dsci 301 blackboard

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Mark G. HaugProbabilityMaking SoundDecisions1) Correlation2) ModelingCollecting DataDescribing DataNatural Phenomena Mark G. HaugOutline of ProbabilityA. PercentagesB. RiskC. Formal probabilityD. Randomness E. Miscellaneous Problems Mark G. HaugA. PercentagesOne simple rule: whenever you are given a percentage (or fraction), always be clear on percentage of what? Mark G. HaugB. RiskRisk is the probability that an event will occur within a given time period. Mark G. HaugWhich do you think is safest based on your own experience and the following information? 1. National Ski Patrol, 1984-97: 34 deaths per 52,250,000 per year2. Denver Post claims: National Safety Council, 1995: 17 drowning deaths per million water-sports participants3. Denver Post claims: National Safety Council, 1995: 7.1 (bicycling?) deaths per million bicyclists4. Skiing magazine claims: National Severe Storms Laboratory: 89 lightning deaths per year in US

A. Snow SkiingB.Water-SportsC. BicyclingD. Lightning Mark G. HaugReview of Risk

1. Measures of risk should be of the same units (per number per time)

2. What is the baseline risk?

3. Does the unit of measurement adequately represent the concern? (E.g., safety is a relative criterion.)

4. Is the risk your risk or some general population? Mark G. HaugC. Formal ProbabilityA preliminary note on probability:

1. Formal (Classical)

2. Empirical (Relative Frequency)

3. Subjective Mark G. HaugC. Formal Probability1. Events2. Outcomes3. Addition Law4. Mutually Exclusive Events5. Complements6. Conditional Probability7. Multiplication Law8. Independent Events9. Combinations Mark G. Haug

pepperonianchovymushroom Mark G. HaugDefinitions in Formal Probability1. Events: Events are the specified results of a situation.2. Outcomes: Outcomes are all of the possible results of a situation.Note: The sum of the probability across all outcomes always adds up to 1. Mark G. HaugDefinitions in Formal Probability1. Events: Events are the specified results of a situation. 2. Outcomes: Outcomes are all of the possible results of a situation.If all of the outcomes are equally likely, then the probability of an event, P(E), is equal to the number of events divided by the number of outcomes.Note: The sum of the probability of each outcome always adds up to 1. Mark G. Haug

= P (pepperoni)= A (anchovy)= M (mushroom)

Example: There areeight slices of pizza.What is P(A) ? (What is theprobability that aslice will contain atleast one anchovy?) Mark G. Haug1946: Malone v. CommonwealthWhich of the following describes the revolver and the shooting sequence?

A.4 chambers, sequentialB.4 chambers, spin after each trigger C.5 chambers, sequentialD.5 chambers, spin after each trigger E.6 chambers, sequentialF.6 chambers, spin after each trigger

Mark G. HaugRead as:P(A or B) = P(A) + P(B) - P(A and B).3. Addition Law:P(AB) = P(A) + P(B) - P(AB) 4. Mutually Exclusive Events:If P(AB) = 0, then events A and B are called mutually exclusive events. Definitions in Formal Probability Mark G. Haug


Example:What is P(AM)=? Mark G. Haug


Are slices with anchovies andslices with mushrooms mutually exclusive?

P(A) + P(M) - P(AM) 5/8 + 4/8 - 2/8 = 7/80.875 No, 2 of the 8 slices, P(AM), contain both anchovies andmushrooms. Mark G. Haug5. Complements: P(A`) = 1 - P(A)Read as: P(Not A) = 1 - P(A)Definitions in Formal Probability Mark G. Haug


Example:What is P(A`) = ? (What is theprobability that aslice will not contain at leastone anchovy?) Mark G. Haug6. Conditional Probability:P(B|A) = [P(AB)] / [P(A)] Definitions in Formal ProbabilityRead as:P(B given A) = P(A and B) / P(A) Mark G. Haug


Example #1:What is P(P|A)=? Mark G. Haug


Example #2:What is P(A|P)=? Mark G. Haug8. Independent Events:Events A and B are independent if (and only if) P(B|A) = P(B). [It also follows that P(A|B) = P(A) when A and B are independent.]7. Multiplication Law:P(AB) = P(A) P(B|A)P(AB) = P(A) P(B)if events A and B are independent.Definitions in Formal Probability Mark G. Haug


Are slices with anchovies and slices with mushrooms independent? Mark G. HaugReading: pp. 74-80 Sections 5.1-5.6

Homework: pp. 84-88Problems 5.1-5.12

101 Special Problems: 1.4, 2.7 Mark G. Haug1973 Candidates: b1 b2 b3 w1 w2 w3

1975 Candidates: b1 b2 b3 b4 w1 w2 w3 w4

Mark G. HaugLEE v. CITY OF RICHMONDUnited States District Court, E.D. Virginia, 1978456 F.Supp. 756

***

The Court concludes that the numbers involved are too small to allow a finding of disparate impact under the Griggs rule. The total of applicants for 1973 and 1975 was fourteen, seven white and seven black.

*** Mark G. HaugHomework: p. 88Problem 5.13

Mark G. HaugHomework: p. 90Problem 5.17

Mark G. Haug100100100100Fire oneExplode?Yes,

then take all 399NoFire anotherExplode?Yes,

then take all 398NoReject 100 Acceptance Sampling Mark G. HaugIf 84% of the shells are duds, what is the probability that approximately 75% or more of all the shells will be accepted?

A. 1/24B.3/24 = 1/8C. 1/6D. 1/4E.1/3F.1/2 Mark G. Haug9. Combinations:

Read as: n choose rExample: How many distinct pairs of people are there in a group of 5 people? (Read as: 5 choose 2) Answer: You can form exactly 10 distinct pairings of people within a group of 5 people. Mark G. Haug9. Combinations: Probability example: If Kathryn, Joseph, Luke, Alexander, and Dominic are the only members of a committee, then what is the probability that Kathryn and Joseph will be selected as co-chairs in a random selection process?

Mark G. Haug9. Combinations: Answer: 1/10 = 0.1.

Kathryn and Joseph are one distinct pairing (the event) of ten equally likely pairings (the outcomes).

