Transcription
Math 365: Elementary StatisticsHomework and Problems (Solutions)Satya MandalSpring 2019, Updated Spring 22, 6 March
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Contents1 The Language and Terminology1.15The Language and Terminology . . . . . . . . . . . . . . . . .52 Measures of Central Tendencyand Measures of Dispersion72.1Mean and Median . . . . . . . . . . . . . . . . . . . . . . . . .72.2Variance and Standard Deviations . . . . . . . . . . . . . . . .93 Probability133.1Basic Concept of Probability . . . . . . . . . . . . . . . . . . . 133.2Probability Table and Equally likely3.3Laws of Probability . . . . . . . . . . . . . . . . . . . . . . . . 153.4Counting Techniques and Probability . . . . . . . . . . . . . . 163.5Conditional Probability and Independence . . . . . . . . . . . 184 Random Variables. . . . . . . . . . . . . . 13234.1Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 234.2Probability Distribution . . . . . . . . . . . . . . . . . . . . . 234.3The Bernoulli and Binomial Experiments . . . . . . . . . . . . 263
4CONTENTS5 Continuous Random Variables295.1Probability Density Function (pdf) . . . . . . . . . . . . . . . 295.2The Normal Random Variable . . . . . . . . . . . . . . . . . . 295.2.15.3Inverse Normal . . . . . . . . . . . . . . . . . . . . . . 31Normal Approximation to Binomial . . . . . . . . . . . . . . . 356 Elements of Sampling Distribution396.1Sampling Distribution . . . . . . . . . . . . . . . . . . . . . . 396.2Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 396.3Distribution of the Sample Proportion . . . . . . . . . . . . . 427 Estimation7.145Point and Interval Estimation . . . . . . . . . . . . . . . . . . 457.1.1Confidence Interval of µ . . . . . . . . . . . . . . . . . 457.1.2The Required Sample Size . . . . . . . . . . . . . . . . 477.2Confidence interval for µ when σ is unknown . . . . . . . . . . 487.3About the Population Proportion . . . . . . . . . . . . . . . . 507.4Confidence Interval of the Variance σ 2 . . . . . . . . . . . . . 518 Comparing Two Populations538.1Confidence Interval of µ1 µ2 . . . . . . . . . . . . . . . . . . 538.2When σ1 and σ2 are unknown . . . . . . . . . . . . . . . . . . 558.3Comparing Two Population Proportions . . . . . . . . . . . . 569 Testing Hypotheses599.1A Significance Test for mean µ when σ is known . . . . . . . . 599.2A Significance Test for mean µ when σ is unknown . . . . . . 619.3Population Proportion . . . . . . . . . . . . . . . . . . . . . . 649.4Testing Hypotheses on Variance σ 2 . . . . . . . . . . . . . . . 66
Chapter 1The Language and Terminology1.1The Language and Terminology1. Consider the following is data on the 9-month salary of mathematicsfaculty members (to the nearest thousand dollars) in the year 1999-00:616753506373505772665052807540527061515676 97 60 57 6554 61 65 70 6844 62 57 51 5246(1.1)Use class width 10K. Complete the following frequency table:Class 39.5 49.5 49.5 59.5 59.5 69.5 69.5 79.5 79.5 89.5 89.5 99.5F req2. Following is the grand total (out of 400) of scores obtained by studentsin a 381389354384362376 350284 298343 352326 82(1.2)
