ࡱ> kmhijs\sZiq Zbjbj I'RY ??cccwww8tw C6?$S2i2i2i2555BBBBBBBAEG^Bc95"599B??i2i2PB???9B?i2ci2B?9B??AJBi2`:¸wE:6BBB0 CB,AI{;AIJBAIcJBP56?57t7Z555BBg>555 C9999AI555555555 : Descriptive Statistics: TermMeaningPopulation FormulaSample FormulaExample {1,16,1,3,9}SortSort values in increasing order{1,1,3,9,16}MeanAverage EMBED Equation.3  EMBED Equation.3 6MedianThe middle value half are below and half are above3ModeThe value with the most appearances1VarianceThe average of the squared deviations between the values and the mean EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 (1-6)2 + (1-6) 2 + (3-6)2 + (9-6)2 + (16-6)2 divided by 5 values = 168/5 = 33.6Standard DeviationThe square root of Variance, thought of as the average deviation from the mean. EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Square root of 33.6 = 5.7966Coefficient of VariationThe variation relative to the value of the mean EMBED Equation.3 5.7966 divided by 6 = 0.9661MinimumThe minimum value1MaximumThe maximum value16RangeMaximum minus Minimum16 1 = 15 Probability Terms: TermMeaningNotationExample* (see footnote)ProbabilityFor any event A, probability is represented within 0 ( P (1.P()0.5Random ExperimentA process leading to at least 2 possible outcomes with uncertainty as to which will occur.Rolling a diceEventA subset of all possible outcomes of an experiment.Events A and BIntersection of EventsLet A and B be two events. Then the intersection of the two events is the event that both A and B occur (logical AND).A(BThe event that a 2 appearsUnion of EventsThe union of the two events is the event that A or B (or both) occurs (logical OR).A(BThe event that a 1, 2, 4, 5 or 6 appearsComplementLet A be an event. The complement of A is the event that A does not occur (logical NOT).EMBED UnknownThe event that an odd number appearsMutually Exclusive EventsA and B are said to be mutually exclusive if at most one of the events A and B can occur.A and B are not mutually exclusive because if a 2 appears, both A and B occurCollectively Exhaustive EventsA and B are said to be collectively exhaustive if at least one of the events A or B must occur.A and B are not collectively exhaustive because if a 3 appears, neither A nor B occurBasic OutcomesThe simple indecomposable possible results of an experiment. One and exactly one of these outcomes must occur. The set of basic outcomes is mutually exclusive and collectively exhaustive.Basic outcomes 1, 2, 3, 4, 5, and 6Sample SpaceThe totality of basic outcomes of an experiment.{1,2,3,4,5,6}* Roll a fair die once. Let A be the event an even number appears, let B be the event a 1, 2 or 5 appears Probability Rules: If events A and B are mutually exclusiveIf events A and B are NOT mutually exclusive Term Equals Area:  EMBED PBrush  Term Equals Venn:  EMBED PBrush   EMBED PBrush  P(A)= P(A)  EMBED PBrush   P(A)= P(A)  EMBED PBrush   EMBED PBrush  P(EMBED Unknown)= 1 - P(A)  EMBED PBrush   P(EMBED Unknown)= 1 - P(A)  EMBED PBrush   EMBED PBrush  P(A(B)= 0  EMBED PBrush   P(A(B)=P(A) * P(B) only if A and B are independent  EMBED PBrush   EMBED PBrush  P(A(B)= P(A) + P(B)  EMBED PBrush   P(A(B)= P(A) + P(B) P(A(B)  EMBED PBrush   EMBED PBrush   P(A|B)= [Bayes' Law: P(A holds given that B holds)] EMBED Unknown  EMBED PBrush  (  EMBED PBrush    EMBED PBrush  (  EMBED PBrush  General probability rules: 1) If P(A|B) = P(A), then A and B are independent events! (for example, rolling dice one after the other). 2) If there are n possible outcomes which are equally likely to occur: P(outcome i occurs) = EMBED Unknown for each i ( [1, 2, ..., n] *Example: Shuffle a deck of cards, and pick one at random. P(chosen card is a 10() = 1/52. 