ࡱ> 574z#` &(bjbj _Vy%xxxx$`,`,`,P,|,-4Ml-82^3336hi<e>MMMMMMMOhRMe?56e?e?Mxx33M9B9B9Be?x33M9Be?M9B9B:Gb>v8M3`- .(7`,?M XMDM0M M.RAR8MR8M e?e?9Be?e?e?e?e?MMApe?e?e?Me?e?e?e?$+d+xxxxxx Chapter  SEQ chapter 2: Descriptive Statistics Prerequisite: Chapter  ch_linear \* MERGEFORMAT 1  SEQ chapter \c2. SEQ section \r 11 Review of Univariate Statistics XE "univariate statistics"  The central tendency of a more or less symmetric distribution of a set of interval, or higher, scaled scores, is often summarized by the arithmetic mean XE "arithmetic mean" \t "See"  XE "mean" \t "See" , which is defined as  EMBED Equation.3 . ( SEQ chapter \c 2. SEQ eqn \r 11) We can use the mean to create a deviation score XE "deviation score" ,  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn2) so named because it quantifies the deviation of the score from the mean. Deviation is often measured by squaring, since it equates negative and positive deviations. The sum of squared deviations, usually just called the sum of squares XE "sum of squares" , is given by  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn3) Another method of calculating the sum of squares was frequently used during the era that preceded computers when students would work with calculating machines,  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn4) Regardless whether one uses Equation ( ss_formula 2.3) or Equation ( ss_formula_hand 2.4), the amount of deviation that exists around the mean in a set of scores can be averaged using the standard deviation XE "standard deviation" , or its square, the variance XE "variance" . The variance is just  EMBED Equation.3  with s being the positive square root of s2. We can take the deviation scores and standardize them, creating, well; standardized scores XE "standardized scores" :  EMBED Equation.3 . ( SEQ chapter \c 2. SEQ eqn5) Next, we define a very important concept, that of the covariance XE "covariance"  of two variables, in this case x and y. The covariance between x and y may be written Cov(x, y). We have  EMBED Equation.3   EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn6) where the EMBED Equation.3 are the deviation scores for the x variable, and the  EMBED Equation.3 are defined analogously for y. Note that with a little semantic gamesmanship, we can say that the variance is the covariance of a variable with itself. The product EMBED Equation.3 is usually called a cross product XE "cross product" .  