6CHAPTER 1. THE LANGUAGE AND TERMINOLOGYWe use class width 50. Complete the frequency table of the datedata in (1.2):ClassF requency51 100101 150151 200201 250251 300301 350351 400If a data value falls on the boundary, count it on the left interval.3. The following is data on the weight (in ounces), at birth, of some 88514091145921569114114899(1.3)Complete the frequency table of the weight distribution data in (1.3):ClassF requency60.5 70.5[70.5 80.580.5 90.590.5 100.5100.5 110.5110.5 120.5120.5 130.5130.5 140.5140.5 150.5150.5 160.5
Chapter 2Measures of Central Tendencyand Measures of Dispersion2.1Mean and Median1. The following is the price (in dollars) of a stock (say, CISCO SYSTEMS) checked by a trader several times on a particular day.139 143 128 138 149 131 143 133(a) Find the mean price (in dollars) observed by the trader.(b) Find the median price observed by the trader (in dollars).2. The following figures refer to the GPA of six students:3.0 3.3 3.1 3.0 3.1 3.1(a) Find the median of the GPA.(b) Find the mean of the GPA.3. The following data give the lifetime (in days) of certain light bulbs.138 952 980 967 992 197 215 1577
8CHAPTER 2. MEASURES OF CENTRAL TENDENCYAND MEASURES OF DISPERSIO(a) Find the mean for the lifetime of these light bulbs.(b) Find the median for the lifetime of the bulbs.4. An athlete ran an event 32 times. The following frequency table givesthe time taken (in seconds) by the athlete to complete the events.Time (in seconds) Frequency11.6411.7511.8611.9712.0612.14Total32(a) Compute the mean time taken by the athlete. (Write up to 4significant digits.)(b) Find the median time taken by the athlete5. The following are the weights (in ounces), at birth, of 30 babies bornin Lawrence Memorial Hospital in May 2000.94 105 124 110 119 137 96 110 120 115104 135 123 129 72 121 117 96 107 8096 123 124 124 134 78 138 106 130 97(a) Compute the mean weight, at birth, of the babies.(b) Using the previous table, compute the median weight, at birth, ofthe babies.6. Following is a frequency table for the hourly wages (paid only in wholedollars) of 99 employees in an industry.Hourly Wages 7 8 9 10 11 12 13 14 15 16 17 18 19Frequency1 4 10 4 9 8 6 5 17 19 3 1 2(a) Compute the mean hourly wage.(b) Using the previous frequency table, compute the median hourlywage.
2.2. VARIANCE AND STANDARD DEVIATIONS97. The following is the frequency table on the number of typos found ina sample of 30 books published by a publisher.Number of Typos 156 158 159 160 162Frequency64569(a) Compute the mean number of typos in a book.(b) Using the previous frequency table, compute the median numberof typos found in a book.8. The following are the lengths (in inches), at birth, of 14 babies born inLawrence Memorial Hospital in May 2000.Length17 18.5 19 20 21.5Frequency 234 32(a) Compute the mean length, at birth, of these babies.(b) Using the previous table, compute the median length, at birth, ofthese babies.2.2Variance and Standard Deviations1. The following is the price (in dollars) of a stock (say, CISCO SYSTEMS) checked by a trader several times on a particular day.138 142 127 137 148 130 142 133(a) Find the variance of the price (to four decimal places).(b) Find the standard deviation of the price (to four decimal places).2. The following figures refer to the GPA of six students:3.0 3.3 3.1 3.0 3.1 3.1(a) Find the variance of the GPA (to four decimal places).(b) Find the standard deviation of the GPA (to four decimal places).