Mark G. HaugReading: p. 83Section 5.10

Homework: pp. 93-94Problem 5.24

From the Materials Published by Harvard Business School: Charles River Jazz Festival Read case and answer Question #1(Answer is on Blackboard HB crjf answers) Mark G. HaugAssume that this week you receive a stock market prediction in the mail. It says the market will go down next week. You wait and see. The market then goes down. The following week, same thing happens: a prediction, which is later confirmed as accurate. This happens seven consecutive times. Then, the mailer asks you to subscribe to his/her newsletter for 52 issues at $500. Mark G. HaugD. Some Curious Aspects of RandomWhat does random mean? Mark G. HaugWhich of the following best describes the meaning of random?

A. chaosB.absence of a patternC. equal likelihood of all outcomesD. independence of outcomes Mark G. HaugIt is difficult to define, but if it is time-dependent, then each random event must be of preceding random events. We infer non-random processes if we observe an unlikely pattern(s) in the process. The absence of a pattern, however, does not guarantee a random process.D. Some Curious Aspects of RandomWhat does random mean? Mark G. HaugLuke Haug dialed 9-1-1 by accident. Was this a random event?

A. Yes, because he was only 2 years old B.No, because there is a patternC. No, because the numbers are not equally likelyD. Yes, because the numbers are equally likelyE. Yes, but Lukes parents should be fined anyway for neglecting their kid Mark G. HaugHow long would you expect the longest sequence of heads (H) to be in a collection of 200 consecutive tosses of a coin? Mark G. HaugIn one season of 30 basketball games, what would you expect is the longest streak of consecutive shots made for a 40% shooter who shoots on average 20 shots per game? (Assume that the Hot Hand does not exist: i.e., shots are independent.)

A. 6B.7C. 8D. 12E.24 Mark G. HaugIn one season of 162 baseball games, what would you expect is the longest streak of consecutive games with one or more hits for a .250 batter who gets on average 5 at bats? (Assume that hits and games are independent.)

A. 7B.11C. 15D. 19E.23 Mark G. HaugE. Miscellaneous Problems1. Bombs on an Airplane2. Birthdays**3. A Family of Four*4. Testing for HIV**5. Lets Make a Deal!* **6. Life and Times of Orchestra Conductors* Treated in Parade Magazine by Marilyn vos Savant.** Treated in The Economist. Mark G. Haug1. Bombs on an AirplaneWhen discovering that the chance of a bomb on an airplane is 1 in 13,000,000, frequent-flyer Frank got nervous. Consulting with you, he learns that the probability of 2 bombs on the same plane is 1 in 169,000,000,000,000, assuming the two bombs are independent. Relieved, Frank now carries a bomb with him when he flies. Mark G. HaugWould Franks bomb be considered independent of another bomb on the airplane?YES.So, whats wrong with Franks logic? Mark G. HaugSo, whats wrong with Franks logic?This is an example of conditional probability. The probability of two bombs on the plane is P(B2 | B1) = P(B2 B1) / P(B1) =P(B2) / P(B1) = (1/169,000,000,000,000) / (1/13,000,000) = 1/13,000,000, or simply 1 in 13,000,000 Mark G. Haug2. BirthdaysWhat is the probability of a matching birthday in any randomly selected group (and most non-random groups) of 25 people?

(Note: Consider the standard 365 days a year -- the months and days -- and disregard the year of birth. Also disregard February 29 for the sake of simplicity.) Mark G. HaugA. 25 / 365B.25! / 365!C. (365-25)! / 365!D. 0.14E.0.28F.0.42G. 0.56H.0.70What is the probability of a matching birthday in any randomly selected group (and most non-random groups) of 25 people? Mark G. Haug101 Special Problems: 1.3 Mark G. HaugA. (1/365)4B.(1/365)3C. (1/90)4D. (1/90)3E.cannot be calculated with this informationWhat is the probability that in a family of fourkids (four singleton births), all four kids have the same birthdate? Boston Globe reported this true story and suggested the probability was about 1 in 18,000,000,000 (1 in 18 billion). Mark G. Haug3. A Family of FourQuestion #2: A husband and wife have two children: one is a boy. Whats the probability that the other is a girl?Question #1: A husband and wife have two children: the oldest one is a boy. Whats the probability that the other is a girl? Mark G. HaugA. 0.50, 0.25B.0.50, 0.33C. 0.50, 0.50D. 0.50, 0.67E.0.50, 0.75Question #2: A husband and wife have two children: one is a boy. Whats the probability that the other is a girl?Question #1: A husband and wife have two children: the oldest one is a boy. Whats the probability that the other is a girl? Mark G. Haug3. A Family of FourQuestion #2: A husband and wife have two children: one is a boy. Whats the probability that the other is a girl?Question #1: A husband and wife have two children: the oldest one is a boy. Whats the probability that the other is a girl? Mark G. HaugQuestion #2: A husband and wife have two children: one is a boy. Whats the probability that the other is a girl? IS THE BUCKET RED OR PURPLE? Mark G. Haug4. Testing for HIV1993: Wellcome- Elisa Test for HIVP( pos. | HIV ) = 0.993 (Sensitivity)P( neg. | Not HIV ) = 0.9999 (Specificity)P( neg. | HIV ) = ? Mark G. Haug1993: Wellcome- Elisa Test for HIVP( pos. | HIV ) = 0.993 (This is sensitivity)P( neg. | Not HIV ) = 0.9999 (This is specificity)P( neg. | HIV ) = ?This is a conditional probability that is simply a complement to a known conditional probability, namely, P( pos. | HIV ). Given that someone has HIV, there are only two possible outcomes from the test: pos. and neg. Mark G. Haug1993: Wellcome- Elisa Test for HIVP( pos. | HIV ) = 0.993 (This is sensitivity)P( neg. | Not HIV ) = 0.9999 (This is specificity)P( pos. | Not HIV ) = ?This is a conditional probability that is simply a complement to P( neg. | Not HIV ). Given that someone does not have HIV, there are only two possible outcomes: pos. and neg. Mark G. Haug1993: Wellcome- Elisa Test for HIVP( pos. | HIV ) = 0.993 (This is sensitivity)P( neg. | Not HIV ) = 0.9999 (This is specificity)Much more interesting: If you tested people at random, what is the probability that someone really has HIV given he tested positive?[P( HIV | pos.) = ?] Mark G. HaugIf you tested people at random, what is the probability that someone really has HIV given that he tested positive?[P( HIV | pos.) = ?]A. 0.00B.0.20C. 0.40D. 0.60E.0.80F.greater than 0.80 but less than or equal to 0.90F.greater than 0.90 but less than or equal to 0.99G.greater than 0.99 Mark G. Haug1993: Wellcome- Elisa Test for HIVP( pos. | HIV ) = 0.993 (This is sensitivity)P( neg. | Not HIV ) = 0.9999 (This is specificity)To answer this question, we need to add a new topic to our discussion of formal probability:10. Bayes Rule:

Mark G. HaugP( HIV | pos. ) = P( pos. | HIV ) = 0.993 (This is sensitivity)P( neg. | Not HIV ) = 0.9999 (This is specificity)

Mark G. HaugWhat if the people being tested are a self-selected group? In other words, people coming to the clinic specifically for the test?P( pos. | HIV ) = 0.993 (This is sensitivity)P( neg. | Not HIV ) = 0.9999 (This is specificity)

IS THE BUCKET RED OR PURPLE? Mark G. HaugHomework Problem:

1) A Manufacturer produces sweatshirts through a standardized process. 2) The process is efficient and cost effective, but as such, it produces 20% defects (cut-outs). 3) A quality management program of thoroughly inspecting every sweatshirt is not cost effective. 4) A quality program of cursory inspection of each sweatshirt is cost effective. 5) A satisfactory sweatshirt will always pass the proposed inspection. 6) Approximately 25% of the defective sweatshirts will also pass the test. Question: P(defective sweatshirt | pass)? .0588 Mark G. HaugReading: p. 80Section 5.7

Note: Consider the tree diagram approach as an aid for setting up these types of problems.

Homework: pp. 91-92Problems 5.19, 5.20 Mark G. Haug5. Lets Make a Deal! Mark G. Haug101 Special Problems: 1.7 Mark G. Haug6. Life and Times of Orchestra Conductors

A study found that the average life expectancy of famous male orchestral conductors was 73.4 years, significantly higher than the life expectancy for all males, which was 68.5 years at the time of the study. Jane Brody in her New York Times health column reported that this was thought to be due to arm exercise. Mark G. HaugOutline of ProbabilityA. PercentagesB. RiskC. Formal probabilityD. Randomness E. Miscellaneous Problems Mark G. HaugProbabilityCollecting DataMaking SoundDecisions1) Correlation2) ModelingDescribing DataNatural Phenomena Mark G. HaugSamplingAn infamous case: the Literary Digest presidential poll of 1936. Mark G. HaugPopular Analysis: Since Republicans were wealthier as a group, they were over-represented in the sample of ten million (i.e., more likely to own phones and automobiles). Thus, the results were biased toward the Republican, Alf Landon. An infamous case: the Literary Digest presidential poll of 1936. Mark G. HaugAnother Analysis: Only 2.3 million of the 10 million responded to the survey. Voluntary response is nearly always biased, since volunteers are typically different than those who dont volunteer. An infamous case: the Literary Digest presidential poll of 1936. Mark G. HaugReading

Harvard Business School Materials:Selection Bias and the Perils of Benchmarking Mark G. HaugLife and Times of Orchestra Conductors

A study found that the average life expectancy of famous male orchestral conductors was 73.4 years, significantly higher than the life expectancy for all males, which was 68.5 years at the time of the study. Jane Brody in her New York Times health column reported that this was thought to be due to arm exercise. Mark G. Haug

1970 DraftLottery Mark G. HaugSampling in AccountingThe Chesapeake and Ohio (C&O) Railroad Company: There were 23,000 waybills for a six month period in a district where freight charges were divided among C&O and another railroad company.C&O can 1) examine all waybills to determine the amount due to C&O, or 2) take a sample of the waybills and make an estimate. Mark G. Haug1. Simple Random Sampling: Randomly select a number of the 23,000 waybills. 2. Stratified Random Sampling: Create strata (categories) of waybills based on some feature (e.g., total $), and then randomly select a number of the waybills within each stratum. Waybills vary from $2 to $200, with most being low and few being high. Mark G. HaugHeres how C&O stratified (based on statistical theory not covered in this course):

Waybill (Total Charges) Proportion Sampled$ 0 to $ 5.00 1%$ 5.01 to $10.00 10%$10.01 to $20.00 20%$20.01 to $40.00 50%$40.01 and over100% Mark G. HaugWaybill (Total Charges) Proportion Sampled$ 0 to $ 5.00 1%$ 5.01 to $10.00 10%$10.01 to $20.00 20%$20.01 to $40.00 50%$40.01 and over100%

This sampling scheme generated a little over 2,000 of the 23,000 waybills (9%). Using the sample data, C&O estimated the portion due to C&O to be $64,568. Mark G. HaugAnalysis of C&Os Accounting:

MethodAmountCost ofDue to C&OMethodStratified Random Sampleof 2,000 WaybillsComplete Examinationof 23,000 WaybillsDifference Mark G. HaugReading: p. 3Section 1.6

Mark G. HaugProbabilityCollecting DataDescribing DataMaking SoundDecisions1) Correlation2) ModelingNatural Phenomena Mark G. HaugOutline of Descriptive Statistics Population vs. Sample1. Qualitative vs. Quantitative Data2. Visually Presenting Data, I3. Measuring the Center of the Data4. Measuring the Variability of the Data, I5. A Special Type of Data: Proportions6. Visually Presenting Data, II7. Measuring the Variability of the Data, II Mark G. HaugOutline of Descriptive StatisticsPopulation versus Sample Mark G. HaugQualitative data is data that describes an attribute rather than a measure.

