ࡱ>     gCwZU Tbjbjnn aaO, ::8: F!Tޖ# ---.}=}=}=1333333$WA}=?;>}=}=}=W::-.Y _J_J_J}=:-$.1_J}=1_J_Ju4,ш. ?>0ޖp?p|шш( p$}=}=_J}=}=}=}=}=WWE}=}=}=ޖ}=}=}=}=p}=}=}=}=}=}=}=}=}= B : Simple Linear Regression: 1. Finding the equation of the line of best fit Objectives: To find the equation of the least squares regression line of y on x. Background and general principle  The aim of regression is to find the linear relationship between two variables. This is in turn translated into a mathematical problem of finding the equation of the line that is closest to all points observed. Consider the scatter plot on the right. One possible line of best fit has been drawn on the diagram. Some of the points lie above the line and some lie below it. The vertical distance each point is above or below the line has been added to the diagram. These distances are called deviations or errors they are symbolised as  EMBED Equation.3 . When drawing in a regression line, the aim is to make the line fit the points as closely as possible. We do this by making the total of the squares of the deviations as small as possible, i.e. we minimise  EMBED Equation.DSMT4 . If a line of best fit is found using this principle, it is called the least-squares regression line. Example 1: A patient is given a drip feed containing a particular chemical and its concentration in his blood is measured, in suitable units, at one hour intervals. The doctors believe that a linear relationship will exist between the variables. Time, x (hours)0123456Concentration, y2.44.35.06.99.111.413.5 We can plot these data on a scatter graph time would be plotted on the horizontal axis (as it is the independent variable). Time is here referred to as a controlled variable, since the experimenter fixed the value of this variable in advance (measurements were taken every hour). Concentration is the dependent variable as the concentration in the blood is likely to vary according to time. The doctor may wish to estimate the concentration of the chemical in the blood after 3.5 hours. She could do this by finding the equation of the line of best fit. There is a formula which gives the equation of the line of best fit. ** The statistical equation of the simple linear regression line, when only the response variable Y is random, is:  EMBED Equation.3  (or in terms of each point:  EMBED Equation.3 ) Here  EMBED Equation.3  is called the intercept,  EMBED Equation.3 the regression slope,  EMBED Equation.3  is the random error with mean 0,  EMBED Equation.3  is the regressor (independent variable), and  EMBED Equation.3 the response variable (dependent variable). ** The least squares regression line is obtained by finding the values of EMBED Equation.3  and EMBED Equation.3 values (denoted in the solutions as EMBED Equation.3 ) that will minimize the sum of the squared vertical distances from all points to the line:  EMBED Equation.3  The solutions are found by solving the equations:  EMBED Equation.3  and  EMBED Equation.3  ** The equation of the fitted least squares regression line is  EMBED Equation.3  (or in terms of each point:  EMBED Equation.3 ) ----- For simplicity of notations, many books denote the fitted regression equation as:  EMBED Equation.3  (* you can see that for some examples, we will use this simpler notation.) where  EMBED Equation.3  and  EMBED Equation.3 . Notations:  EMBED Equation.3 ;  EMBED Equation.3 ;  EMBED Equation.3  and  EMBED Equation.3  are the mean values of x and y respectively. Note 1: Please notice that in finding the least squares regression line, we do not need to assume any distribution for the random errors  EMBED Equation.3 . However, for statistical inference on the model