10CHAPTER 2. MEASURES OF CENTRAL TENDENCYAND MEASURES OF DISPERSI3. The following data give the life time (in days) of certain light bulbs.938 952 980 967 992 997 915 957(a) Find the variance for the life time of these bulbs (to four decimalplaces).(b) Find the standard deviation for these bulbs (to four decimal places).4. An athlete ran an event 32 times. The following frequency table givesthe time taken (in seconds) by the athlete to complete the events.Time (in seconds) Frequency11.6411.7511.8611.9712.0612.14Total32(a) Compute the variance for the times taken by the athlete (to fourdecimal places).(b) Find the standard deviation for the times taken by the athlete (tofour decimal places).5. The following are the weights (in ounces), at birth, of 30 babies bornin Lawrence Memorial Hospital in May 2000.94 105 124 110 119 137 96 110 120 115104 135 123 129 72 121 117 96 107 8096 123 124 124 134 78 138 106 130 97(a) Compute the variance of the weight, at birth, of the babies.(b) Compute the standard deviation of the weight, at birth, of thebabies.6. Following is a frequency table for the hourly wages (paid only in wholedollars) of 99 employees in an industry.Hourly Wages 7 8 9 10 11 12 13 14 15 16 17 18 19Frequency11 4 10 4 9 8 6 5 17 19 3 1 2
2.2. VARIANCE AND STANDARD DEVIATIONS11(a) Compute the variance of the hourly wage.(b) Compute the standard deviation of the hourly wage.7. The following is a frequency table on the number of typos found in asample of 30 books published by a publisher.Number of Typos 156 158 159 160 162Frequency64569(a) Compute the variance of the number of typos in a book.(b) Compute the standard deviation of the number of typos in a book.8. The following are the lengths (in inches), at birth, of 14 babies born inLawrence Memorial Hospital in May 2000.Length17 18.5 19 20 21.5Frequency 234 32(a) Compute the variance of the length, at birth, of these babies.(b) Compute the standard deviation of the length, at birth, of thesebabies.
12CHAPTER 2. MEASURES OF CENTRAL TENDENCYAND MEASURES OF DISPERSI
Chapter 3Probability3.1Basic Concept of ProbabilityNone3.2Probability Table and Equally likely1. The following table gives the probability distribution of a loaded die.Face123456Probability 0.20 0.15 0.15 0.10 0.05 0.35Find the probability that an even number face will show up when youroll the die.2. A die is rolled twice. What is the probability that the two numbers areunequal?Solution: n(S) 36Let E that the two numbers are unequalSo, (not E) (1, 1, ), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)So, n(not E) 613
14CHAPTER 3. PROBABILITYSo, n(E) n(S) n(notE) 36 6 30P (E) n(E) 30 .8333n(S)363. A student wants to pick a school based on the recent grade distributionof the school. Following is the grade distribution of the previous yearin a school:GradesA B C D FPercentage of Students 19 33 31 14 3What is the probability that a student will receive a C-grade or higherin this school?4. A container has 11 red balls, 15 green balls, 10 white balls, and 8 yellowballs. A child picks up a ball from the container. Find the probabilitythat the ball is red.5. A die is rolled twice. What is the probability that the product of thetwo numbers is 11?6. A magician asks you to pick a card from a deck of 52 cards. What isthe probability that you will pick a heart?7. You toss a coin 4 times. What is the total number of outcomes?8. The sex of 4 unborn children is to be determined in a lab today. Whatis the probability that there will be at least 2 girls?Solution: n(S) 24 16Let E be the even that there would be at least two girls(not E) means 0 or 1 girlSo, (not E) {BBBB, GBBB, BGBB, BBGB, BBBG}n(not E) 5n(E) n(S) n(not E) 16-5 11So, P (E) n(E) 11/16n(S)
3.3. LAWS OF PROBABILITY3.315Laws of Probability1. Suppose E and F are two events. It is given that P(E) .22, P(notF) .65, and P(E and F) .13. Determine the probability that either Eor F occur.Solution: First, P (F ) 1 P (not F ) 1 .65 .35P (E or F ) P (E) P (F ) P (E and F ) .22 .35 .13 .442. Suppose E and F are two events. It is given that P(not E) .33, P(notF) .54, and P(E or F) .68. Determine the probability that both Eand F occur.3. Use the information in Question 2. Determine the probability thatneither E nor F occur.4. The proportion of students who own a vehicle is 0.55, and the proportion of students who live in a dorm is 0.41. If the proportion of studentswho either own a vehicle or live in a dorm is 0.83, find the proportionof students who live in a dorm and own a vehicle.5. The probability that it does not rain in Arizona in June is 0.22. Findthe probability that it rains at least once in the month of June inArizona.Solution: Let E event that it rains at least once in June.(not E) event that it does not does not rain in JuneP (not E) .22P (E) 1 P (not E) 1 .22 .786. The probabilities that a particular office phone rings 0, 1, 2, 3 times inhalf hour are respectively: .05, .15, .21, .42. Find the probability thatthe phone rings at least 4 times between 3:30-4:00 pm.Solution: E event that it rings at least 4 times.So, (not E) {0.1, 2, 3}.P (not E) P (0, 1, , 2, 3) P (0) P (1) P (2) P (3) .05 .15 .21 .42 .83.So, P (E) 1 P (not E) 1 .83 .17.