Quantitative data is data that measures, rather than describes an attribute.1. Qualitative vs. Quantitative Data Mark G. Haug 0 8 16 24 32 40 48 56 64 72 0 8 16 24 32 40 48 56 64 72134422213 Mark G. Haug 0 8 16 24 32 40 48 56 64 72134422213With a histogram, you can estimate probabilities. For example, whats the estimated probability that a lightbulb lasts longer than 40 seconds? Mark G. HaugWhat is the estimated probability that a lightbulb will last more than 64 seconds given that it survives for more than 40 seconds? P(Y>64|Y>40) = ? A. 3/24B.9/24C. 3/9 0 8 16 24 32 40 48 56 64 72134422213 Mark G. HaugReading: pp. 10-12Sections 2.1-2.3

Homework: p. 21Problem 2.2

Mark G. Haug

Mark G. HaugDeaths from CholeraAug 18-24Aug 25-31Sep 1-7Sep 8-14Sep 15-21Sep 22-28100200300400500Pump Handle RemovedSeptember 8 Mark G. Haug

Mark G. Haug3. Measuring the Center of the Data a. Meanb. Medianc. Mode Mark G. Haug3. Measuring the Center of the Data The mean is the sum of the observations divided by the number of the observations.

For example, the mean of the light bulb data is the sum of: 6, 9, 11, 14, 16, 19, 20, 22, 24, 26, 28, 29, 31, 34, 36, 44, 46, 50, 54, 56, 63, 66, 70, 70 divided by 24: (844) / (24) 35. Mark G. Haug3. Measuring the Center of the Data Population Mean:

Sample Mean:

Mark G. Haug3. Measuring the Center of the Data The median is simply the middle observation after the observations have been arranged in increasing (or decreasing) order. For example, the median of the light bulb data: 6, 9, 11, 14, 16, 19, 20, 22, 24, 26, 28, 29, 31, 34, 36, 44, 46, 50, 54, 56, 63, 66, 70, 70.Take the average of 29 and 31 since they are both in the middle: 30. Mark G. Haug3. Measuring the Center of the Data The mode is the observation that appears most frequently.

For example, the mode of the light bulb data: 6, 9, 11, 14, 16, 19, 20, 22, 24, 26, 28, 29, 31, 34, 36, 44, 46, 50, 54, 56, 63, 66, 70, 70.70 appears more frequently than any other number, so 70 is the mode. Mark G. HaugReading: pp. 47-48Section 3.8

Homework: p.49Problem 3.1 Mark G. Haug4. Measuring the Variability of the Data, Ia. Rangeb. Variancec. Standard Deviation Mark G. Haug4. Measuring the Variability of the Data, IThe range is simply the highest value minus the lowest value. For the light bulb data:

6, 9, 11, 14, 16, 19, 20, 22, 24, 26, 28, 29, 31, 34, 36, 44, 46, 50, 54, 56, 63, 66, 70, 70

the range is 70 - 6 = 64. Mark G. Haug4. Measuring the Variability of the Data, IThe variance and the standard deviation are the most abstract descriptive statistics to this point. The variance is the mean of the squared differences between each data point and the mean. The standard deviation is the square root of the variance. Mark G. Haug4. Measuring the Variability of the Data, I

Mark G. Haug4. Measuring the Variability of the Data, IThe variance and the standard deviation are given as follows:Sample Variance:

Sample Standard Deviation:

Mark G. Haug4. Measuring the Variability of the Data, IFor the light bulb data:

6, 9, 11, 14, 16, 19, 20, 22, 24, 26, 28, 29, 31, 34, 36, 44, 46, 50, 54, 56, 63, 66, 70, 70

sample mean is 35, and n is 24 :

Mark G. Haug6, 9, 11, 14, 16, 19, 20, 22, 24, 26, 28, 29, 31, 34, 36, 44, 46, 50, 54, 56, 63, 66, 70, 70

Mark G. HaugReading: p. 60 Section 4.6

Homework: p. 67Problem 4.6 Mark G. HaugThe Empirical Rule: For data that yield symmetric and mound shaped histograms (normally distributed): .340.340.135.135.025.025

Mark G. Haug 0 8 16 24 32 40 48 56 64 72134422213? Mark G. HaugExample: Suppose data exhibit a symmetric and mound shaped histogram, with a sample mean of 100 and a standard deviation of 15. P(X>115) ?P(X p-value Mark G. HaugAssociation (Continued)

Classical Experimentation Clinical Trials Cohort Studies Case-Control Studies Mark G. HaugClassical ExperimentationTheory:My new super-polymer rubber tirelasts longer than other tires.HO: super-polymer = other Mark G. HaugRandom samplesClassical Experimentation: DiagramOther tiresNew tiresApply statistical methods to determineif there is a significant difference Mark G. HaugClinical TrialsTheory:Aspirin is effective against heart attacks.HO: aspirin = placeboORHO: No Association Mark G. HaugClinical Trials: DiagramThe populationof people whomay suffer fromheart attacksVolunteersRandomly assignvolunteers into groups:placebo and aspirinIs there a difference? Mark G. HaugHeartNo HeartAttackAttack

Aspirin 10410,933Placebo 18910,842Conclusion: Since 2=26 > 3.84, we conclude that there is an association between aspirin intake and heart attacks. Mark G. HaugVolunteers124VolunteersStep 1 DietStratified RandomizationPre-TestCholesterolPre-TestCholesterolPost-TestCholesterol6 weeksof cornflakesPost-TestCholesterol6 weeksof Cheerios Mark G. Haug

Mark G. HaugCohort Study (Prospective Study)Theory:Women in executive positionsare at risk for breast cancer.HO: No Association Mark G. HaugCohort Study: DiagramThe entirepopulation ofwomen withoutbreast cancerVolunteersFollowvolunteersover timeIs there a difference inthe proportionof women who are are executives?Volunteers withoutbreast cancerVolunteers withbreast cancer Mark G. HaugBreastNo BreastCancerCancerExecutive 10 1,605Non-Exec.1,770560,010 Mark G. HaugBreastNo BreastCancerCancerExecutive 10 1,605Non-Exec. 1,770 560,010 Relative Risk (RR):

Mark G. HaugCase-Control Study (Retrospective Study)Theory:Asbestos causeslung cancer.HO: No Association Mark G. HaugCase-Control Study: DiagramWas there a difference in asbestos exposure between these two groups prior to this study?VolunteersThe entire populationof peoplePeople with lung cancerPeople withoutlung cancer Mark G. HaugCase-Control StudyLungNo Lung CancerCancerNo Exposure 71 102Exposure 44 17 Mark G. HaugCase-Control StudyLungNo Lung CancerCancerNo Exposure 71 102Exposure 44 17Odds Ratio (OR):

Mark G. HaugReading:

Blackboard: Short Story from Mark Twain.doc Mark G. HaugCausation

1. Clinical Trials: YES, but limited2. Cohort Studies: NO3. Case-Control Studies: NO Mark G. HaugThree types of an association1. Causation: A causes B

2. Common Response: Changes in A and B are caused by changes in C, which may or may not be known.

3. Confounding: Changes in B may be caused by changes in A and by changes in C, which may or may not be known. Mark G. Haug1. Causation: A causes BABA finding of association between A and B Mark G. Haug2. Common Response: Changes in A and B are caused by changes in C, which may or may not be known.ABCA finding of association between A and B Mark G. Haug3. Confounding: Changes in B may be caused by changes in A and by changes in C, which may or may not be known.ABCA finding of association between A and BAssociationbetween A and C Mark G. Haug(Finding Causation in an Association) Mark G. HaugCorrelationWhen talking about correlation in statistics, we are usually talking about a relationship that is linearExample:WeightHeight Mark G. HaugThe correlation coefficient, (rho) -- called r for sample data -- can take on any value from -1 to 1:-101Perfect inverselinear relationshipNo linear relationshipPerfectlinear relationshipPartialinverselinear relationshipPartial linear relationship Mark G. HaugYX Mark G. HaugYX Mark G. HaugYX Mark G. HaugYX Mark G. HaugYX Mark G. HaugYX Mark G. HaugYX Mark G. HaugYX Mark G. Haug

r = -0.271970 DraftLottery Mark G. HaugReading: p. 264Section 14.2 Mark G. HaugThe (linear) correlation coefficient, ryx:

Mark G. HaugThe (linear) correlation coefficient, ryx:An Example:

Mark G. Haug

xy(x2)(xy)(y2) Mark G. Haug

Mark G. HaugHO: = 0

The degrees of freedom for t is (n-2) Mark G. HaugThree types of correlation1. Causation

2. Common Response

3. Confounding Mark G. HaugCovarianceij = ijijMarket Asset AAsset B Condition Good$1.16$1.01Average$1.10$1.10Poor$1.04$1.19Adapted from Elton & Gruber,Modern Portfolio Theory and Investment Analysis, 5e Mark G. HaugProbabilityMaking SoundDecisionsCorrelationModelingCollecting DataDescribing DataNatural Phenomena Mark G. HaugRegression

Simple Linear Regression (SLR)Three Types of Uncertainty MethodCriterion for Establishing SLR (and MLR below)ExcelStatisticsMultiple Linear RegressionExcelStatistics - the bright sideStatistics - the dark sideApplicationsThe Market Model and BETATransformationsCalculating the Rate of Return for an Investment PortfolioOperations StrategyCautionary Tale about Where do babies come from?Operational Costs for Commercial AirlinesAdvertising Case Mark G. HaugSimple Linear RegressionWeightHeight

Mark G. Haugxy10b+b=y = b + m xy = a + b xInterceptSlope Mark G. HaugSimple Linear RegressionWeightHeight

Mark G. HaugRegression

Simple Linear Regression (SLR)Three Types of Uncertainty MethodCriterion for Establishing SLR (and MLR below)ExcelStatisticsMultiple Linear RegressionExcelStatistics - the bright sideStatistics - the dark sideApplicationsThe Market Model and BETATransformationsCalculating the Rate of Return for an Investment PortfolioOperations StrategyCautionary Tale about Where do babies come from?Operational Costs for Commercial AirlinesAdvertising Case Mark G. HaugSalesTime Mark G. HaugReading: The Use and Misuse of StatisticsRead Section 3 Mark G. HaugRegression


Mark G. HaugSimple Linear RegressionIf you estimate 0 and 1 with the following formulae, you will satisfy the least squares criterion.

Mark G. HaugSimple Linear Regression

Mark G. HaugSimple Linear Regression

Mark G. HaugSimple Linear RegressionWeightHeight



Mark G. HaugOrdinary Least Squares (OLS) Regression

Difference between actual weight and predicted weight.

Difference squared. Why?

Sum of each difference squared. Mark G. HaugRegression

Simple Linear Regression (SLR)Three Types of Uncertainty MethodCriterion for Establishing SLR (and MLR below)ExcelStatisticsMultiple Linear RegressionExcelStatistics - the bright sideStatistics - the dark sideApplicationsThe Market Model and BETATransformationsCalculating the Rate of Return for an Investment PortfolioOperations StrategyCautionary Tale about Where do babies come from?Operational Costs for Commercial AirlinesAdvertising Case Mark G. HaugRegression

Simple Linear Regression (SLR)Three Types of Uncertainty MethodCriterion for Establishing SLR (and MLR below)ExcelStatisticsMultiple Linear RegressionExcelStatistics - the bright sideStatistics - the dark sideApplicationsThe Market Model and BETATransformationsCalculating the Rate of Return for an Investment PortfolioOperations StrategyCautionary Tale about Where do babies come from?Operational Costs for Commercial AirlinesAdvertising Case Mark G. Haug

Mark G. Haug

Adjusted R Square: The amount of variation in Y explained by X. In this case, 79% of the variation in Y (weight) can be explained by X (height).

Standard Error: a measure of the variability between the OLS line and the data points.

Significance F: This is the p-value for the null hypothesis that the model as a whole is irrelevant. Mark G. Haug

Coefficients are the estimated values for 0 and 1. In this case 0 is Intercept and 1 is Height.