parameters ( EMBED Equation.3  and EMBED Equation.3 ), it is assumed in our class that the errors have the following three properties: Normally distributed errors Homoscedasticity (constant error variance  EMBED Equation.3 for Y at all levels of X) Independent errors (usually checked when data collected over time or space) ***The above three properties can be summarized as:  EMBED Equation.3 ,  EMBED Equation.3  Note 2: Please notice that the least squares regression is only suitable when the random errors exist in the dependent variable Y only. If the regression X is also random it is then referred to as the Errors in Variable (EIV) regression. One can find a good summary of the EIV regression in section 12.2 of the book: Statistical Inference (2nd edition) by George Casella and Roger Berger. We can work out the equation for our example as follows:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  So,  EMBED Equation.3  and  EMBED Equation.3 . So the equation of the regression line is  EMBED Equation.3 = 1.985 + 1.843x. To work out the concentration after 3.5 hours:  EMBED Equation.3  = 1.985 + 1.843 3.5 = 8.44 (3sf) If you want to find how long it would be before the concentration reaches 8 units, we substitute  EMBED Equation.3 = 8 into the regression equation: 8 = 1.985 + 1.843x Solving this we get: x = 3.26 hours Note: It would not be sensible to predict the concentration after 8 hours from this equation we dont know whether the relationship will continue to be linear. The process of trying to predict a value from outside the range of your data is called extrapolation. Example 2: The heights and weights of a sample of 11 students are: Height (m) h1.361.471.541.561.591.631.661.671.691.741.81Weight (kg) w5250676269745987777367 [ EMBED Equation.3 ] a) Calculate the regression line of w on h. b) Use the regression line to estimate the weight of someone whose height is 1.6m. Note: Both height and weight are referred to as random variables their values could not have been predicted before the data were collected. If the sampling were repeated again, different values would be obtained for the heights and weights. Solution: a) We begin by finding the mean of each variable:  EMBED Equation.3  Next we find the sums of squares:  EMBED Equation.3  The equation of the least squares regression line is:  EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3  So the equation of the regression line of w on h is:  EMBED Equation.3  = -22.4 + 55.5h b) To find the weight for someone that is 1.6m high:  EMBED Equation.3  = -22.4 + 55.51.6 = 66.4 kg Simple Linear Regression: 2. Measures of Variation Objectives: measures of variation, the goodness-of-fit measure, and the correlation coefficient Sums of Squares Total sum of squares = Regression sum of squares + Error sum of squares  EMBED Equation.3  (Total variation = Explained variation + Unexplained variation) Total sum of squares (Total Variation):  EMBED Equation.3  Regression sum of squares (Explained Variation by the Regression):  EMBED Equation.3  Error sum of squares (Unexplained Variation):  EMBED Equation.3   Coefficients of Determination and Correlation Coefficient of Determination it is a measure of the regression goodness-of-fit It also represents the proportion of variation in Y explained by the regression on X  EMBED Equation.3 ;  EMBED Equation.3  Pearson (Product-Moment) Correlation Coefficient -- measure of the direction and strength of the linear association between Y and X The sample correlation is denoted by r and is closely related to the coefficient of determination as follows:  EMBED Equation.3 ;  EMBED Equation.3  The sample correlation is indeed defined by the