3) If event A is composed of n equally likely basic outcomes: P(A) =EMBED Unknown *Example: Suppose we toss two dice. Let A denote the event that the sum of the two dice is 9. P(A) = 4/36 = 1/9, because there are 4 out of 36 basic outcomes that will sum 9.P(A(B) = P(A|B) * P(B)P(A(B) = P(B|A) * P(A) P(A)= P(A(B) + P(A(EMBED Unknown) = P(A|B)P(B) + P(A|EMBED Unknown)P(EMBED Unknown )  EMBED PBrush  +  EMBED PBrush    EMBED PBrush  +  EMBED PBrush  *Example: Take a deck of 52 cards. Take out 2 cards sequentially, but dont look at the first. The probability that the second card you chose was a ( is the probability of choosing a ( (event A) after choosing a ( (event B), plus the probability of choosing a ( (event A) after not choosing a ( (event B), which equals (12/51)(13/52) + (13/51)(39/52) = 1/4 = 0.25.  Random Variables and Distributions: To calculate the Expected Value  EMBED Equation.3 , use the following table: ( * ( = (EventPayoffProbabilityWeighted Payoff[name of first event][payoff of first event in $][probability of first event 0(P(1][product of Payoff * Probability][name of second event][payoff of second event in $][probability of second event 0(P(1][product of Payoff * Probability][name of third event][payoff of third event in $][probability of third event 0(P(1][product of Payoff * Probability]* See example in BOOK 1 page 54Total (Expected Payoff):[total of all Weighted Payoffs above] To calculate the Variance Var(X) =EMBED Unknown and Standard Deviation EMBED Unknown, use: ( - ( = ( ^2= ( * ( = (EventPayoffExpected PayoffError(Error)2ProbabilityWeighted (Error)2[1st event][1st payoff][Total from above][1st payoff minus Expected Payoff]1st Error squared1st events probability1st (Error)2 * 1st events probability[2nd event][2nd payoff][Total from above][2nd payoff minus Expected Payoff]2nd Error squared2nd events probability2nd (Error)2 * 2nd events probability[3rd event][3rd payoff][Total from above][3rd payoff minus Expected Payoff]3rd Error squared3rd events probability3rd (Error)2 * 3rd events probabilityVariance:[total of above]Std. Deviation:[square root of Variance]Counting Rules: TermMeaningFormulaExample Basic Counting RuleThe number of ways to pick x things out of a set of n (with no regard to order). The probability is calculated as 1/x of the result. EMBED UnknownThe number of ways to pick 4 specific cards out of a deck of 52 is: 52!/((4!)(48!)) = 270,725, and the probability is 1/270,725 = 0.000003694Bernoulli ProcessFor a sequence of n trials, each with an outcome of either success or failure, each with a probability of p to succeed the probability to get x successes is equal to the Basic Counting Rule formula (above) times px(1-p)n-x.  EMBED Equation.3  EMBED Equation.3 If an airline takes 20 reservations, and there is a 0.9 probability that each passenger will show up, then the probability that exactly 16 passengers will show is: EMBED Unknown (0.9)16(0.1)4 = 0.08978Bernoulli Expected ValueThe expected value of a Bernoulli Process, given n trials and p probability. E(X) = npIn the example above, the number of people expected to show is: (20)(0.9) = 18Bernoulli VarianceThe variance of a Bernoulli Process, given n trials and p probability. Var(X) = np(1 - p)In the example above, the Bernoulli Variance is (20)(0.9)(0.1) = 1.8Bernoulli Standard DeviationThe standard deviation of a Bernoulli Process: ((X) = EMBED UnknownIn the example above, the Bernoulli Standard Deviation is EMBED Unknown= 1.34Linear Transformation RuleIf X is random and Y=aX+b, then the following formulas apply:E(Y) = a*E(X) + b Var (Y) = a2*Var(X) ( (Y) = |a|*((X) Uniform Distribution:  