The p-value for each coefficient is the result of a hypothesis test for HO: i = 0 where i represents i=0 (in the case of the intercept), i represents i=1 (in the case of the slope), and any other subscript that is used in the model (covered later). Mark G. Haug

p-value for slope coefficient will be equal to p-value for model

for simple linear regression (SLR) only Mark G. HaugReading: pp. 265-267, 268-269, 271Sections 14.3-14.6, 14.10, 14.14

Homework: p. 272Problem 14.1 Data:Do 1, 2, and 3 by hand & 1 and 3 by Excel:Calculate correlation coefficientDetermine whether correlation coefficient is statistically significantDetermine the simple linear regression equation for these data Mark G. HaugRegression




Mark G. Haug

Mark G. Haug

Height Only

Height and Waist Mark G. HaugHomework: Blackboard HW Regression Calories.doc Mark G. HaugRegression


Height Only

Height and Waist Mark G. HaugRegression


Mark G. Haug

Mark G. Haug

Mark G. Haug



Simple Linear Regression (SLR)Three Types of Uncertainty MethodCriterion for Establishing SLR (and MLR below)ExcelStatisticsMultiple Linear RegressionExcelStatistics - the bright sideStatistics - the dark sideApplicationsThe Market Model and BETATransformationsCalculating the Rate of Return for an Investment PortfolioOperations StrategyCautionary Tale about Where do babies come from?Operational Costs for Commercial AirlinesAdvertising Case Mark G. HaugRi = The return on a stock, stock ii = Represents the component of the stock i return that is independent of the marketi = Represents the component of the stock i return that is dependent on the market (The BETA)Rm = The return of the stock market

i = Random variable that represents the uncertainty in the component of the stock i return that is independent of the market Mark G. HaugHomework:

Blackboard: data for semilog and market model hw.xls(problem #2 only) Mark G. HaugRegression



Mark G. Haug

Pt The principal after t periods of time.

P0 The principal after t=0 periods of time.Here t=0, meaning this is the initial amount invested.

iThe fixed rate of return for the investment.

tThe number of periods of time. Mark G. Haug

Mark G. Haug

Mark G. Haug

Mark G. HaugHomework:

Blackboard: data for semilog and market model hw.xls(problem #1 only) Mark G. HaugRegression

Simple Linear Regression (SLR)Three Types of Uncertainty MethodCriterion for Establishing SLR (and MLR below)ExcelStatisticsMultiple Linear RegressionExcelStatistics - the bright sideStatistics - the dark sideApplicationsThe Market Model and BETATransformationsCalculating the Rate of Return for an Investment PortfolioOperations StrategyCautionary Tale about Where do babies come from?Operational Costs for Commercial AirlinesAdvertising Case Mark G. HaugTemperatureWinning TimeColdHot Mark G. Haug

Mark G. Haug

Mark G. HaugTemperatureWinning Time Mark G. Haug

Mark G. Haug

Mark G. HaugHomework:

National Highway Traffic Safety Administrationaccident rate as a function of drivers age:

f(y) = 60.0 - 2.28(y) + 0.0232(y2) y is age, 16 y 85

Find the estimated age when accident rate is a minimum.

Answer: 49 Mark G. HaugRegression


Mark G. Haug



More seats equate to greater costs per hour:1. Larger aircraft (fixed)a. cost of airplaneb. larger staff to operatec. higher insuranced. etc2. Heavier--thus more fuel (fixed and variable)3. In flight service (variable) Mark G. Haug

Cost = $1,136.34 + $14.67 (No. of Seats)

Mark G. Haug

More seats equate to greater costs per hour:1. Larger aircraft (fixed)a. cost of airplaneb. larger staff to operatec. higher insuranced. etc2. Heavier--thus more fuel (fixed and variable)3. In flight service (variable) Mark G. Haug

Mark G. Haug



More Data: 88 Observations Total Mark G. HaugRegression

Simple Linear Regression (SLR)Three Types of Uncertainty MethodCriterion for Establishing SLR (and MLR below)ExcelStatisticsMultiple Linear RegressionExcelStatistics - the bright sideStatistics - the dark sideApplicationsThe Market Model and BETATransformationsCalculating the Rate of Return for an Investment PortfolioOperations StrategyCautionary Tale about Where do babies come from?Operational Costs for Commercial AirlinesAdvertising CaseChart321321226122622721019914816820313398

Median of Ranks Within Each MonthApproximate Median of Ranks Within Each Month(1970 Draft Lottery)

Sheet1MonthMedian of Ranks Within Each MonthJanuary213February212March261April226May227June210July199August148September168October203November133December98

Sheet1000000000000

Median of Ranks Within Each MonthApproximate Median of Ranks Within Each Month(1970 Draft Lottery)

Sheet2

Sheet3

Motor Vehicle Mfrs. Ass=n of USA v. E.P.A.

768 F. 2d 385

(D.C. Cir., 1985)Nitrous Oxide Emissions from 16 Cars:

Car

Base Fuel (B)Petrocoal (P)Difference (P-B)

Sign of Difference1

1.195

1.385

0.190

+

2

1.185

1.230

0.045

+

3

0.755

0.755

0.000

tie

4

0.715

0.775

0.060

+

5

1.805

2.024

0.219

+

6

1.807

1.792

-0.015

-

7

2.207

2.387

0.180

+

8

0.301

0.532

0.231

+

9

0.687

0.875

0.188

+

10

0.498

0.541

0.043

+

11

1.843

2.186

0.343

+

12

0.838

0.809

-0.029

-

13

0.720

0.900

0.180

+

14

0.580

0.600

0.020

+

15

0.630

0.720

0.090

+

16

1.440

1.040

-0.400

-

Mean

1.075

1.159

0.0841

S. Dev.

0.5796

0.6134

0.1672

Motor Vehicle Mfrs. Ass=n of USA v. E.P.A.

768 F. 2d 385

(D.C. Cir., 1985)Nitrous Oxide Emissions from 16 Cars:

Car

Base Fuel (B)Petrocoal (P)Difference (P-B)

Sign of Difference1

1.195

1.385

0.190

+

2

1.185

1.230

0.045

+

3

0.755

0.755

0.000

tie

4

0.715

0.775

0.060

+

5

1.805

2.024

0.219

+

6

1.807

1.792

-0.015

-

7

2.207

2.387

0.180

+

8

0.301

0.532

0.231

+

9

0.687

0.875

0.188

+

10

0.498

0.541

0.043

+

11

1.843

2.186

0.343

+

12

0.838

0.809

-0.029

-

13

0.720

0.900

0.180

+

14

0.580

0.600

0.020

+

15

0.630

0.720

0.090

+

16

1.440

1.040

-0.400

-

Mean

1.075

1.159

0.0841

S. Dev.

0.5796

0.6134

0.1672

Sheet1Height (ft.)Weight (lbs.)5.81455.51504.91006.21955.92055.41256.12155.6165

Sheet2

Sheet3

Sheet1XY5.814533.64210258415.515030.25225008254.910024.01100004906.219538.443802512095.920534.814202512105.412529.16156256756.121537.214622513125.616531.362722592445.41300258.8822265074855.6750162.500074850.420040.3556