following formula:  The corresponding population correlation between Y and X is denoted by  and defined by:  Therefore one can see that in the population correlation definition, both X and Y are assumed to be random. When the joint distribution of X and Y is bivariate normal, one can perform the following t-test to test whether the population correlation is zero: Hypotheses H0:  = 0 (no correlation) HA:  `" 0 (correlation exists) Test statistic  Note: One can show that this t-test is indeed the same t-test in testing the regression slope 1 = 0 shown in the following section. Note: The sample correlation is not an unbiased estimator of the population correlation. You can study this and other properties from the wiki site:  HYPERLINK "http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient" http://en.wikipedia.org/wiki/Pearson_product-moment_correlation_coefficient Example 3: The following example tabulates the relations between trunk diameter and tree height. Tree HeightTrunk Diameteryxxyy2x23582801225644994412401812771897294933619810893660137803600169217147441494511495202512151126122601144(=321(=73(=3142(=14111(=713 Scatter plot:     r = 0.886 ! relatively strong positive linear association between x and y Significance Test for Correlation Is there evidence of a linear relationship between tree height and trunk diameter at the .05 level of significance? H0:  = 0 (No correlation) H1:  `" 0 (correlation exists)  At the significance level  = 0.05, we reject the null hypothesis because  EMBED Equation.3  and conclude that there is a linear relationship between tree height and trunk diameter. SAS for Correlation Data tree; Input height trunk; Datalines; 35 8 49 9 27 7 33 6 60 13 21 7 45 11 51 12 ; Run; Proc Corr data = tree; Var height trunk; Run; Note: See the following website for more examples and interpretations of the output plus how to draw the scatter plot (proc Gplot) in SAS:  HYPERLINK "http://www.ats.ucla.edu/stat/sas/output/corr.htm" http://www.ats.ucla.edu/stat/sas/output/corr.htm Standard Error of the Estimate (Residual Standard Deviation) The mean of the random error  is equal to zero. An estimator of the standard deviation of the error  is given by  EMBED Equation.3  Simple Linear Regression: 3. Inferences Concerning the Slope Objectives: measures of variation, the goodness-of-fit measure, and the correlation coefficient t-test Test used to determine whether the population based slope parameter ( EMBED Equation.COEE2 ) is equal to a pre-determined value (often, but not necessarily 0). Tests can be one-sided (pre-determined direction) or two-sided (either direction). 2-sided t-test: H0: 1 = 0 (no linear relationship) H1: 1 `"0 (linear relationship does exist) Test statistic:  EMBED Equation.3  Where  EMBED Equation.3  At the significance level , we reject the null hypothesis if  EMBED Equation.3  (Note: one can also conduct the one-sided tests if necessary.) F-test (based on k independent variables) A test based directly on sum of squares that tests the specific hypotheses of whether the slope parameter is 0 (2-sided). The book describes the general case of k predictor variables, for simple linear regression, k = 1.  