.00000369 31. A multiple-choice test has five questions, each with five choices for the answer. Only one of the choices is correct. You randomly guess the answer to each question. What is the probability that you do not answer any of the questions correctly? 31. P(all five questions answers incorrect) = (4/5)(4/5)(4/5)(4/5)(4/5) = 0.32768 32. The probability it will rain is 40% each day over a three-day period. What is the probability it will rain at least one of the three days? 32. P(rain at least one day) = 1 - P(no rain all three days)= 1 - (0.60)(0.60)(0.60) = 0.7845 Chapter 3 Practice Problems 32. A card is selected at random from a standard deck. Find each probability. 13 4 1 16 0.308 (a) Randomly selecting a diamond or a 7. _________________________ 52 52 52 52 (b) Randomly selecting a red suit or a queen. (c) Randomly selecting a 3 or a face card 26 4 2 28 0.538 52 52 52 52 _________________________ 4 12 16 0.308 52 52 52 _________________________ 33. You roll one die. Find each probability. (a) Rolling a 6 or a number greater than 4. 1 2 1 2 0.333 6 6 6 6 _________________________ (b) Rolling a number less than 5 or an odd number. 4 3 2 5 0.833 _________________________ 6 6 6 6 (c) Rolling a 3 or an even number. 1 3 4 0.667 6 6 6 _________________________ Chapter 3 Practice Problems 34. Determine the probability that an 8-sided die, numbered 1-8 is rolled and the results of the roll is an even number or a number that is greater than 6. 4 2 1 5 0.625 ___________________________________ 8 8 8 8 Chapter 3 Practice Problems 35. Gender Male Female Total Level Associates 260 405 665 Of Bachelor’s 595 804 1399 Degree Master’s 230 329 559 Doctorate 25 23 48 Total 1110 1561 2671 (a) earned a master’s degree or is a female (b) earned an associate degree and is a male 559 1561 329 0.671 2671 2671 2671 _________________ 260 0.097 2671 _________________ 804 0.575 1399 (c) is a female given that the person earned a bachelor’s degree_________________ Chapter 3 Practice Problems 36. Using a standard 52-card deck for each situation, determine the probability. (a) Drawing a five and a spade. (b) Drawing a five of spades. 4 1 4 0.019 52 4 208 1 0.019 52 (c) Drawing a spade and then without replacement, drawing club. 13 13 169 0.064 52 51 2652 (d) Drawing a 3 of clubs and then without replacement, drawing another club. 1 12 12 0.005 52 51 2652 (e) Drawing a 4 of diamonds or a heart on the first draw. 1 13 14 0.269 52 52 52 Chapter 3 Practice Problems 37. How many ways can a jury of five men and three women be selected from twelve men and ten women? (12C5)(10C3) = 95,040 38. How many different permutations of the letters in the word PROBABILITY are there? 11!/(2!2!) = 9,979,200 39. A student must answer six questions on an exam that contains twelve questions. a) How many ways can the student do this? b) How many ways are there if the student must answer the first and last question? (a) 12C6 = 924; (b) 10C4 = 210 Chapter 3 Practice Problems 40. In the California State lottery, you must select six numbers from fifty-two numbers to win the big prize. The numbers do not have to be in a particular order. What is the probability that you will win the big prize if you buy one ticket? 1 52C6 = 1/20,358,520 = 0.0000000491 41. From a group of 40 people, a jury of 12 people is selected. In how many different ways can a jury of 12 people be selected? 5,586,853,480 40 C12 40! 5,586,853,480 (40 12)!(12!) Chapter 3 Practice Problems 42. An area code consists of three digits. How many area codes are possible if (a) there are no restrictions and (b) if the first digit can not be 1 or 0. (c) What is the probability of selecting an area code at random that ends in an odd number if the first digit cannot be a 1 or a 0? 42. (a) 10x10x10= 1000 (b) 10x10x8= 800 (c) 0.5 43. A shipment of 10 microwave ovens contains two defective units. In how many ways can a restaurant buy three of these units and receive (a) no defective units, (b) one defective unit, and (c) at least two non-defective units? (d) What is the probability of the restaurant buying at least two defective units? 43. (a) 56 (b) 56 (c) 112 (d) 0.067 (a) (8C3 )(2C0 ) = (56 )(1 )= 56 (b) (8C2 )(2C1 ) = (28 )(2 )= 56 (c) At least two good units one or fewer defective units. 56 + 56 = 112 (d) Pat least 2 defective units = 8 C1 2 C2 8 1 0.067 120 10 C3