16CHAPTER 3. PROBABILITY7. The probability of a student going to a bar on Wednesday is 3/5. Theprobability of the student going to class on Thursday is 4/5. If theprobability that the student either goes to the bar or goes to class thenext day is 9/10, what is the probability that the student will do theboth?8. The probability that a student will major in mathematics is .13 andthe probability that a student will have a major in engineering is .31.The probability that a student will double major in mathematics andengineering is .07. What is the probability that a student will majorin mathematics or engineering?3.4Counting Techniques and Probability1. Compute 8 C2 .2. Suppose 2 scholarships, 1000 each, is to be given to 2 students. Thereare 9 applicants. How many choices of 2 awardees are possible?Solution: Each scholarship is worth same amount.So, order of selection does not count.It is an unordered selection.So, answer 9 C2 n!/[(n r)!r!] 9!/[7!2!] 9 8/[1 2] 36.3. Suppose 2 scholarships, one for 2000 and another for 1000, will begiven to 2 students. There are 9 applicants. How many choices of 2awardees are possible?Solution: Scholarships have distinct values.So, order of selection has to be taken into account.So, Answer 9 P2 n!/(n r)! 9!/7! 8 9 72.4. In an annual sports meet, 9 students compete in 2 events (the 100 meterand 1000 meter race). How many ways can two winners can be picked?Solution: Repetitions are allowed here.Use multiplication principle:
3.4. COUNTING TECHNIQUES AND PROBABILITY17First event : 9 choice2nd event : 9 choiceAnswer product 9 9 815. A coin is tossed 6 times. What is the total number of possible outcomes?Solution: answer 2 2 2 2 2 2 646. The sex of 8 unborn children is to be determined today in a lab. Whatis the probability that 2 particular children (say the child of Elizabethand the child of Maria) will be boys?Solution: Each child could be B or G.n(S) 2 2 2 2 2 2 2 2 28n(E) 1 1 2 2 2 2 2 2 266 228 .25P (E) n(E)n(S)7. A committee of 8 is to be formed from a group of 7 mothers, 6 fathers,and 8 children. What is the probability that the committee will haveexactly 4 fathers?Solution: The sample space consists of all possible committees of 8,from total of n 7 6 8 21 people. Committee selection is anunordered selection.The number of such committees n(S) 21 C8 .Let E be the event that there would be 4 fathers(and hence 4 others from 7 8 15).So, n(E) (6 C4 )(15 C4 ) (6 C421)(C158 C4 ) .So, P (E) n(E)n(S)8. The 6 seats in the first row are to be assigned to a class of 17 womenand 20 men. What is the probability that all the seats will be assignedto women.Solution: Seat assignment is an ordered selection.Total number of such assignment isn(S) 37 P6 .