SEQ chapter \c2. SEQ section2 Matri XE "matrices" x Expressions for Descriptive Statistics In this section we will return to our data matrix, X, with n observations and m variables,  EMBED Equation.3  We now define the mean vector XE "mean vector"  EMBED Equation.3 , such that  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn7) You might note that here we are beginning to see some of the advantages of matrix notation. For example, look at the second line of the above equation. The piece 1'X expresses the operation of adding each of the columns of the X matrix and putting them in a row vector. How many more symbols would it take to express this using scalar notation using the summation operator (? The mean vector can then be used to create the deviation score matrix, as below.  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn8) We would say of the D matrix that it is column-centered XE "column centering" , as we have used the column means to center each column around zero. Now lets reconsider the matrix X'X. This matrix is known as the raw XE "raw cross products matrix"  XE "sum of squares and cross products, raw"  XE "sum of squares and cross products, uncorrected"  XE "uncorrected sum of squares and cross products" , or uncorrected XE "uncorrected SSCP"  XE "SSCP, raw" , sum of squares and cross products matrix. Often the latter part of this name is abbreviated SSCP XE "SSCP matrix" . We will use the symbol B for the raw SSCP matrix:  EMBED Equation.3 . ( SEQ chapter \c 2. SEQ eqn9) In addition, we have seen this matrix expressed row by row and column by column in Equations ( cross_prod_1 1.26) and ( cross_prod_2 1.27). The uncorrected SSCP matrix can be corrected for the mean of each variable in X. Of course, it is then called the corrected SSCP XE "corrected sum of square and cross products"  XE "sum of square and cross products, corrected"  XE "corrected SSCP matrix"  XE "SSCP matrix, corrected"  matrix at that point: A = D(D ( SEQ chapter \c 2. SEQ eqn10) A EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn11) Note that Equation ( sscp_formula 2.10) is analogous to the classic statement of the sum of squares in Equation ( ss_formula 2.3) while the second version in Equation ( sscp_formula_hand 2.11) resembles the hand calculator formula found in Equation ( ss_formula_hand 2.4). The correction for the mean in the formula for the corrected SSCP matrix A can be expressed in a variety of other ways:  EMBED Equation.3  Now, we come to one of the most important matrices in all of statistics, namely the variance-covariance matrix XE "variance-covariance matrix"  XE "matrix, variance-covariance"  XE "variance matrix" , often just called the variance matrix. It is created by multiplying the scalar 1/(n-1) times A, i. e.  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn12) This is the unbiased formula for S. From time to time we might have occasion to see the maximum likelihood formula which uses n instead of n - 1. The covariance matrix is a symmetric matrix, square, with as many rows (and columns) as there are variables. We can think of it as summarizing the relationships between the variables. As such, we must remember that the covariance