Analysis of Variance (based on k Predictor Variables for simple linear regression, k = 1) Source df Sum of Squares Mean Square F Regression k SSR MSR=SSR/k Fobs=MSR/MSE Error n-k-1 SSE MSE=SSE/(n-k-1) --- Total n-1 SST --- ---  100(1-a)% Confidence Interval for the slope parameter, b1: If entire interval is positive, conclude b1>0 (Positive association) If interval contains 0, conclude (do not reject) b1=0 (No association) If entire interval is negative, conclude b1<0 (Negative association) Example 4: A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet). A random sample of 10 houses is selected Dependent variable (y) = house price in $1000s Independent variable (x) = square feet House Price in $1000s (y)Square Feet (x)2451400312160027917003081875199110021915504052350324245031914252551700 Solution: Regression analysis output:   b1 measures the estimated change in the average value of Y as a result of a one-unit change in X Here, b1 = .10977 tells us that the average value of a house increases by .10977($1000) = $109.77, on average, for each additional one square foot of size  This means that 58.08% of the variation in house prices is explained by variation in square feet.  The standard error (estimated standard deviation of the random error) is also given in the output (above). t test for a population slope Is there a linear relationship between x and y? Null and alternative hypotheses H0: 1 = 0 (no linear relationship) H1: 1 `" 0 (linear relationship does exist) Test statistic: At the significance level  = 0.05, we reject the null hypothesis because  EMBED Equation.3  and conclude that there is sufficient evidence that square footage affects house price.  Confidence Interval Estimate of the Slope: The 95% confidence interval for the slope is (0.0337, 0.1858) Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house size This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the .05 level of significance Predict the price for a house with 2000 square feet:  The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,85 Example 5 (SAS): What is the relationship between Mothers Estriol level & Birthweight using the following data? Estriol Birthweight (mg/24h) (g/1000) 1 1 2 1 3 2 4 2 5 4 Data BW; /*Reading data in SAS*/ input estriol birthw@@; datalines; 1 1 2 1 3 2 4 2 5 4 ; run; PROC REG data=BW; /*Fitting linear regression models*/ model birthw=estriol; run; Finance Application: Market Model One of the most important applications of linear regression is the market model. It is assumed that rate of return on a stock (R) is linearly related to the rate of return on the overall market. R = 0 + 1Rm + R: Rate of return on a particular stock Rm: Rate of return on some major stock index 1: The beta coefficient measures how sensitive the stock s rate of return is to changes in the level of the overall market. Example: Here we estimate the market model for Nortel, a stock traded in the Toronto Stock Exchange. Data consisted of monthly percentage return for Nortel and monthly percentage return for all the stocks.  TSE (estimated regression slope): This is a measure of the stocks market related risk. In this sample, for each 1% increase in the TSE return, the average increase in Nortels return is .8877%. R Square (R2) This is a measure of the total risk embedded in the Nortel stock, that is market-related. Specifically, 31.37% of the variation in Nortels return are explained by the variation in the TSEs returns. Linear Regression in Matrix Form Data:  EMBED Equation.3  The multiple linear regression model in scalar form is  EMBED Equation.3  The above linear regression can also be represented in the vector/matrix form. Let  EMBED Equation.3 . Then,  EMBED Equation.3  = EMBED Equation.3 . Estimation: Least square method: The least square method is to find the estimate of  EMBED Equation.3  minimizing the sum of square of residual,  EMBED Equation.3   EMBED Equation.3  since  EMBED Equation.3 . Expanding  EMBED Equation.3  yields  EMBED Equation.3   EMBED Equation.3  Note: For two matrices A and B,  EMBED Equation.3  and  EMBED Equation.3  Similar to the procedure in finding the minimum of a function in calculus, the least square estimate b can