18CHAPTER 3. PROBABILITYLet E be the event that all seat would be assigned to women.n(E) 17 P6P6 17.So, P (E) n(E)n(S)37 P63.5Conditional Probability and Independence1. For two events, E and F, it is given that the probabilities P (E) 0.6,P (F E) 0.3, and P (F ) 0.7. What is the probability that E and Foccur simultaneously?Solution: P (E and F ) P (F E)P (E) (0.3) (0.6).2. For two events, E and F, it is given that the probabilities P (E) 0.5, P (F and E) 0.3, and P (F ) 0.9. What is the conditionalprobability that F occurred, given that E has occurred?Solution: P (F E) P (F and E)/P (E) 0.3/.053. Following are some data from a hospital emergency room:(a) The probability that a patient in the emergency room will havehealth insurance is 0.8.(b) The probability that a patient in the emergency room will survivethe treatment 0.9.(c) The probability that a patient in the emergency room will havehealth insurance and will also survive is 0.6.What is the conditional probability that a patient in the emergencyroom will survive, given that he/she has health insurance.Solution: Let H the event that the patient has Health insurance.Let E the event that the patient will survive.Given P (H) 0.8., P (E) 0.9, P (H and E) 0.6E 0.6P (E H) H Pand(H)0.84. Following are some observations on the stock market:(a) The probability that the Dow Jones Industrial Average (DJIA)will go up today is 0.75.
3.5. CONDITIONAL PROBABILITY AND INDEPENDENCE19(b) The probability that the value of your portfolio will go up todayis 0.80.(c) The probability that both the DJIA will go up and the value ofyour portfolio will go up today is 0.6.What is the conditional probability that the value of your portfoliowent up today, given that the DJIA went up today?Solution: Let D the event that DJIA will go up.Let E the event that your portfolio will go up.Given P (D) 0.75., P (E) 0.80, P (D and E) 0.60.6E 0.75.P (E D) D Pand(D)5. Following are some data on allergy treatment:(a) The probability that a person will get the allergy in a season is0.55.(b) The probability that a person has taken an allergy treatment andgets an allergy in the season is 0.15.(c) The probability that a person has taken the treatment is 0.6.What is the conditional probability that a person will get an allergy,given that she/he has taken the allergy treatment?Solution: Let E the event one would get the allergy in a season.Let F the event that one would have taken an allergy treatment.Given P (E) 0.55., P (F ) 0.6, P (F and E) 0.15F 0.15P (E F ) E Pand(F )0.66.Consider the following circuit with the radio: It is given that the probability that switch A is closed is 0.35 and
20CHAPTER 3. PROBABILITYthe probability that switch B is closed is 0.40. It is also known thatthe 2 switches function independently. What is the probability thatthe radio is playing?Solution: E event that switch-A is on.F event that switch-B is on. We compute P (E orF ).We have P (E) 0.35, P (F ) 0.40.P (E and F ) P (E)P (F ) .35 40 .14.(Because of Independence.)P (E or F ) P (E) P (F ) P (E and F ) .35 .40 .14 .61.7. An airplane has 2 engines and the engines function independently. Theprobability that the first engine fails during a flight is 0.1 and theprobability that the second engine fails is 0.1. What is the probabilitythat both will fail during a flight?Solution: E event that Engine-1 is fails.F event thatEngine-1 is fails.We have P (E) 0.1, P (F ) 0.1.We compute P (E and F )P (E)P (F ) (.1) (.1)8. The probability that you will receive a wrong number call this weekis 0.2. The probability that you will receive a sales call this week is0.5. The probability that you will receive a survey call this week is 0.3.What is the probability that you will receive one of each this week?(Assume independence.)Solution: Let E be the event that you receive a wrong number callLet F be the event that you receive a sales callLet G be the event that you receive a survey callP (E and F and G) P (E)P (F )P (G) (0.2.) (0.5) (0.3)9. The probability that you will receive a wrong number call this week is0.1. The probability that you will receive a sales call this week is 0.6.What is the probability that you will receive either a wrong numbercall or a sales call this week? (Assume independence.)Solution: Let E be the event that you receive a wrong number callLet F be the event that you receive a sales call
3.5. CONDITIONAL PROBABILITY AND INDEPENDENCE21P (E) 0.1, P (F ) 0.6.P (E or F ) P (E) P (F ) P (E and F ) P (E) P (F ) P (E)P (F ) 0.1 0.6 (0.1) 0.6).10. During your trip to Florida, the probability that the weather in Floridawill be good is 0.9 and the probability that it will be cold in Lawrenceis 0.3. What is the probability that, during your trip, you will havegood weather in Florida and it will be cold in Lawrence? (Assumeindependence.)Solution: Let E be the event that weather in Florida will be goodLet F be the event that it will be cold in LawrenceP (E and F ) (0.9) (0.3).