between variable 1 and variable 2 is the same as the covariance between variable 2 and variable 1. The matrix S has  EMBED Equation.3 unique elements and  EMBED Equation.3 unique off-diagonal elements (of course there are m diagonal elements). We should also point out that EMBED Equation.3 is the number of m things taken two at a time. Previously we had mean-centered X using its column means to create the matrix D of deviation scores. Now we will further standardize our variables by creating Z scores. Define ( as the matrix consisting of diagonal elements of S. We define the function Diag XE "Diag function" () for this purpose:  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn13) Next, we need to invert the ( matrix, and take the square root of the diagonal elements. We can use the following notation in this case:  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn14) The notion of taking the square root does not exactly generalize to matrices [see Equation ( eigen_sqrt 3.38)]. However, with a diagonal matrix, one can create a unique square root by taking the square roots of all the diagonal XE "diagonal"  elements. With non-diagonal matrices there is no unique way to decompose a matrix into two identical components. In any case, the matrix (-1/2 will now prove useful to us in creating Z scores. When you postmultiply a matrix by a diagonal matrix, you operate on the columns of the premultiplying matrix. That is what we will do to D:  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn15) which creates a matrix full of z scores. Note that just as postmultiplication by a diagonal XE "diagonal"  matrix operates on the columns of the premultiplying matrix, premultiplying by a diagonal matrix operates on the rows of the postmultiplying matrix. Now we are ready to create the matrix of correlations, R. The correlation matrix is the covariance matrix of the z scores,  EMBED Equation.3  ( SEQ chapter \c 2. SEQ eqn16) Since the correlation of x and y is the same as the correlation between y and x, R, like S, is a symmetric matrix. As such we will have occasion to write it like  EMBED Equation.3  leaving off the upper triangular part. We can also do this for S.      PAGE 18 Chapter  SEQ chapter \c 2 Descriptive Statistics  PAGE 19  PAGE 8  PAGE 1  23?IJefghijyz{|}~þöö×uuhUP h $jh 5CJOJQJU\h 5CJOJQJ\&h 56CJOJQJ\mHnHuh 56CJOJQJ\$jh 56CJOJQJU\hEh mHnHujh U h 5h (h 6CJOJQJ^JaJmHnHu&jh 6CJOJQJU^JaJh 6CJOJQJ^JaJ23i G H z { $ Ba$ $ Ba$ $ !Ba$'(%([ j k p   ʷʏʏʖʏʏʖ|riih mHnHujh EHUj4SB h UVjh U h 6jh 6U h 6h h 5CJOJQJ\$jh 5CJOJQJU\ h #h 5CJOJQJ\^JaJh 5CJOJQJ\ h 5CJOJQJ^JaJ) , - 2 A C D I J ] ^ _ ` b c s t u v w x f t u z | } ׹׹߮׹׹ߙjh EHUj/B h UVjh EHUjR/B h UVh mHnHuj|h EHUj_tB h UVjh Uh  h 6 h jh 6U h 68        { 45ùjcB h UVh CJEHH*aJjV h EHUj0B h UV h 6 h jh 6U h 6h mHnHuh jh U<  78l.1JK[\ $ Ba$ $ Ba$'127ACD$%&'\鹯餚鏅j6h EHUjWB h UVjmh EHUj\6B h UVjh EHUjB h UV h 6 h jh 6U h 6h mHnHuh jh Uj h EHU4\]pqrs./01ERSXeghlm|}~Ĺ󚌚xxkX$jh 5CJOJQJU\h 5CJOJQJ\&h 56CJOJQJ\mHnHuh 56CJOJQJ\$jh 56CJOJQJU\ h 6 h jh 6U h 6jh EHUj6B h UVjh EHUjxB h UVh jh U"23FGHI]hiny{|}䎳|rg]jgh EH4UjtB h UVjfh EHUj_B h UV h 6jh 6U h 6j.h EHUj}B h UVjh U h 5h h 5CJOJQJ\$jh 5CJOJQJU\ h h 5CJOJQJ\h 5\$VW$%*:<=!OQSXźź h 6 h jh 6U h 6j#h EH,Uj~B h UV jSh  h 5h mHnHujh Uh Dpq(!)! $ Ba$"&',79:TUrs !#$+,:;<=?@(*,1FȾӵӵӵӵh mHnHujz(h EHUjHB h UVjh U h 5 h 6 h jh 6U h 6h GFHJOeghdeqrstuv伲j4,h EHUjiB h UVh mHnHujh U jh  h 5h  h 6jh 6U h F  WX0249TVX]lno 9: ꩟jtB h UVjn5h EHUj8+B h UV h 6 h jh 6U h 6j1h EHUjF h UV h 5h h mHnHujh U:34GHIJ  F G !!!!*!+!>!?!@!A!C!D!T!U!V!W!X!Y!a!b!d!e!ɿ魢鋁xxh mHnHuj>h EHxUjxB h UV h 6 h jh 6U h 6 jDh 5 h 5jK<h EHUj+tB h UVj:h EHUjtB h UVh jh Uj7h EHU/)!h!i!!!4"5"$$$$$%%D&E&&&''('A'B'''''''' $ Ba$e!!!!! " " " """" "!"""#"$","-"/"0"2"3"""""""""##,#-####}$~$$$$$$$$$$$$$$$$$$$%%-%.%%%j=Eh EHUj28+B h UV h H* h jh U h 5h mHnHujAh EHUjkNB h UVjh U jDh 5h ?%F&G&Z&[&\&]&_&`&p&q&r&s&t&u&}&~&&&&&&&)'*'='>'?'@''''''''''''''''''''''''''''''''''''Ƽh CJOJQJaJjhUUhUj"Ph EHUjtB h UV h 5h mHnHujLh EH\UjlB h UVjh Uh ?'''''''''''''((((((((( ( ( ( ( (((('''''''''&(hUjhUUhUmHnHu (((((((((((((((( (!("(#($(%(&( $ Ba$/ 0&P/ =!"#$%` ,&P/ =!"#$%` |Dd \  c $A? ?#" `2 :`1v) l0̄X86҂7z?[q\jaF&B-%;_(mf=哈XYӥnv$W*xթ?h<'/{?+43B*`^1z.xV1mbp=ChпP;"S^;&Ӿ:P1:*sJ~V|@$?95uo_q*η|Ϧι礸˃:s|yuj )?ԋwAIݎOҟUQ'4u &܊VFfb"LMyۃ)a2g$Zt'\X kֶϞ,Z`ODd J  C A? "2}/j-^X `!}/j-^X`@ :SxڍR=KP=IkB 88[1`B Z']tut_ppg <.p=~ &qXy9iL$I4Jf֘9J/޼y˨ QE%KQ\!kҜvڃְ# >TSkd/%Gy {r 7{6܄^-Ww3S<2=oHkcNFtv peX/Vă)nОƇ6>?っ/3Ŧ?M7*u*Y)\[2bzE}ZH:Ӈѳ;Q $t7UWpDd ,J  C A? "2dk$e,2].A`!dk$e,2].Ԩtxcdd`  !"#$%&'()*+-./0123{69[:;=<>?@BACDEGFHIJKLNMOPQSRTUVWZXY\]^_`bacedfgihjlkmnopqrstvuwyx|}~Root Entry F)78DData ,RWordDocument_VObjectPool(7)7_11164925961'F(7s(7Ole CompObjfObjInfo  #&'*/234567:=@CFILOPQRSV[^_`abcdefilmnopqrstuvwx{~ FMicrosoft Equation 3.0 DS Equation Equation.39qvOX 2x=1nx iin " FMicrosoft Equation 3.0 DS EqEquation Native k_1114955871Oc Fs(7s(7Ole CompObj fuation Equation.39q:E d i =x i "2x, FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  a_1118318418Fs(7s(7Ole  CompObj fObjInfoEquation Native _1118318557Fs(7s(7° a=(x i "2x) 2in " or=d i2in " . FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native  G a=x i2in " "x iin " () 2 n. FMicrosoft Equation 3.0 DS Equation Equation.39q< s 2 =_1118318816Fs(7s(7Ole CompObjfObjInfoEquation Native X_1107356771Fs(7s(7Ole !CompObj "f1n"1a FMicrosoft Equation 3.0 DS Equation Equation.39qp@mICompObj24?fuation Equation.39q(ԞIDI d y i  FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo5AEquation Native BD_11183203288F(7(7Ole DCompObj79EfObjInfo:GEquation Native Hd_1120304638TJ=F(7(7H d x i  d y i  FMicrosoft Equation 3.0 DS Equation