be found by solving the equation based on the first derivative of  EMBED Equation.3 ,  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The fitted regression equation is  EMBED Equation.3 . The fitted values (in vector):  EMBED Equation.3  The residuals (in vector):  EMBED Equation.3  Note: (i)  EMBED Equation.3  where  EMBED Equation.3  and  EMBED Equation.3 . (ii)  EMBED Equation.3  where  EMBED Equation.3  is any symmetric  EMBED Equation.3  matrix. Note: Since  EMBED Equation.3 ,  EMBED Equation.3  is a symmetric matrix. Also,  EMBED Equation.3 ,  EMBED Equation.3  is a symmetric matrix. Note:  EMBED Equation.3  is called the normal equation. Note:  EMBED Equation.3   EMBED Equation.3 . Therefore, if there is intercept, then the first column of X is  EMBED Equation.3 . Then,  EMBED Equation.3  Note: for the linear regression model without the intercept,  EMBED Equation.3  might not KLM  Y n ļvofof^fofoVHoA huhuhT@h.}5nHo(tHhT@hly5hT@nHtHhT@nHo(tH huhlyhu5CJaJ"jhU CJUaJmHnHuhly5CJaJhJhly5CJaJhs6hU CJaJhU hU B*phh4CJaJ"hPkhT(5CJ aJ nHo(tHh5CJ aJ nHo(tHh4h45CJ aJ hT(5CJ aJ nHo(tHLM : L M X D E U W Y $$Ifa$gddu$Ifx-DM gdGx-DM gdG -DM gdG -DM gdU  -D`M 3gd : ; ? 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tHuYvYwYxYYYYYYYYYYʳiI.5h?h?5CJKHPJ\]aJmH nHsH tH>h?h?5B*CJKHPJ\]aJmH nHphsH tH1jvh?h'bEHKHPJUmH nHsH tH:jd] h?hFOCJPJUVaJmH nHo(sH tH$h?h?KHPJmH nHsH tH-jh?h?KHPJUmH nHsH tH1jwth?h?EHKHPJUmH nHsH tH7j> h?h?CJPJUaJmH nHo(sH tH YYYYYYYYPPPPũjhP55h?hFO5CJKHPJ\]aJmH nHsH tH/h?5CJKHPJ\]aJmH nHsH tHU8h?h?5CJKHPJ\]aJmH nHo(sH tHBj{h?h?5CJEHKHPJU\]aJmH nHsH tH7j> h?h?CJPJUaJmH nHo(sH tH5h?h?5CJKHPJ\]aJmH nHsH tH>jh?h?5CJKHPJU\]aJmH nHsH tH be equal to 0. Properties of the least square estimate: Two useful results: Let  EMBED Equation.3  be a  EMBED Equation.3  random vector,  EMBED Equation.3  is a  EMBED Equation.3  matrix and  EMBED Equation.3  is a  EMBED Equation.3  vector. Let  EMBED Equation.3  and  EMBED Equation.3 . Then (a)  EMBED Equation.3 . (b)  EMBED Equation.3  Note:  EMBED Equation.3  The properties of least square estimate: 1.  EMBED Equation.3  2. The variance covariance matrix of the least square estimate b is  EMBED Equation.3  [Derivation:]  EMBED Equation.3  since  EMBED Equation.3 . Also,  EMBED Equation.3  since  EMBED Equation.3  Example 6: Heller Company manufactures lawn mowers and related lawn equipment. The managers believe the quantity of lawn mowers sold depends on the price of the mower and the price of a competitors mower. We have the following data: Competitors Price  EMBED Equation.3 Hellers Price  EMBED Equation.3 Quantity sold  EMBED Equation.3 1201001021401101001909012013015077155210461751509312525026145270691803006515025085The regression model for the above data is  EMBED Equation.3 . The data in matrix form are  EMBED Equation.3 . The least square estimate  EMBED Equation.3  is  EMBED Equation.3 . The fitted regression equation is  EMBED Equation.3 . The fitted equation implies an increase in the competitors price of 1 unit is associated with an increase of 0.414 unit in expected quantity sold and an increase in its own price of 1 unit is associated with a decrease of 0.269 unit in expected quantity sold. Suppose now we want to predict the quantity sold in a city where Heller prices it mower at $160 and the competitor prices its mower at $170. The quantity sold predicted is  EMBED Equation.3 . Example 7: We show how to use the matrix approach to obtain the least square estimate and its expected value and the variance. Let  EMBED Equation.3 . Then,  EMBED Equation.3  Thus,  EMBED Equation.3  ,  EMBED Equation.3  and  EMBED Equation.3  = EMBED Equation.3 , since  EMBED Equation.3  Therefore,  