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1 TLI\TSTICS FOR INTRODUGTORY COURSESJ Statistics - A set of tools for collecting,oreanizing, presenting, and analyzingnumerical facts or . Descriptive Statistics - procedures used toorganize and present data in a convenient,useable. and communicable Inferential Statistics - procedures employedto arrive at broader generalizations orinferences from sample data to STATISTIC - A number describing a samplecharacteristic. Results from the manipulationof sample data according to certain DATA - Characteristics or numbers thatare collected by POPULATION - A complete set of actualor potential PARAMETER - A number describing apopulation characteristic; typically, inferredfrom sample SAMPLE - A subset of the populationselected according to some RANDOM SAMPLE - A subset selectedin such a way that each member of thepopulation has an equal opportunity to beselected.

2 Numbers in afair lotteryJ VARIABLE - A phenomenon that may takeon different MEAN -The ooint in a distribution of measurementsabout which the summed deviations are equal to value of a sample or MEAN SAMPLE MEANp: +!,*,o:#2*,Note: The mean ls very sensltlve to extreme measure-ments that are not balanced on both WEIGHTED MEAN - Sum of a set of observationsmultiplied by their respective weights, divided by thesum of the weights: 9, *, *,WEIGHTED MEAN -L-,\r*'where xr, : weight,'x, - observation; G : number ofobservaiion grdups.'Calculated from a or gr6upings in a frequency In the FrequencVDistribution below, the meun : culculatbd by- using frequencies for the grouped, use closs midpoints Jbr MEDIAN - Observation or potenlial observation in aset that divides the set so that the same number ofobservations lie on each side of it.

3 For an odd numberof values. it is the middle value; for an even number itis the average of the middle In the Frequency Distribution table below, themedian is MODE - Observation that occurs with the greatesttiequency. Ex. In the Frequency Distributioln nblebelow. the mode is SUM OF SOUARES fSSr- Der iations tiomthe mean. squared and summed: , (I r,),PopulationSS:I(Xi )'or Ixi'- t N_ r, \,)2 Sample SS:I(xi -x)2or Ixi2---O VARIANCE - The average of square differ-ences between observations and their SAMPLEVARIANCEVARIANCES FOH GBOUPED DATAPOPUIATION SAMPLE^{G-'{Go2:*i t,(r,-p )t s2=;1i tilm'-x;2lI ;_r t=1D STANDARD DEVIATION - Square root ofthe variance:Ex. Pop. o -nYIUfiz)D BAR GRAPH - A form of graph that usesbars to indicate the frequency of occurrenceof Histogram - a form of bar graph used rr ithinterval or ratio-scaled Interval Scale- a quantitative scale thatpermits the use of arithmetic operations.}}

4 Thezero point in the scale is Scale- same as interval scale exceplthat there is a true zero FREOUENCY CURVE - A form of graphrepresenting a frequency distribution in the formof a continuous line that traces a Cumulative Frequency Curve - a continuousline that traces a histogram where bars in all thelower classes are stacked up in the adjacenthigher class. It cannot have a negative slop .o Normal curve - bell-shaped Skewed curve - departs from symmetry andtails-off at one DATAS hows the number of times each observationoccurs when the values ofa variable are arrangedin order according to their GROTJPED FREOUENCY EilSTRIBUTION- A frequency distribution in which the valuesofthe variable have been grouped into il {il, I a rr I.)'A .l b]|, K I 3artl LQxfxtxfxt100183117411f65o991ut111117511 1166198085176116711gl086o771116819611871 7AI6911195088111111179117011119408911180 171093I118111721192091182I73111tr CUMULATUE FREOUENCY -A distribution which shows the to-tal frequency through the upper real limit ofeach CUMUIATIVE PERCENTAGE distribution which shows the to-tal percentage through the upper real limit ofeach class.}

5 !I!llrfGl:I il {. 'tlzCLASSfICum f" + CURVE^/T\./\-t-att?\CLASS f CLASS t98-10015100 SKEWED CURVE--\/\-/LEFT\J-\Probability of occurrence^t at -Number of outcomafamring EwntAoif'ent'l Ant=@D SAMPLE SPACE - All possible outcomes of TYPE OF EVENTSo Exhaustive - two or more events are said to be exhaustiveif all possible outcomes are , P (A or B ) - -two or more events are said to be non-exhaustive if they do not exhaust all possible Exclusive - Events that cannot occursimultaneously:p (A and B) = 0; and p (A or B) = p (A) + p (B).Ex. males, femalesoNon-Mutually Exclusive - Event-s that can occursimultaneously: p (A orB) = P(A) +p(B) - p(A and B)'&x. males, brown - Events whose probability is unaffectedby occurrence or nonoccurrence of each other: p(A lB) =p(A); ptB In)= p(e); and p(A and B) = p(A) p(B).}