22CHAPTER 3. PROBABILITY
Chapter 4Random Variables4.1Random VariablesNone4.2Probability Distribution1. The following table gives the probability distribution of the startingsalary X earned by new graduates from some university (to the nearest 10K).X x 01234567p(x)0.05 0.15 0.22 0.22 0.17 0.10 .05 .04What is the expected starting salary of a new graduate?PSolution: E(X) µ xi p(xi ) 0 .05 1 .15 2 .22 3 .22 4 .17 5 .10 6 .05 7 .04 3.012. For the starting salary X in Question 1, what is the standard deviationof the staring salary?23
24CHAPTER 4. RANDOM VARIABLESSolution:P Variance2σ x2i p(xi ) µ2 [0 .05 1 .15 4 .22 9 .22 16 .17 25 .10 36 .05 49 .04] (3.01)2 2.9299 So, standard Deviation σ σ 2 2.9299 1.71173. Maria is a plumber who works for 3 different employers. EmployerA pays her 120 a day, employer B pays her 70 dollars a day, andemployer C pays her 180 a day. She works for whoever calls her first.The probability that employer A calls her first is 0.4, the probabilitythat employer B calls first is .2, and the probability that employer Ccalls her first is 0.3 (the probability that no one calls is .1). What isthe expected income of Maria per day?Solution: Let X Income of Maria on a day.So, X takes values 0, 120, 70, 180. The probability function is givenby p(0) .1, p(120) .4, p(70) .2, p(180) .3.In a tabular form, the distibution of Maria’s income is given byX x 0 120 70 180p(x).1 .4 .2 .3PSo, mean E(X) µ xi p(xi ) 0 p(0) 120 p(120) 70 p(70) 180 p(180) 0 0 120 .4 70 .2 180 .3 1164. Maria is the plumber in Question 3. What is the standard deviation ofthe daily income of Maria?PSolution: Variance σ 2 s x2i p(xi ) µ2 [0 0 14400 .4 4900 .2 32400 .3] (116)2 3004 So, standard Deviation σ σ 2 30045. The number X of typos in a website has the following probability distribution.X x 012345p(x).14 0.27 0.27 0.18 0.09 0.05What is the expected number of typos in a website?
4.2. PROBABILITY DISTRIBUTION25Solution:The meanPµ xi p(xi ) 0 p(0) 1 p(1) 2 p(2) 3 p(3) 4 p(4) 5 p(5) 0 .14 1 .27 2 .27 3 .18 4 .09 5 .05 1.966. Let X be the number of typos, as in Question 5. What is the standarddeviation of X?PSolution: Variance σ 2 x2i p(xi ) µ2 (0 p(0) 1 p(1) 4 p(2) 9 p(3) 16 p(4) 25 p(5)] (1.96)2 (0 .14 1 .27 4 .27 9 .18 16 .09 25 .05) (1.96)2 5.66 4.8416 1.8184 So, standard Deviation σ σ 2 1.81847. Let X be the number of typos, as in Question 5. What is the probabilitythat there would be at least 3 typos is a book?Solution:P (3 X) P (X 3) P (X 4) P (X 5) 0.18 0.09 0.058. Let X represents the number of students in a class, in a school district.The following is the distribution of X:X x 11 12 13 14 15 16 17 18 19 20 21p(x).08 .09 .11 .13 .14 .13 .10 .09 .06 .05 .02What is the expected number of students in a class?Solution: TheP mean,E(X) µ xi p(xi ) ( 2) .3 ( 1) .2 0 .2 1 .1 2 .1 3 .1 29. Refer to Question 8. Compute the standard deviation σ.10. A Gambling house has a loaded die. One pays 3 to play each time.One gets back number of dollards equal to the face that turns up. Thefollowing is the probability distribution of the loaded die, together withthe win X.F ace12 3 4 5 6p(x).3 .2 .2 .1 .1 .1W in X x 2 1 0 1 2 3Compute the expected win E(X)?