EMBED PBrush Term/MeaningFormula Expected Value EMBED Equation.3 EMBED Unknown Variance EMBED Equation.3 EMBED Unknown Standard Deviation EMBED Equation.3 EMBED Unknown Probability that X falls between c and d EMBED Equation.3  EMBED Equation.3  Normal Distribution:  EMBED PBrush z0.000.010.020.030.040.050.060.070.080.090.0.0000.0040.0080.0120.0160.0199.0239.0279.0319.03590.1.0398.0438.0478.0517.0557.0596.0636.0675.0714.07530.2.0793.0832.0871.0910.0948.0987.1026.1064.1103.11410.3.1179.1217.1255.1293.1331.1368.1406.1443.1480.15170.4.1554.1591.1628.1664.1700.1736.1772.1808.1844.18790.5.1915.1950.1985.2019.2054.2088.2123.2157.2190.22240.6.2257.2291.2324.2357.2389.2422.2454.2486.2517.25490.7.2580.2611.2642.2673.2704.2734.2764.2794.2823.28520.8.2881.2910.2939.2967.2995.3023.3051.3078.3106.3133Probability Density Function: EMBED Unknown where ( ( 3.1416 and e ( 2.71830.9.3159.3186.3212.3238.3264.3289.3315.3340.3365.33891.0.3413.3438.3461.3485.3508.3531.3554.3577.3599.36211.1.3643.3665.3686.3708.3729.3749.3770.3790.3810.38301.2.3849.3869.3888.3907.3925.3944.3962.3980.3997.40151.3.4032.4049.4066.4082.4099.4115.4131.4147.4162.4177Standard Deviations away from the mean: EMBED Unknown (Z and ( are swappable!)1.4.4192.4207.4222.4236.4251.4265.4279.4292.4306.43191.5.4332.4345.4357.4370.4382.4394.4406.4418.4429.44411.6.4452.4463.4474.4484.4495.4505.4515.4525.4535.45451.7.4554.4564.4573.4582.4591.4599.4608.4616.4625.4633P(a ( X ( b) = area under fX(x) between a and b: EMBED UnknownEMBED Unknown1.8.4641.4649.4656.4664.4671.4678.4686.4693.4699.47061.9.4713.4719.4726.4732.4738.4744.4750.4756.4761.47672.0.4772.4778.4783.4788.4793.4798.4803.4808.4812.48172.1.4821.4826.4830.4834.4838.4842.4846.4850.4854.4857Standard Normal Table - seven usage scenarios:2.2.4861.4864.4868.4871.4875.4878.4881.4884.4887.4890  EMBED PBrush  =  EMBED PBrush  +  EMBED PBrush   EMBED PBrush  =  EMBED PBrush  +  EMBED PBrush 2.3.4893.4896.4898.4901.4904.4906.4909.4911.4913.49162.4.4918.4920.4922.4925.4927.4929.4931.4932.4934.4936  EMBED PBrush  =  EMBED PBrush  -  EMBED PBrush   EMBED PBrush  =  EMBED PBrush  -  EMBED PBrush 2.5.4938.4940.4941.4943.4945.4946.4948.4949.4951.49522.6.4953.4955.4956.4957.4959.4960.4961.4962.4963.4964  EMBED PBrush  =  EMBED PBrush  -  EMBED PBrush   EMBED PBrush  =  EMBED PBrush  -  EMBED PBrush 2.7.4965.4966.4967.4968.4969.4970.4971.4972.4973.49742.8.4974.4975.4976.4977.4977.4978.4979.4979.4980.4981  EMBED PBrush  =  EMBED PBrush  +  EMBED PBrush  2.9.4981.4982.4982.4983.4984.4984.4985.4985.4986.49863.0.4987.4987.4987.4988.4988.4989.4989.4989.4990.4990Correlation: If X and Y are two different sets of data, their correlation is represented by Corr(XY), rXY, or (XY (rho). If Y increases as X increases, 0 < ( XY < 1. If Y decreases as X increases, -1 < ( XY < 0. The extremes ( XY = 1 and ( XY = -1 indicated perfect correlation info about one results in an exact prediction about the other. If X and Y are completely uncorrelated, (XY = 0. The Covariance of X and Y, Cov(XY) , has the same sign as ( XY, has unusual units and is usually a means to find ( XY. TermFormulaNotesCorrelation EMBED Equation.3 Used with Covariance formulas belowCovariance (2 formulas) EMBED Equation.3  (difficult to calculate)Sum of the products of all sample pairs distance from their respective means multiplied by their respective probabilities EMBED Equation.3 Sum of the products of all sample pairs multiplied by their respective probabilities, minus the product of both meansFinding Covariance given Correlation EMBED Equation.3  Portfolio Analysis: TermFormulaExample*Mean of any Portfolio S EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 = (8.0%)+ (11.0%) = 8.75% UncorrelatedPortfolio Variance(2 EMBED Equation.3 (2 = ()2(0.5)2 + ()2(6.0)2 = 2.3906Portfolio Standard Deviation( EMBED Equation.3 ( = 1.5462 CorrelatedPortfolio Variance EMBED Equation.3 Portfolio Standard Deviation EMBED Equation.3 * Portfolio S composed of Stock A (mean return: 8.0%, standard deviation: 0.5%) and Stock B (11.0%, 6.0% respectively) The Central Limit Theorem Normal distribution can be used to approximate binominals of more than 30 trials (n(30):Continuity Correction Unlike continuous (normal) distributions (i.e. $, time), discrete binomial distribution of integers (i.e. # people) must be corrected:TermFormulaOld cutoffNew cutoff EMBED PBrush MeanE(X) = npP(X>20)P(X>20.5)VarianceVar(X) = np(1 - p)P(X<20)P(X<19.5)Standard Deviation((X) = EMBED UnknownP(X(20)P(X(19.5)P(X(20)P(X(20.5) Sampling Distribution of the Mean If the Xi's are normally distributed (or n ( 30), then  EMBED Equation.3  is normally distributed with:Sampling Distribution of a Proportion If, for a proportion, n ( 30 then  EMBED Equation.3  is normally distributed with:TermFormulaTermFormulaMean(MeanpStandard Error of the Mean EMBED Equation.3 Standard Deviation EMBED Equation.3  Confidence Intervals: ParameterConfidence IntervalUsageSample(1 EMBED Equation.3  EMBED Equation.3 NormalKnown ( EMBED Equation.3  EMBED Equation.3 LargeUnknown (2 EMBED Equation.3  EMBED Equation.3 NormalSmallUnknown (3 EMBED Equation.3  EMBED Equation.3 BinomialLarge4 EMBED Equation.3  EMBED Equation.3 NormalMatched pairs5 EMBED Equation.3 EMBED Equation.3NormalKnown (, Independent Samples EMBED Equation.3 EMBED Equation.3Large6 EMBED Equation.3 EMBED Equation.3BinomialLarge Formulae Guidet-tabled. f.0.1000.0500.0250.0100.00513.0786.31412.70631.82163.65621.8862.9204.3036.9659.92531.6382.3533.1824.5415.84141.5332.1322.7763.7474.60451.4762.0152.5713.3654.03261.4401.9432.4473.1433.70771.4151.8952.3652.9983.49981.3971.8602.3062.8963.35591.3831.8332.2622.8213.250101.3721.8122.2282.7643.169111.3631.7962.2012.7183.106121.3561.7822.1792.6813.055131.3501.7712.1602.6503.012141.3451.7612.1452.6242.977151.3411.7532.1312.6022.947161.3371.7462.1202.5832.921Confidence Level to Z-Value Guide171.3331.7402.1102.5672.898Confidence LevelZ(/2 (2-Tail)Z( (1-Tail)181.3301.7342.1012.5522.87880%( = 20%1.280.84191.3281.7292.0932.5392.86190%( = 10%1.6451.28201.3251.7252.0862.5282.84595%( = 5%1.961.645211.3231.7212.0802.5182.83199%( = 1%2.5752.325221.3211.7172.0742.5082.819c( = 1.0-cZ(c/2)z(c-0.5)231.3191.7142.0692.5002.807241.3181.7112.0642.4922.797Determining the Appropriate Sample Size251.3161.7082.0602.4852.787TermNormal Distribution FormulaProportion Formula261.3151.7062.0562.4792.779Sample Size (for +/- e)EMBED Equation.3 EMBED Equation.3  271.3141.7032.0522.4732.771281.3131.7012.0482.4672.763Hypothesis Testing: Two-tailedLower-tailUpper-tailTest TypeTest StatisticHaCritical ValueHaCritical ValueHaCritical ValueSingle ( (n ( 30) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Single ( (n ( 30) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Single p (n ( 30) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Diff. between two (s EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Diff. between two ps EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  Classic Hypothesis Testing ProcedureStepDescriptionExample1Formulate Two HypothesesThe hypotheses ought to be mutually exclusive and collectively exhaustive. The hypothesis to be tested (the null hypothesis) always contains an equals sign, referring to some proposed value of a population parameter. The alternative hypothesis never contains an equals sign, but can be either