.MCAD 308000000 \  docDocumentMmcObject[ d2_graph_format graphData% axisFormat)L)Ltrace2D&&&&&&&&& & & & & &&& dim_formatTmasslengthtimecharge temperature luminosity substanceNumericalFormatQdii shpRectVmcDocumentObjectState\ mcPageModelK????mcHeaderFooterI@I |P CHeaderFooterJ@J{n}@J@JMbP?MbP? TextState? TextStyle>@ ArialSerial_ParPropDefaultW?Normal>@Arial@W? Heading 1>@ Arial@W? Heading 2 >@ Arial@W? Heading 3 >@ Arial@W3 Paragraph >@ Arial@W3List >@ Arial@W>Indent >@Times New Roman@W/Title>@Times New Roman@W/Subtitle font_style_listO font_styleP  Variables MathSoftText@P  ConstantsTimes New Roman@P TextArial@P Greek VariablesSymbol@P User 1Arial@P User 2 Courier New@P User 3System@P User 4Script@P User 5Roman@P User 6Modern@P User 7Times New Roman@P SymbolsSymbol@P Current Selection FontArial@P Undefined Font@P HeaderArial@P FooterArial@P Rotated Math Font MathSoftText TextRegion* docRegionGshpBoxU CB ;;6;6 CharacterMap-RangeMap;@aAPPLIED QUANTITATIVE RESEARCH PAD/PDD/UST601 COMPUTER LAB#1: Tutorial and The Normal Distribution ChrPropMap7a RangeElem<a ChrPropData8 RangeData=| MathSoftText0,0,128 ParPropMap9a<?:@W?<@@<=@A:@W?>@@@@1@B<@C EmbedData2EmbedObj3@D EmbedObjPtr4/\@E<\@F0@NormalArial @GeqRegionB@Uh@Htree @ p@I@@@H@J@@@@I@K@@t@J3.3@L@@@J4.1@M@@@I@N@@+@@MSerial_DisplayNodeX@O@@@M@P@B@Uh@Q@@ p@R@@@Q@S@@@@R@T@@t@S2.4@U@@@S3.5@V@@@R@W@@+@@V@X@X@@@V@Y@B@Uh@Z@@ p@[@@@Z@\@@{@@[@]@@@\2@^@@@[@_@@+@@^@X@`@@@^@a@B@Uh0G@@b@@ p@c@@@b@d@@@@c@e@@t@d2.3@f@@K@d@g@@@f4.5@h@@@c@i@@+@@h@X@j@@@h@k@B@Uh`Hwp@l@@ p@m@@@l@n@@@@m@o@@@@n@p@@p@@o@q@@@p@r@@t@q3.1@s@@@q4.2@t@@@o4.3@u@@@n6.5@v@@@m@w@@+@@v@X@x@@@v@y@B@Uh@z@@ p@{@@@z@|@@@@{@}@@@@|@~@@t@}7@@@p@}@@@@@@@t@2.3@@@@8.6@@@@|3@@@@{@@@+@@@X@@@@@@B@Uh @@@ p@@@@@@@@@@@@K@@@@@@3.7@@@{@@@@@@@@K@@@@@@2.3@@@@8.6@@@@@@@+@@@X@@@@@*@U)>30 -COPYRIGHT @KAUFMAN, 19997@<@8@~9@<@:@W?1@</@<@0@NormalArial @*@UQ6`..K.K-CWWe proceed to the computation and graphing of basic statistics. Perhaps the most important probability distribution function is the normal. MathCAD provides this function, which it calls dnorm(x,m,s) where: x is the variable whose values are normally distributed; m is the population mean; s is the population standard deviation. We graph here the normal distribution function of mean -2 and standard deviation 3, dnorm(x, -2, 3), then two more distributions differing in mean and standard deviation: Click on the graph palette. Then click on the X-Y Plot button. On the x-axis placeholder type x and on the y-axis placeholder type dnorm(x,-2,3). Then select the whole name of the function by hitting the spacebar. Hit , (coma). A new placeholder appears, where you type dnorm (x,0,3). Repeat this process and type dnorm(x,1,3). 7TW@<@8@@ MathSoftText128,0,0@<@8@@ Arial128,0,0@@<@8@@ Arial128,0,0@@<@8@n MathSoftText128,0,0@@<@8@n MathSoftText128,0,0@@<@8@@ Arial128,0,0@@<@8@n MathSoftText128,0,0@@<@8@l  MathSoftText128,0,0@@<@8@@ Symbol128,0,0@@<@8@@  MathSoftText128,0,0@@<@8@@ Symbol128,0,0@@<@8@@  MathSoftText128,0,0@@< @8@@ MathSoftText128,0,0@@<@8@@  MathSoftText128,0,0@@<9@8@@ MathSoftText128,0,0@@<@8@@ Symbol128,0,0@@<@8@@ MathSoftText128,0,0@@<@8@@ Symbol128,0,0@@<&@8@@ MathSoftText128,0,0@@<@8@@ Arial128,0,0@@<@8@@ Arial128,0,0@@@@A<1A?