Equation.39q[( X=x 11 x 12 "x Ole JCompObj<>KfObjInfo?MEquation Native Nw1m x 21 x 22 "x 2m """"x n1 x n2 "x nm []={x ij }. FMicrosoft Equation 3.0 DS Equation Equation.39q_1107525983BF(7(7Ole TCompObjACUfObjInfoDWEquation Native X-_1114958288GF(7(7Ole YCompObjFHZf <^ "x FMicrosoft Equation 3.0 DS Equation Equation.39q:B8M "x=2x 1 2x 2 "2x m []=1n, ObjInfoI\Equation Native ]^_1120304660YLF(7(7Ole g1 "1 n X=1n11"1[]x 11 x 12 "x 1m x 21 x 22 "x 2m """"x n1 x n2 "x nm [].CompObjKMhfObjInfoNjEquation Native k`_1107708133@|QF(7(7 FMicrosoft Equation 3.0 DS Equation Equation.39qDAt D=X", n 1 1 "Xd 11 d 12 "d 1m d 21 d 22 "d 2m """"d n1 d n2 "d nm []=X"11"1[]"x 1 "x 2 ""x m []=X"2x 1 2x 2 "2x m 2x 1 2x 2 "2x m """"2x 1 2x 2 "2x m [] FMicrosoft Equation 3.0 DS Equation Equation.39q<^ B="XX=x i1 x i1 " x i1 x i2 "Ole yCompObjPRzfObjInfoS|Equation Native } "x i1 x im " x i2 x i1 " x i2 x i2 " "x i2 x im " """"x im x i1 " x im x i2 " "x im x im " []_11183333846rVF(7(7Ole CompObjUWfObjInfoX FMicrosoft Equation 3.0 DS Equation Equation.39q»XM =B"1nx i1 " () 2 x i1 " ()x i2 " ()Equation Native _1186568710[F(7(7Ole CompObjZ\f"x i1 " ()x im " ()x i2 " ()x i1 " ()x i2 " () 2 "x i2 " ()x im " ()""""x im " ()x i1 " ()x im " ()x i2 " ()"x im " () 2 [] FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo]Equation Native _1110128656`F(7(7Ole ( A=B"1n, m "X n , n 1 1 (), m 2X n , n 1 1 ()  2 =B"1n"X1()"1X()=B"1n"X(1"1)X=B"1n"X11"111"1""""11"1[]X=B""x("1X). FMicrosoft Equation 3.0 DS Equation Equation.39q6 S=1n"CompObj_afObjInfobEquation Native R_1114961868EmeF(7(71A FMicrosoft Equation 3.0 DS Equation Equation.39q:.PM )m(m+1)2Ole CompObjdffObjInfogEquation Native J_1114961911jF(7(7Ole CompObjikfObjInfol FMicrosoft Equation 3.0 DS Equation Equation.39q:.h1 )m(m"1)2 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native J_1114961963hoF(7(7Ole CompObjnpfObjInfoqEquation Native J_1118337185tF(7(7Ole :.h1 )m(m"1)2 FMicrosoft Equation 3.0 DS Equation Equation.39ql =DiagCompObjsufObjInfovEquation Native _1112435657yF(7(7(S)=s 12 0"00s 22 "0""""00"s m2 [] FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjxzfObjInfo{Equation Native 8,nII  "1/2 =1/ s 12 0"001/ s 22 "0""""00"1/ s m2 [] FMicrosoft Equation 3.0 DS Equation Equation.39q_1110128690^w~F(7(7Ole CompObj}fObjInfo Z=D "1/2 =d 11 d 12 "d 1m d 21 d 22 "d 2m """"d n1 d n2 "d nm []1/ s 12 0"0Equation Native _1118334109F(7(7Ole CompObjf01/ s 22 "0""""00"1/ s m2 []=d 11  s 12 d 12  s 22 "d 1m  s m2 d 21  s 12 d 22  s 22 "d 2m  s m2 """"d n1  s 12 d n2  s 22 "d nm  s m2 [] FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _1114962047F.(7.(7Ole pPM R=1n"1"ZZ= "1/2 S "1/2 =1r 12 "r 1m r 21 1"r 2m """"r m1 r m2 "1[] FMicrosoft Equation 3.0 DS Equation Equation.39q:©P R=1r 21 1"""r m1 r m2 "1[]CompObjfObjInfoEquation Native 1TableR`a!0 ĜL  312Ec21BUs30=`d` A?dm@TFnĒʂT.