a one-sided or two-sided inequality.H0: ( = 0 HA: ( < 02Select a Test StatisticThe test statistic is a standardized estimate of the difference between our sample and some hypothesized population parameter. It answers the question: If the null hypothesis were true, how many standard deviations is our sample away from where we expected it to be? EMBED Equation.3 3Derive a Decision RuleThe decision rule consists of regions of rejection and non-rejection, defined by critical values of the test statistic. It is used to establish the probable truth or falsity of the null hypothesis.We reject H0 if EMBED Equation.3.4Calculate the Value of the Test Statistic; Invoke the Decision Rule in light of the Test StatisticEither reject the null hypothesis (if the test statistic falls into the rejection region) or do not reject the null hypothesis (if the test statistic does not fall into the rejection region. EMBED Equation.3  EMBED Equation.3  Regression: StatisticSymbolRegression StatisticsIndependent VariablesX1,XkMultiple R0.9568R Square0.9155Dependent Variable (a random variable)YAdjusted R Square0.9015Standard Error6.6220Dependent Variable (an individual observation among sample)YiObservations15ANOVAIntercept (or constant); an unknown population parameter EMBED Equation.3 dfSSMSFSignificance FRegression25704.02732852.013765.03910.0000Estimated intercept; an estimate of  EMBED Equation.3  EMBED Equation.3 Residual12526.208743.8507Total146230.2360Slope (or coefficient) for Independent Variable 1 (unknown) EMBED Equation.3 CoefficientsStandard Errort StatP-valueIntercept-20.37229.8139-2.07580.0601Estimated slope for Independent Variable 1; an estimate of  EMBED Equation.3   EMBED Equation.3 Size (100 sq ft)4.31170.410410.50590.0000Lot Size (1000 sq ft)4.71770.76466.17050.0000 Statistic (Mapped to Output Above)SymbolFormulaStatistic (Mapped to Output Above)SymbolFormulaDependent Variable (sample mean of n observations) EMBED Equation.3  EMBED Equation.3 0.9155 R-square (Coefficient of Determination) EMBED Equation.3  EMBED Equation.3 Dependent Variable (estimated value for a given vector of independent variables) EMBED Equation.3  EMBED Equation.3 0.9568 Multiple R (Coefficient of Multiple Correlation) EMBED Equation.3  EMBED Equation.3 Error for observation i. The unexplained difference between the actual value of Yi and the prediction for Yi based on our regression model. EMBED Equation.3  EMBED Equation.3 0.9015 Adjusted R-square EMBED Equation.3  EMBED Equation.3 6230.2360 Total Sum of Squares EMBED Equation.3  (or SST) EMBED Equation.3   EMBED Equation.3 6.6220 Standard Error (a.k.a. Standard Error of the Estimate) EMBED Equation.3  EMBED Equation.3 526.2087 Sum of Squares due to Error EMBED Equation.3  EMBED Equation.3 -2.0758 t-statistic for testing  EMBED Equation.3 vs.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 43.8507 Mean Squares due to Error EMBED Equation.3  EMBED Equation.3 0.0601 p-value for testing  EMBED Equation.3  vs.  EMBED Equation.3 p-value EMBED Equation.3 5704.0273 Sum of Squares due to Regression EMBED Equation.3  EMBED Equation.3 65.0391 F EMBED Equation.3  EMBED Equation.3 2852.0137 Mean Squares due to Regression EMBED Equation.3  EMBED Equation.3  EMBED Equation.3       Statistics Cheat Sheets  PAGE 1137 Yoavi Liedersdorf (MBA03) \]b     < > @ J ȷ򛑀rjthO}EHOJQJU j;T< hO}UVmHnHuhO}EHOJQJjΟhO}EHOJQJUj,hO}EHOJQJU jG< hO}UVmHnHuhO}OJQJjhO}OJQJU hO}5\hO}5CJ\ hO}CJhO}hO}56\]'%8G\]Dkd$$Iflr"3 S"*G              064 la $$Ifa$ $$Ifa$]bIkd$$Iflr"3 S"*G         064 la$If $$Ifa$  RI? $If $$Ifa$kd$$Iflr"3 S"*G    064 la$If     < = > @ RI 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