Sheet2

Sheet3

Sheet1X - 5.675Y - 162.5(X-5.675)(Y-162.5)5.81450.13-18-2.190.01565.5150-0.18-132.190.03064.9100-0.77-6348.440.60066.21950.533317.060.27565.92050.23439.560.05065.4125-0.27-3810.310.07566.12150.435322.310.18065.6165-0.083-0.190.0056mean = 5.675mean = 162.5107.50001.2350

Sheet2

Sheet3

Sheet1SUMMARY OUTPUTRegression StatisticsMultiple R0.9059882386R Square0.8208146885Adjusted R Square0.79095047Standard Error18.4513439014Observations8ANOVAdfSSMSFSignificance FRegression19357.28744939279357.287449392727.48488752350.0019335297Residual62042.7125506073340.4520917679Total711400CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%Intercept-331.477732793594.4493081682-3.50958349220.0126778238-562.5870332952-100.3684322919-562.5870332952-100.3684322919Height87.04453441316.60330418725.24260312470.001933529746.4176829183127.671385907746.4176829183127.6713859077

ht-wtHeightWeightPredicted Wt [ Wt = -331 + 87 (Ht) ]DifferenceDiff Sq5.8145173.3828.38805.465.5150Coefficients147.27-2.737.474.9100-331.477732793595.04-4.9624.606.219587.044534413208.2013.20174.205.9205182.09-22.91525.105.4125138.5613.56183.956.1215199.49-15.51240.445.6165155.97-9.0381.512042.71

caloriesCaloriesFatCarbsProtein110222312002811609201903152600.501410002702402.546810002401609162110222220053722401319121601367100024215010152201.50214032611407172110123215062032005382130619234020393200538210002701101240110124018062944202339151701.53351002012012511200292800210600.51411201.525360283150524214062021806294120032016082121201252180629421053281405223130520312002721708241900183604.523600.501310002601101240190730122034010110123118082332404.5457902.551319082641201252140717213042221901443100201302.526121012242000019082651801376110025210002601200281180823322034253902233152404.5429803.510127064692304419110123210081720053732206385604.52312012821406211803.5100210341418062941101232200537223073572801435414081721200.526212032317021022101447902.5171701.541021073071101.52312401425412002921207015110871.522813263340173413160816414032722602.5471111012321507183150718323014252190826590022015012972000.54352001.5417901202200538215032753601345142201223619082659002201809222200131921101241602831501015214032711806294440195116902.51341300.5301110222115072016011311201251140327215091541201.52521404.5213602832002217062532000.54351903.5357140520320060361406220160333023013253120322213052031003.5171110026214051941608164250738816081642301325337020381012032311001.52211809213180921390218212032221300320250738813071431100251270153241806294800181402.551150129715014261100.5262140816219082651702370160527212014001709211601.510125073881001232907351302.520614032721407172804101600.513270211110002322005372160919214071721500380702.5832801237614061942507388200928211002364101075512051282602.547111205192700.51255001202602.547111908264160111511100253702111160040119015572602.547119011921000270210244515072013001042111407172700152150818114032723601345142201.54762602.54711150622214062111300311902181100220116023341300331100027024083211601.51021200291210052032014341515052421305203130422234015371626013314

gasolineyeartotal gasprice indexper capitaprice indexprice indexprice indexaggregateaggregateaggregateconsumptionof gasolinereal disposableof new carsof used carsfor publicprice indexprice indexprice indexindexincome indextransportationdurable goodsnon-durableconsumer servYRTGCIPIGPCRDIIPINCPIUCPIPTAPIDGAPINDGAPICS19601.001.001.001.001.001.001.001.001.0019611.01.991.011.001.041.041.011.011.0219621.06.991.041.001.131.081.031.021.0419631.09.991.06.991.151.091.041.041.0619641.15.991.11.991.201.111.061.051.0819651.201.031.16.971.191.131.061.071.1019661.271.051.21.951.161.181.071.111.1319671.321.081.24.961.201.231.091.131.1719681.411.101.28.981.231.291.131.181.2219691.511.131.311.001.231.391.161.241.2819701.601.141.351.031.251.591.191.291.3519711.681.151.381.071.321.701.231.341.4319721.751.161.421.061.321.771.251.381.4919731.831.281.501.061.411.791.271.501.5719741.741.731.471.121.471.831.361.731.7019751.791.851.481.221.751.961.481.861.8419761.861.921.521.302.012.151.571.931.9819771.922.031.551.372.192.251.642.032.1519782.012.121.611.472.232.321.732.172.3119791.922.871.631.592.402.471.852.422.5019801.753.991.611.722.493.112.012.702.7819811.744.441.621.823.073.852.162.933.0719821.764.211.611.893.554.272.253.023.3119831.854.071.651.943.944.482.343.083.5219841.894.011.732.004.494.762.343.173.7019851.904.041.752.064.544.972.353.253.8819862.083.161.792.144.345.262.373.234.05

storksCountryStorks (Pairs)Birth Rate (1000/yr)Albania10083Austria30087Belgium1118Bulgaria5000117Denmark959France140774Germany3300901Greece2500106Holland4188Hungary5000124Italy5551Poland30000610Portugal1500120Romania5000367Spain8000439Switzerland15082Turkey250001576

sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832

aircraftAll Numbers Are Based Upon AveragesModelPassenger SeatsFlight Speed (miles/hr)Length of Flight (miles)Cost per Hour (dollars)B747-10041051828826567B747-40040053950637075B747-200/30036952932317790L-1011-100/20030549813635081B-77729151324514194DC-10-1028649814935092DC-10-4028450419634684DC-10-3027251623795859A300-60026646711265123MD-1126052432536335L1011-50022252329954764B767-300ER21649523313616

marathonYearTempMenWomen197875132.2152.5197980131.7147.6198050129.7145.7198154128.2145.5198252129.5147.2198359129.0147.0198479134.9149.5198572131.6148.6198665131.1148.1198764131.0150.3198867128.3148.1198956128.0145.5199073132.7150.8199157129.5147.5199251129.5144.7199373130.1146.4199470131.4147.6199562131.0148.1199649129.9148.3199761128.2148.7199855128.8145.3