(X'#=`yqDd  b  c $A? ?3"`?2]썣R":̼`!]썣R":̼V(`zxڝMhA~41)k"VmAՈڂ֦`ZŸC詧B<[ Rԓ(68_;9n{oofg AE QFCXG՗q>Kuum]b\a:z,G o.]2_6k=DAsvŭ B'&{7Gq~9<RH M]ŮKSI~ZZWY&=kO=F;yT6{sI-_r9\8R\㦿lgb`_ZXlPkzQ]\pYm"qLhlu]U:`jk>k#PI)ewt!mr7q|IZ|6U4U{#m +bj?6F/5FCqu޾xC|P:kl$O&N+{<^xޕ֯d~NⱶJ.fda^bokZu0/Y)澘's?NK55 DO?*Ώ:sM?|:kc)M\SM}|_orN4[iOx 䉯8~@jM#]#v3'o?pLD_4Dd ,@J   C A ? "2JVAZVAL&z`!VAZVALٮ xcdd`` @bD"L1JE `x0 Yjl R A@V ^Դjx|K2B* R. `W01d++&10\D(0f1gBMp F>MjҁĻb#$F3I) \Prl8L~3 0y{!Ĥ\Y\ 1 u`uDDd ,@J   C A ? " 2JsUXi&b`!sUXi xcdd`` @bD"L1JE `x0 Yjl R A@V ^Դjx|K2B* R. `W01d++&10D(03gBMp F>MjҁĻb#$F3I) \Prl8L~3 0y{!Ĥ\Y\ 1 u`u&}E-(Dd D@b   c $A ? ?3"`? 2rXm2QANJ`!FXm2QA xڝQ=KA6~.BDO B 렵8.pngm Hgqew{),vgٙݷ6``lÏ%67"v):Ͱuuly1ᕠS1o2`? PIgEz}>ܑJ8FKt^$(66Iׯy΄^[XpҜǐS.:BN>65B#Ƅ&XQC{׶ggvb?|wqp6+?\8Dd Db   c $A ? ?3"`? 2/Pr06r$[Y/W!ڄo}"Dt$﫼5;C*>L>d&QT*G#ﻎo!3yp$%+\T'fVU gMO=U%3G!3'~:`EU+n&N!Bwӎ\;B?'ރ|?YDd \   c $A ? ?#" ` 2Q_*}o -`!%_*}o f `Ƚ!xcdd``$d@9`,&FF(`T)QQRcgbR  h7T obIFHeA*PD.#lp0KȺvo * ZZ |\*g`0ˈKю䷊J.f CF&&\A kg!M;v} ;\Dd $ b  c $A? ?3"`? 2g PN|mMC`!; PN|mM#@ xڥMhAvS3ͦ1J)M,DjIc[Hb-XA^cSa_ĝhE^T8m-04Oןn,)Tc}+~[څdIOD/c,4Vo8!:֭; i5zF9"sDnjS_/7_s|3 {|?{P**!Y}d-LٌwL2rӞvU0ma)^b|í:8]36w,nY+wZ" '\~,u{dc}OCl<22[̋}!; N_ {/s)eȼ˼sܹ_f*u.x])$s/F&LJ?ߦ;e٢!.<9EZKrYcF)Lu5[߫A ,ĉxgS @qb2D0/(r*oߑJ;n>~;ٝHߌ&6IJiQKӛͦhmrQ,#3y7l}I4֍8'= L~5\ I0RIFM5[^:^s̑f…}b,~Al}7uiV/NT ?Nak:׫kã0wJ'x4YRk+زƗϧ%GNŐYܭ{Uj%ȷ/.c9C^Pg9r0K>WΘ>Ro(e~~~6+ᾮh2b%x***s9޺zu*UuJ٣r3r~'޺wv£N]6͔LQ_t=MRrs?>nG.|fN-$I۠h~FTmWh1~]n"c1~,d_ckrs[@1{hryv߽SZ~ q(v6Q@r]Au?U{;{Irc_?BmI^-iZ;*%GI9v͗7?g9gDd |b  c $A? ?3"`?2+0 @c:ӨhAx,`!0 @c:ӨhA"` @{5xڥ[HQϜ9;3+뺻Yn30l"QAhQJPdfº=IP솔S=/aQ/ҋt$4n͙ʬ|9v%"oo$Hˡ_*M$Y-R)ZZfsg!ЧJlx0KRi+*|$&ߔ-?Vh1!8F6M|X,!ߋXB)%oX`}׊e-z+[W}#VoEH_X]T!bX]2Z<;KbuouowI;:O{Ho.{ȳ Bvj&ddQ3{ȳ Bvj&Q1\m׭h#%k:Ap7:fEԚgD_Ҽ˨hx&Ón?5vK?AؼЇ.8{_XƯ<%58 ;T3~1}~՝쌥6\<fCa7!́<7)!C. @eSJa\u -&bjV.)j 9׻Kiޏ.F7frیsy6Dia}#㤑k__|fjk_DKBϚײCu:,]cnW%ejw.}/USmb+.v_.b5WY7հ@uaO.Y~||sfs}9קsmrۚymS4p5NecgGC#^1Gy4c\;Ɓ!x! "CPA}" 'P0~EUxX<%oo1y_.SĔ{ .*||Q e_/'2Wa M0_5,A⓸:QQk<'j=T{m>z@?]:vRnp& C+W[ #7M2^}bW @NG3l[T`EN/w2/0,iir\ 0@`wן2ds|Il櫵oʌ!mDu^|A% SB+Vyg~u"JSNzs =PDd \  c $A? ?#" `2 S9X} W7|5`!t S9X} W7V@@8 Bxcdd``f 2 ĜL0##0KQ* W&d3H1)fYA<6@P5< %! 8 :@u!f0m`tb5, wf:r ,L ! ~ Ay b'Xr`8XA~s @JT'0f> J? σՙ!XeBܤrP8spU^&`suǥm%4nv0edbR ,.IA> 1h:(G33X?_GDd ,\  c $A? ?#" `2Z~wvdfs8`!kZ~wvdf ~ 9xcdd``fg 2 ĜL0##0KQ* WQMRcgbR @øjx|K2B* R8 :@u!f0y3XL݌X@tDFz&br<1@pT9LF]F\N F0# b ȅ| 8ߚ"}L@(\P5$w`] 9.p肽KF&&\A D,Ġtb-PFDd ,\  c $A? ?#" `2E$xC~ErI:`!jE$xC~E ~ 8xcdd``fg 2 ĜL0##0KQ* WQMRcgbR @øjx|K2B* R8 :@u!f0y3(XL݌Y@tDFz&br<1@pT9LF]F\ F0 oe ȅ| 8ߚ"}L@(\P8ͅ}L`;_0Э䂆8tf%#RpeqIj.ŠV "|bt:Q ]LgPGDd ,\  c $A? ?#" `2θk+ʴ=7Ҷs<`!kθk+ʴ=7Ҷ ~ 9xcdd``fg 2 ĜL0##0KQ* WQMRcgbR @øjx|K2B* R8 :@u!f0y3(XL݌Y@tDFz&br<1@pT9LF]F\ F0 oe ȅ| 8ߚ"}L@(\Pp>&0ɝ/@FVrAC `o3@``パI)$5bPd+> 1h:(.&`~P1Dd X0\  c $A? ?#" `2k( VS G>`!?( VS D 7H xڥkALv2ٙ†DJ`Z71Dŋ1^I!hJXԜ=b17Րfֺ~؇tˤtPj;67!n*Ž@\9vq7^T#SʺMPH,1O*s.