26CHAPTER 4. RANDOM VARIABLES4.3The Bernoulli and Binomial Experiments1. The statistics of campus interviews show that for every 10 interviews 4job offers are made. Out of 66 interviews, what is the probability thatat least 30 offers would be made? (Use TI-84, Silver Edition.)4Solution: Here n 66, p 10 .4P (30 X) 1 P (X 29) 1 binomialcdf (66, .4, 29) 1 .7829 .21712. Refer to Question 1. What would be the expected number of job offersmade?Solution: mean µ E(X) np 66 .4 26.43. Refer to Question 1. What would be the standard deviation σ of thenumber of job offers made?Solution:StandardpDeviationpσ np(1 p) 66 .4 (1 .6) 3.97994. Assume that 20 percent of the babies get fever after immunizationshoots. If 130 babies are immunized in a clinic, what is the probabilitythat at most 30 will get fever?Solution: Here n 130, p .2P (X 30) binomialcdf (130, .2, 30) .83845. Refer to Question 4. What is the expected number of babies that wouldget fever?Solution: mean µ E(X) np 130 .2 266. Refer to Question 4. What would be the standard deviation σ ?Solution:StandardpDeviationpσ np(1 p) 130 .2 (1 .2) 4.56077. It is assumed by the coach that a swimmer can finish an event withintarget time 45 percent times during practice. Suppose the swimmer
4.3. THE BERNOULLI AND BINOMIAL EXPERIMENTS27swims 80 times. What is the probability that he/she will finish withinthe target time at least 40 times?Solution: Here n 80, p .45P (40 X) 1 P (X 39) 1 binomialcdf (80, .45, 39) 1 .7846 .21548. Refer to Question 8. What is the expected number E(X) of times thathe/she will finish within target time?Solution: mean µ E(X) np 80 .459. Refer to Question 8. What is the standard deviation σ of the numberof times X that he/she will finish within target time?Solution:StandardpDeviationpσ np(1 p) 80 .45 (1 .45)10. It was reported, that 20 percent of the population in a region wereinfected by AIDS-HIV. A sample of 200 people from this region wereexamined for AIDS-HIV. What was the expected number of infectedpeople?