@@A;3A@@@AAA@@@A@AB@@@AAAC@@vAB10AD@@KABAE@@AD10AF@@AAAG@@@AFAH@@AFAI@@A@xAJ 3)N)NNORMAL DISTRIBUTION DENSITY&&&&&&&&& & & & & &&&AK*@U?//?/?-@The normal distribution function is highest at the mean, and is symetrical about the mean. Next, we graph several normal distributions of various standard deviations. 7AL<AM8AK~ MathSoftText128,0,09AN<AO:@W?1AP</AQ<AR0@NormalArial AS*@Ui>sp -COPYRIGHT @KAUFMAN, 19997AT<AU8AS~9AV<AW:@W?1AX</AY<AZ0@NormalArial A[@B@U6~A\@@ pA]@@A\A^@@@A]A_@@@A^A`@@@A_Aa@@tA`1Ab@@A`0Ac@@A_Ad@@@AcAe@@AcAf@@ A^Ag@@ @AfAh@@@AgAi@@dAhdnormAj@@pAhAk@@ AjAl@@ @AkAm@@dAlxAn@@Al0Ao@@Ak0.5Ap@@AgAq@@dApdnormAr@@pApAs@@ ArAt@@ @AsAu@@dAtxAv@@At0Aw@@As1Ax@@AfAy@@dAxdnormAz@@pAxA{@@ AzA|@@ @A{A}@@dA|xA~@@A|0A@@A{2A@@A]A@@@AA@@@AA@@vA10A@@KAA@@A10A@@AA@@@AA@@AA@@AxA 4)N)NNORMAL DISTRIBUTION DENSITY&&&&&&&&& & & & & &&&A*@U<44?4?-@From these three curves centered on m=0 (that differ only in standard deviation s), we note that as the standard deviation increases, the normal distribution becomes wider and flatter.7$A<$A8A~ MathSoftText128,0,0A<A8A| Symbol128,0,0AA<*A8A~ MathSoftText128,0,0AA<A8A| Symbol128,0,0AA<A8Al Symbol128,0,0AA<(A8A~ MathSoftText128,0,0AA<A8A~ 128,0,0AA<"A8A~ MathSoftText128,0,0AA<A8A~ MathSoftText128,0,0AAA9A<A:@W?1A</A<A0@NormalArial A*@UB ::?:?-@pnorm(X,m,s) returns the cumulative probability distribution, of the probability that the variable x is less or equal to a specified value X. Geometrically, it is the area under the normal curve to the left of X.7A<A8Al  MathSoftText128,0,0A<A8Al 128,0,0AA<A8Al  MathSoftText128,0,0AA<A8Al Symbol128,0,0AA<A8Al  MathSoftText128,0,0AA<A8Al Symbol128,0,0AA<A8Al  MathSoftText128,0,0AA<A8Al 128,0,0AA<#A8A~ MathSoftText128,0,0AA<A8A~ MathSoftText128,0,0AA< A8A~ MathSoftText128,0,0AA<A8A~ MathSoftText128,0,0AA<A8A~ MathSoftText128,0,0AA<A8A~ MathSoftText128,0,0AA<A8A~ MathSoftText128,0,0AA<A8Al  MathSoftText128,0,0AA<'A8A~ MathSoftText128,0,0AA<A8Al  MathSoftText128,0,0AA<A8A~ MathSoftText128,0,0AA<A8A~ 128,0,0AA<A8A~ 128,0,0AA  B@@ pB@@BB@@@BB@@@BB@@@BB@@vB0.133B@@BB@@tB2.259B@@BB@@tB10B@@KBB@@B5B@@BB @@@BB!@@BB"@@BB#@@dB"dnormB$@@pB"B%@@ B$B&@@ @B%B'@@dB&xB(@@B&2.5B)@@B%3B*@@BB+@@@B*B,@@@B+B-@@vB,10B.@@KB,B/@@B.10B0@@B+B1@@K@B0B2@@B10.5B3@@B05B4@@B*xB5 5)^)NNORMAL DISTRIBUTION DENSITY&&&&&&&&& & & & & &&&B6@B@U />  B7@@ pB8@@B7B9@@@B8B:@@@B9B;@@@B:B<@@vB;0.994B=@@B;B>@@tB=1.545B?@@B=B@@@tB?10BA@@KB?BB@@BA5BC@@B:BD@@@BCBE@@BCBF@@B9BG@@dBFpnormBH@@pBFBI@@ BHBJ@@ @BIBK@@dBJXBL@@BJ2.5BM@@BI3BN@@B8BO@@@BNBP@@@BOBQ@@vBP10BR@@KBPBS@@BR10BT@@BOBU@@@BTBV@@BTBW@@BNXBX 2")N)NCUMULATIVE NORMAL DISTRIBUTION&&&&&&&&& & & & & &&&BY*@UI = X --i-i-A`We graphed above the cumulative normal distribution of mean 2.5 and standard deviation 3, pnorm(X,2.5,3) against values of X. The result is an S-shaped curve. For small values of X, the probability of finding variable values below X is low, close to 0; for large X values, the probability of finding variable values below X becomes large, close to 1.7_Z`BZ