6 Ex. gender and eye colorSDependent - Events whose probability changesdeoendlns upon the occurrence or non-occurrence ofeachother: p{.I I bl dilfers lrom AA): p(B lA) differs fromp(B); and p(A and B): p(A) p(BlA): p(B) AAIB)Ex. rsce and eye colonC JOINT PROBABILITIES - Probability that2 otmore events occur MARGINAL PROBABILITIES or Uncondi-tional Probabilities = summation of probabilities'D CONDITIONAL PROBABILITIES - Probabilityof I given the existence of ,S, written, p (Al$.fl EXAMPLE- Given the numbers I to 9 asobservations in a sample space:.Events mutually exclusive and exhaustive'Example: p (all odd numb ers) ; p ( all eu-e n nurnbers ).Evenls mutualty exclusive but not exhaustive-Example: p (an eien number); p (the numbers 7 and 5).Events ni:ither mutually exclusive or exhaustive-Example: p (an even number or a 2)fl SAMPLING DISTRIBUTION - A theoreticalprobability distribution of a statistic that wouldiesult from drawing all possible samples of agiven size from some STAIUDARD EBROROF THE MEANA theoretical standard deviation of sample mean of agiven sample si4e, drawn from some speciJied based on a very large, known population, thestandard error is: 6_ _ o"r_ ^l nEWhen estimated from a sample drawn from very largepopulation, the standard error is.}

7 LThe dispersion of sample means decreases as samplesize is S^t-'fnRANDOM VARIABLESA mapping or function that assigns one and'onlv one-numerical value to eachoutcome in an DISCRETE RANDOM VARIABLES - In-volves rules or probability models for assign-ing or generating only distinct values (not frac-tional measurements).C BINOMIAL DISTRIBUTION - A modelfor the sum of a series of n independent trialswhere trial results in a 0 (failure) or I (suc-cess). Ex. Coin to"t p(r) =(!)n'l-trl"-'where p(s) is the probability of s success in ntrials with a constant n probability per trials,and where (,1\= , n!-"- "'-'- ts/ s!(n-s)!Binomial mean: !: nxBinomial variance: o': n, (l - tr)As n increases, the Binomial approaches theNormal HYPERGEOMETRIC DISTRIBUTION -A model for the sum of a series of n trials whereeach trial results in a 0 or I and is drawn from asmall population with N elements split betweenN1 successes and N2 failures.)

8 Then the probabil-ity of splitting the n trials between xl successesand x2 failures is: Nl! {_z!p(xlandtrr:W't 4tlv-r;lrHypergeometric mean : pt :E(xi - +and variance: o2 : ffit+][p]D POISSON DISTRIBUTION - A model forthe number of occurrences of an event x :0,1,2,.., when the probability of occurrenceis small, but the number of opportunities forthe occurrence is large, for x : 0,1,2, and)v > 0 . otherwise P(x) =. $t=ffPoisson mean and rariance: , r c ontinuo u s t' a ri u b I e s..fi'e q u e n t' i e s u re p re s s e din terms areus under u t' CONTINUOUS RANDOM VARIABLES- Variable that may take on any value along anuninterrupted interval of a NORMAL DISTRIBUTION - bell cun'e;a distribution whose values cluster symmetri-cally around the mean (also median and mode).)}

9 F(x)=-1, (x-P)212o2o"t'2xwheref (x): frequency. givenrzalueo : standard deviatlon of thedistributionlt : approximately I 111qapproximately : the mean of the distributionx : any score in the distributionD STANDARD NORMAL DISTRIBUTION- A normal random variable Z. that has a meanof0. and standard deviation of Z-VALUES - The number of standard devia-tions a specific observation lies from the mean:': x- 11tr LEVEL OF SIGNIFICANCE -Aprobabilinvalue considered rare in the sampling distrib under the null hypothesis where one iswilling to acknowledge the operation of chancefactors. Common significance levels are 170,50 ,l0o . Alpha (a) level : the lowest levefor which the null hypothesis can be significance level determinesthe critical region.[| NULL HYPOTHESIS (flr) - A statementthat specifies hypothesized value(s) for one ormore of the population parameter.]

10 LBx. Hs= acoin is unbiased. That isp : ]tr ALTERNATM HYPOTHESIS (.r/1) - Astatement that specifies that the populationparameter is some value other than the onespecified underthe null trypothesis. [Ex. I1r: a coinis biased That isp * NONDIRECTIONAL HYPOTHESIS -an alternative hypothesis (H1) that states onllthat the population parameter is different fromthe one ipicified under H 6. Ex. [1 f lt + !t0 Two-Tailed Probability Value is employed whenthe alternative hypothesis is DIRECTIONAL HYPOTHESIS - analternative hypothesis that states the direction rnwhich the population parameter differs fiom theone specified under 11* Ex. Ilt: Ir > pn r-tr H f lr ' t1 One-Tailed Probability Value is employed u'henthe alternative hypothesis is NOTION OF INDIRECT PROOF - Stnctinterpretation ofhypothesis testing reveals that thc'null hypothesis can never be proved.]