viaticallifeagegenderillnessinitial monthexpectancy1 = female1 = cancer1 = january1 = greater than0 = male2 = respiratory2 = february0 = less than3 = cardiovascular3 = march4 = other(and so forth)NETAGEGENDERILLNESSMONTH18612307802508701418312604602105912120771190611412166125180145070023051122060038192132064037071121193125036128064036062131018913120691330780212047027065124071031207201100850350521350740111189125068043087117065014180129048035190145037111004603307204507302101931230571380640370520310050041204401605713806002120690310076118080028061116052132077025079034048041076133066032181124071124085029049021108504708212909403907403606003408002309713904302207501105112817612306613110690410063048049022059113083131218702100950210611180520180570251991310871190680410057022076113180121105003407204805413120750270690260830340660180800341921310580380761390780430660430680331821340860130620230731490610370631331790430891230630150870340671390590210066149095033074012198126082114079039073138069017082023060033069011007514110900210080149072022057026196147






sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832









sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832









sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832









sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832




Sheet3SUMMARY OUTPUTRegression StatisticsMultiple R0.91R Square0.82Adjusted R Square0.79Standard Error18.45Observations8ANOVAdfSSMSFSignificance FRegression19357.28749357.287427.48490.0019Residual62042.7126340.4521Total711400CoefficientsStandard Errort StatP-valueL 95%U 95%L 95%U 95%Intercept-331.4894.4493-3.50960.0127-562.59-100.37-562.59-100.37Height87.0416.60335.24260.001946.42127.6746.42127.67

ht-wtHeightWeight5.81455.51504.91006.21955.92055.41256.12155.6165

sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832


storksCountryStorks (Pairs)Human Birth Rate (1000/yr)Albania10083Austria30087Belgium1118Bulgaria5000117Denmark959France140774Germany3300901Greece2500106Holland4188Hungary5000124Italy5551Poland30000610Portugal1500120Romania5000367Spain8000439Switzerland15082Turkey250001576


cbcObservationNetworkMonthDayRatingFactStarsPrevious RatingCompetition1BBS1115.60114.214.52BBS1710.81015.317.23BBS1714.10113.814.44BBS1116.81112.815.35BBS2114.31112.413.36BBS2117.11112.915.17BBS318.90010.814.98BBS3716.21013.311.69BBS479.40112.312.810BBS5110.20110.715.611BBS579.40010.714.512BBS5112.10110.115.613BBS5110.7108.617.014BBS9715.0109.88.215BBS9710.20011.713.516BBS9710.30110.115.217BBS10710.80110.913.118BBS10714.41015.912.619BBS11714.41112.114.220BBS11713.61011.411.921ABN1714.60019.314.422ABN1210.80116.315.223ABN1716.20020.114.424ABN1212.80014.813.125ABN1716.00119.313.526ABN2718.90117.813.027ABN2214.01114.313.828ABN3719.51116.211.829ABN3214.71013.815.730ABN3716.30118.011.431ABN3715.81017.713.332ABN3717.10117.111.333ABN3211.50013.813.134ABN3716.01015.311.835ABN3211.70116.614.336ABN4214.20013.611.437ABN4711.20014.314.438ABN4210.90012.413.039ABN4713.30113.110.140ABN4715.51017.012.441ABN4216.61013.611.842ABN5716.31016.512.843ABN5715.80115.711.344ABN5213.31010.712.845ABN9715.40117.310.946ABN9214.70015.513.947ABN9715.50017.412.648ABN9214.71015.314.049ABN10715.91018.410.550ABN10713.81024.712.151ABN10210.00114.212.952ABN11712.90116.918.653ABN11215.41015.912.454ABN11714.50219.414.255ABN12718.80216.714.756ABN12216.70014.910.157ABN12212.80016.312.058ABN12716.80115.710.159CBC1714.0018.214.860CBC1111.31013.013.261CBC1113.60013.715.162CBC2712.9108.816.063CBC2113.21013.117.064CBC2716.0106.915.865CBC2114.61113.817.466CBC2716.60116.814.467CBC3117.51014.814.268CBC3711.60010.014.069CBC478.9008.613.070CBC4115.60013.316.871CBC479.2016.812.172CBC4111.80012.912.073CBC4711.0005.314.774CBC419.51013.017.375CBC9711.60010.112.876CBC9113.31013.120.377CBC9113.61014.118.378CBC10112.40013.620.279CBC10113.81010.216.680CBC10711.91011.812.281CBC10114.60014.914.982CBC11115.81113.417.283CBC11115.40113.616.884CBC11112.80012.714.685CBC12712.80012.018.686CBC12115.10014.115.587CBC12111.40111.216.488CBC12119.11012.615.4



sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832







sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832







sp500YearS&P 5001976$10,0001977$9,2811978$9,8861979$11,7101980$15,5091981$14,7531982$17,9241983$21,9491984$23,3151985$30,6921986$36,4071987$38,2921988$44,6101989$58,7021990$56,8711991$74,1591992$79,8041993$87,832




cbcObservationNetworkMonthDayRatingFactStarsPrevious RatingCompetition1BBS1115.60114.214.52BBS1710.81015.317.23BBS1714.10113.814.44BBS1116.81112.815.35BBS2114.31112.413.36BBS2117.11112.915.17BBS318.90010.814.98BBS3716.21013.311.69BBS479.40112.312.810BBS5110.20110.715.611BBS579.40010.714.512BBS5112.10110.115.613BBS5110.7108.617.014BBS9715.0109.88.215BBS9710.20011.713.516BBS9710.30110.115.217BBS10710.80110.913.118BBS10714.41015.912.619BBS11714.41112.114.220BBS11713.61011.411.921ABN1714.60019.314.422ABN1210.80116.315.223ABN1716.20020.114.424ABN1212.80014.813.125ABN1716.00119.313.526ABN2718.90117.813.027ABN2214.01114.313.828ABN3719.51116.211.829ABN3214.71013.815.730ABN3716.30118.011.431ABN3715.81017.713.332ABN3717.10117.111.333ABN3211.50013.813.134ABN3716.01015.311.835ABN3211.70116.614.336ABN4214.20013.611.437ABN4711.20014.314.438ABN4210.90012.413.039ABN4713.30113.110.140

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