-:1ߓ,-sy1ʘRk7>zY }֡Dd dJ  C A? "2~RH41Ûb+ A`!~RH41Ûb+ \A^ x}MhA|gF U"Vۂ %/I!X1,^E/B^QP"<qƦ3f3v̾0 Q9⦄D N'RI_PS4Kӆ9vAGA xE^!yWԨ4LjW e9X lP@4A Yh?mwV&*I2&ecl.ڨ#=uPȞ{f䯊<$˙}aѻ3OLNIW=pQ_V)Ws s !F'=hTqE(e[s/߶Χ藈GNP=0Z $=i9YTMx&r^Wr='dfMx-~!1Bb˶qKC=%٩Bڭi27Qo).nJ=vxjH\\U}指ҟVѭ7K#.K`.9RkmOVca|qƓ%wBʼ/b>R`nKa1_K`>Yy~he^879﵏r~*e51bc`1cޫa>*ƼW|h^0G1wE0'uW4- d?> ̋":+^NzYK!u7{g߃*eM3钐|gaig}k~g[T"0s{Ģ/װ_zcu%!-MC )>/s}HVT!=`_ 堞ϛz>o_¸*|sx9[*<Ƌ|п*绑TSiQ9Q^ǼPyVQz)X?hK |5Dy,軣HOUW7WWk^eQ2|_eK~?snȯ |Yo8xP̯q~yjpmH@㯙8/5xo&\1nK]xۭcIs+4Xb~FdH_z#*Qq瘨?=KQECL-fÄ+(;J<%Ib=^MYx(C;ֈȇDd 4 b  c $A? ?3"`?2 (tJxL`! (tJx X(xڭ=l@ǟmJB 7*H 7 ! bH i#%J'21 A(bBT(*6XX0UaRės tz]YHӤNí9g:Vw0MV vBܢ@ί^*%짳H`Aǘ&|oCOĬaj}5Xm!7YB\ :$+tW5qS)–n3C*mX0NT^U;?<9x䆥gXHD7tcuE=':{_]O:-+DZ|GqҬ;Y/V}q3E6x&f[OOaX>UVrEOD ~%RߣPzWg>0aAHɫ∼M2gdQg ~9%^}_9yM!놪7T}K`uݓ=pw3ir=>Q8b5%ooXm^B/~5'^z!7xz7U>_̿.|ۀŁD.BB8[-0"/iw|}:_l^Dd \  c $A? ?#" `2, }J:\9fP`! }J:\9Z `xڥK@%i-ED0*UqS@բQi 18;PG:V8 {w@CG- #$- (ꑾhçKPk^X0)שuAm OŢ2&z)d1~O2QLxw7RLkx9` 4OhiE7_jDqCeMWITa'-H?ŒC} 5]&QU?$ߑsqR4Z†[8K%m-h__w͖HtKq9.00x2="Eoy<35M-ܡ}+c@> Heading 2,Heading 2 Char,Heading 2 Char2 Char,Heading 2 Char1 Char Char,Heading 2 Char Char Char Char,Heading 2 Char1 Char Char Char Char,Heading 2 Char Char Char Char Char Char,Heading 2 Char2 Char Char1 Char Char Char Char,Heading 2 Char Char1 Char$<@& 56CJOJQJ\]^JaJVV Heading 3$<@&5CJOJQJ\^JaJDA@D Default Paragraph FontVi@V  Table Normal :V 44 la (k@(No List 8"8 Caption xx5\&& TOC 1.. TOC 2 ^6U!6 Hyperlink >*B*ph424 Header  !4 @B4 Footer  !H@RH  Balloon TextCJOJQJ^JaJ2a2 Heading 2 Char1,Heading 2 Char Char,Heading 2 Char2 Char Char,Heading 2 Char1 Char Char Char,Heading 2 Char Char Char Char Char,Heading 2 Char1 Char Char Char Char Char,Heading 2 Char Char Char Char Char Char Char,Heading 2 Char Char1 Char Char1056CJOJQJ\]^J_HaJmH sH tH q Heading 2 Char Char Char1,Heading 2 Char2 Char Char Char,Heading 2 Char Char Char1 Char Char Char,Heading 2 Char Char Char1 Char Char Char Char,Heading 2 Char1 Char,Heading 2 Char4056CJOJQJ\]^J_HaJmH sH tH  Heading 2 Char2 Char Char2,Heading 2 Char Char1 Char Char,Heading 2 Char2 Char Char1 Char Char2,Heading 2 Char Char Char3 Char Char Char1,Heading 2 Char2 Char Char1 Char Char1 Char Char1,Heading 2 Char Char Char3056CJOJQJ\]^J_HaJmH sH tH "" Heading 2 Char3 Char Char,Heading 2 Char2 Char Char2 Char,Heading 2 Char Char1 Char Char1 Char,Heading 2 Char2 Char Char1 Char Char2 Char,Heading 2 Char Char1 Char Char Char Char Char,Heading 2 Char2 Char Char1 Char Char1 Char Char1 Char056CJOJQJ\]^J_HaJmH sH tH ,, Heading 2 Char2 Char1 Char1,Heading 2 Char Char Char3 Char,Heading 2 Char3 Char Char Char Char Char,Heading 2 Char2 Char Char2 Char Char Char Char,Heading 2 Char Char1 Char Char1 Char Char Char1 Char,Heading 2 Char2 Char2,Heading 2 Char Char3056CJOJQJ\]^J_HaJmH sH tH    Heading 2 Char2 