28CHAPTER 4. RANDOM VARIABLES
Chapter 5Continuous Random Variables5.1Probability Density Function (pdf)None5.2The Normal Random VariableUse TI-84 "Normalcdf" function, under DISTR tab.1. Suppose Z is the standard normal random variable. Find the probability P ( 1.23 Z).Solution: P ( 1.23 Z) normalcdf ( 1.23, 5) .89072. Suppose Z is the standard normal random variable. Find the probability P ( 1.18 Z 1.87).Solution: P ( 1.18 Z 1.87) nomalcdf ( 1.18, 1.87) .85013. Suppose X is a normal random variable with mean µ 11.5 and standard deviation σ 2.3. Find the probability P (X 14).29
30CHAPTER 5. CONTINUOUS RANDOM VARIABLESSolution: 14 µP (X 14) P X µσσ P Z 14 11.5 P (Z 1.0870) nomalcdf ( 5, 1.0870) .86152.34. Monthly consumption of electricity X (in KWH, in a winter month) bythe households in a county has a normal distribution with mean 875KWH and standard deviation 155 KWH. What proportions of households consumes less than 1000 KWH?Solution: 1000 µP (X 1000) P X µσσ P Z 1000 875 P (Z .8065) nomalcdf ( 5, 1.0870) .79001555. The monthly cell phone minutes used by an individual in a city hasnormal distribution with mean 2700 minutes and standard deviation325 minutes. What proportion of calls would last more than 3000minutes?Solution: X µP (3000 X) P 3000 µσσ Z P (.9231 Z) normalcdf (.9231, 5) .1780 P 3000 27003256. The amount of time a student spends to take a test has normal distributions with mean 45 minutes and standard deviation 13 minutes.What proportion of students would finish in 60 minutes?Solution: 60 µP (X 60) P X µ σσ P Z 60 45 P (Z 1.1538) nomalcdf ( 5, 1.1538) .8757137. The annual rainfall X in a region is normally distributed with meanµ 62 cm and standard deviation σ 9 cm. What is the probabilitythat rainfall will be between 55 cm and 70 cm?Solution: X µ 70 µ P 55 62 Z 70 62P (55 X 70) P 55 µσσσ99 P ( .7778 Z .8889) nomalcdf ( .7778, .8889) .59468. The birth weight X of babies is normally distributed with mean µ 113ounces and standard deviation σ 19 ounces. What proportion ofbabies will have birth weight below 150 ounces?
5.2. THE NORMAL RANDOM VARIABLE31Solution: 150 µP (X 150) P X µσσ P Z 150 113 P (Z 1.9474) nomalcdf ( 5, 1.9474) .9743199. The length X of babies at birth is normally distributed with meanµ 18.5 inches and standard deviation σ 2.3 inch. What proportionof babies are smaller than 21 inches ?Solution: 21 µP (X 21) P X µσσ P (Z 1.0870) nomalcdf ( 5, 1.0870) .8615 P Z 21 18.52.310. The weight X of salmon caught in a river is normally distributed withmean µ 24 pounds and standard deviation σ 7 pounds. Youcan keep only those above 15 pounds. What proportion of the salmoncaught can be kept?Solution: X µ P (15 X) P 15 µσσ P 15 24 Z P ( 1.4286 Z) normalcdf ( 1.4286, 5) .923475.2.1Inverse NormalUse TI-84 "invNorm" function, under DISTR tab.1. Suppose Z is the standard normal random variable. It is given thatP (Z c) 0.2300. What is the value of c?Solution:Answer invN orm(.23) .73882. Suppose Z is the standard normal random variable. It is given thatP (d Z) 0.9500. What is the value of d?Solution:P (d Z) .95P (Z d) .05Answer invN orm(.05) 1.6449
32CHAPTER 5. CONTINUOUS RANDOM VARIABLES3. Suppose X is a normal random variable with mean µ 3.5 poundsand standard deviation σ 1.2 pounds. Given that the probabilityP (c X) 0.2550, what is the value of c? Solution: µ 3.5, σ 1.2P (c X) 0.2550P (X c) 1 .2550 .745 .745 (This step iscalled Standardization)P Z c µσP (Z .6588) .745 Because invN orm(.745) .6588Compare:c µc 3.5 .6588 c 3.5 1.2 .6588 4.2906σ1.24. The weight X of babies (of a fixed age) is normally distributed withmean µ 212 ounces and standard deviation σ 25 ounces. Doctorswould be concerned (not necessarily alarmed) if a baby is among thelower 5 percent in weight. Find the cut-off weight l, below which thedoctors will be concerned.Solution: µ 212, σ 25,P (
Math 365: Elementary Statistics Homework and Problems (Solutions) Satya Mandal Spring 2019, Updated Spring 22, 6 March. 2. Contents . Chapter 2 Measures of Central Tendency and Measures of Dispersion 2.1 MeanandMedian 1.The following is the price (in dollars) of a stock (say, CISCO SYS-