Char1,Heading 2 Char Char1 Char1,Heading 2 Char2 Char1 Char1 Char,Heading 2 Char Char1 Char1 Char Char,Heading 2 Char2 Char1 Char1 Char Char Char,Heading 2 Char Char1 Char1 Char Char Char1 Char,Heading 2 Char Char3 Char1056CJOJQJ\]^J_HaJmH sH tH && Heading 2 Char5 Char,Heading 2 Char Char1 Char2,Heading 2 Char2 Char1 Char1 Char1,Heading 2 Char Char1 Char1 Char Char1,Heading 2 Char2 Char1 Char1 Char Char Char3,Heading 2 Char Char1 Char1 Char Char Char1 Char3,Heading 2 Char2 Char3 Char056CJOJQJ\]^J_HaJmH sH tH  Heading 2 Char5,Heading 2 Char Char1,Heading 2 Char5 Char Char,Heading 2 Char Char1 Char1 Char,Heading 2 Char2 Char1 Char1 Char Char,Heading 2 Char Char1 Char1 Char Char Char1,Heading 2 Char2 Char1 Char1 Char Char Char2 Char056CJOJQJ\]^J_HaJmH sH tH   !"#$&         !!""##$  !"#$'      !"#& V1V23iGHz{  78l . 1 J K [ \ pq()hi45DE'(AB' @0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0@0Z!MuX!MX!HM00000000@0@0@0@000 ' 0\00 7]]]]]ixxx{ \Fe!%'&(!"#$&') )!'(&( %(*%(Iegiy{}I]_bsuw|$&\pr . 0 l | ~ 2 F H | r#+:?dqu 3GI*>@CTVXad  !#,/FZ\_prt})=?&    :  :  :  :  ::  ::  :::  :::  :  :    :  ::  ::::  :  :  :  : !24OVY]dfkrt{! !!!P"$LdHh7>c"$؇t+d"$6e ܮ83"$=1m^g)R3b$:&];oq~>Se"$wC3齦1k3b$jݽ ٗ\"$t]IBW2r$|ka@O+8ki"$C+G>%<"$apzcؽN0"$.g$eV@0(  B S  ?& ? _Ref44595516 _Toc45528399 _Toc45529544 _Toc45530544 _Toc45637768 _Toc45875641 _Toc45890282 _Toc46108693 _Toc46135170 _Toc46135856 _Toc47000293 _Toc47014632 _Toc47015293 _Toc47015628 _Toc47180860 _Toc47278321 _Toc47278515 _Toc112830214ch_descriptive _Toc34228913 _Toc45528400 _Toc45529545 _Toc45530545 _Toc45637769 _Toc45875642 _Toc45890283 _Toc46108694 _Toc46135171 _Toc46135857 _Toc47000294 _Toc47014633 _Toc47015294 _Toc47015629 _Toc47180861 _Toc47278322 _Toc47278516 _Toc112830215 ss_formulass_formula_hand _Toc34228921 _Toc45528401 _Toc45529546 _Toc45530546 _Toc45637770 _Toc45875643 _Toc45890284 _Toc46108695 _Toc46135172 _Toc46135858 _Toc47000295 _Toc47014634 _Toc47015295 _Toc47015630 _Toc47180862 _Toc47278323 _Toc47278517 _Toc112830216 def_raw_sscp sscp_formulasscp_formula_handcov_matdef_diagmat_sqrtizzzzzzzzzzzzzzzzzl } } } } } } } } } } } } } } } } } C'   !"#$%&'()*+,-./0123456789:;<=>111111111111111111 e0'  $ '   M V v | BI' 333333333333) ' ' $XHtc_ޥ' hh^h`hH) ^`hH) 88^8`hH) ^`hH() ^`hH() pp^p`hH()   ^ `hH. @ @ ^@ `hH.   ^ `hH.h^`OJQJo(hHh^`OJQJ^Jo(hHohpp^p`OJQJo(hHh@ @ ^@ `OJQJo(hHh^`OJQJ^Jo(hHoh^`OJQJo(hHh^`OJQJo(hHh^`OJQJ^Jo(hHohPP^P`OJQJo(hH$XHcM x0$XHeM x0$XH$hM x0tc_cM ) dMax0)XdM")dM()dM()eM%()8eM&.peM'w0.eM(.,fM .dfMax0.fM".fM) gM()DgM%()|gM&()gM'w0()gM(()phM .hMax0..hM"...$iM....hiM .....iM% ......iM& .......LjM'w0 ........jM( .........          U3' 11111@& @@UnknownGz Times New Roman5Symbol3& z Arial9Rockwell5& zaTahoma?5 z Courier New;Wingdings#1hwgӘu 99!4dww3QHP ?  2%Chapter 1 Basic Mathematical ConceptsCollege of BusinessCharles Hofacker