EMBED Equation.3   EMBED Equation.3  Also,  EMBED Equation.3  and  EMBED Equation.3  Acknowledgement: In compiling this lecture notes, we have revised some materials from the following websites:  HYPERLINK "http://www.schoolworkout.co.uk/documents/s1/Regression.doc" www.schoolworkout.co.uk/documents/s1/Regression.doc www.stat.ufl.edu/~winner/mar5621/mar5621.doc www.fordham.edu/economics/Vinod/correl-regr.ppt www2.thu.edu.tw/~wenwei/Courses/regression/ch6.1.doc www.msu.edu/~fuw/teaching/Fu_Ch11_linear_regression.ppt www.stanford.edu/class/msande247s/kchap17.ppt Quiz. Due Tuesday at the beginning of the lecture Please prove that: Total sum of squares = Regression sum of squares + Error sum of squares; that is,  EMBED Equation.3  Please derive the method of moment estimators of the regression model parameters      PAGE \* MERGEFORMAT 18  EMBED FXStat.V1  1  EMBED FXStat.V1  These could all be found on a calculator (if you enter the data into a calculator).  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"2(ӥ繎 }wD4/UK)!yE'~?ɁߵITY4sdo@[VyוY5.2<)Eƣy\ ʙtJ`VyFq,Gَ`XpH}l_`%ViU])J{fx.8ӓuފ>H=FJj9 7M'K*JDd b  c $A? ?3"`?2Y{ g'7+p`!hY{ g'7+J`! x3F6xڥ]HSQ9wwΜӶnv]STJz"‚&1BYm҃CEObjInfoContents_ *_1206601203KR1J;% % Ole 1.0.0.9NoneNone0Times New Roman0Times New Roman0Times New Roman0Times New Roman<<<ddd<<<ddd 0Times New Roman Height (m) Weight (kg)0Times New Roman$@$@$$????HLABCA(\?1.36Q?1.47p= ף?1.54(\?1.56q= ףp?1.59Gz?1.63(\?1.66Q?1.67 ףp= ?1.69 ףp= ?1.74 (\?1.81BJ@52I@50P@67O@62@Q@69R@74M@59U@87@S@77 @R@73 P@67CDEFGH I J K L MNOPQRSTUVWXYZ Row NumberRow?@@@@@@ @"@ $@ &@ (@ *@ ,@.@0@1@2@3@4@5@6@7@8@9@:@;@<@=@>@?@@@ @@!A@"A@#B@$B@%C@&C@'D@(D@)E@*E@+F@,F@-G@.G@/H@0H@1I@2I@3J@4J@5K@6K@7L@8L@9M@:M@;N@<N@=O@>O@?P@@@P@AP@BP@CQ@D@Q@EQ@FQ@GR@H@R@IR@JR@KS@L@S@MS@NS@OT@P@T@QT@RT@SU@T@U@UU@VU@WV@X@V@YV@ZV@[W@\@W@]W@^W@_X@`@X@aX@bX@cY@d@Y@eY@fY@gZ@h@Z@iZ@jZ@k[@l@[@m[@n[@o\@p@\@q\@r\@s]@t@]@u]@v]@w^@x@^@y^@z^@{_@|@_@}_@~_@`@ `@@`@``@`@`@`@`@a@ a@@a@`a@a@a@a@a@b@ b@@b@`b@b@b@b@b@c@ c@@c@`c@c@c@c@c@d@ d@@d@`d@d@d@d@d@e@ e@@e@`e@e@e@e@e@f@ f@@f@`f@f@f@f@f@g@ g@@g@`g@g@g@g@g@h@ h@@h@`h@h@h@h@h@i@ i@@i@`i@i@i@i@i@j@ j@@j@`j@j@j@j@j@k@ k@@k@`k@k@k@k@k@l@ l@@l@`l@l@l@l@l@m@ m@@m@`m@m@m@m@m@n@ n@@n@`n@n@n@n@n@o@ o@@o@`o@o@o@o@o@p@p@ p@0p@@p@Pp@`p@pp@p@p@ p@ p@ p@ p@ p@p@q@q@ q@0q@@q@Pq@`q@pq@q@q@q@q@q@q@q@q@r@ r@! 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7T2w@5wP!f-2  -ySymbolw 7T2w@5wP!f-2 !-y2 -y2 =y Times New RomanT2w@5wP!f-2  ny2 +HyTimes New RomanT2w@5wP!f-2 ~ ty & "SystemP!f 8l-NANIá61 t 0 =r 1"r 2 n"2  + H 0 ~ t n"2 FMicrosoft ʽ 3.0 İ DS_1446450477F% % Ole ^CompObj_iObjInfoa Equation Equation.39q   .  ` & & MathType@-@$ -,&-.`vv'  -  &OlePres000b^Equation Native |a_1446450656F% % Ole - .0Times New RomanT2w@5w{!f- 2 z0.8862 c ]. 2 c (321) 2 c ][8(14111) 2 c h (73) 2 c [8(713) 2   (73)(321)) 2 8(3142)2 Y](2 Yy)2 Y()2 Y))2 Yy) 2 Y][n(2 Y x)2 Y ()2 Y))2 Yx) 2 Y[n((2 yn2 xn2 D xy2  ny2 :ry Times New RomanT2w@5w{!f-2 2y2  2y2 92y2 2y2 S2y2  2ySymbolwM HT2w@5w{!f-2 *=y2 c r-y2 c N-y2  -y2 *=y2 Yk-y2 Y -y2 -y2 *=ySymbolw# T2w@5w{!f-2 Ky2  y2 =y2  y2 p y2 /y2 y & "System{!f 8l-NANIE% r=nxy"xy  "  "  "  [n(x 2 )"(x) 2 ][n(y 2 )"(y) 2 ]  "  "  "  "  =8(3142)"(73)(321) [8(713)"(73) 2 ][8(14111)"(321) 2 ]  =0.886 FMicrosoft ʽ 3.0 İ DSCompObjiObjInfoOlePres000Equation Native  Equation Equation.39q* @ .  &8 & MathType-u>u-Bs -s-D D D-u u-  - ( -0 D D D >Times New RomanT2w@5wSf- 2 `4.682 2.2  8. 2  .8862  18 2 n .8862 282 tn82 8r82 .182 n7r82 `@t8 Times New RomanT2w@5wSf-2 7I282 7282 08Symbolw" UjT2w@5wSf-2 `=82  -82  -82 `J=82 -82 -82 `=8 & "SystemSf 8l-NANIif t 0 =r 1"r 2 n"2  =.886 1".886 2 8"2  =4.68 FMicrosoft Equation 2.0 DS EquationEquation.COEE29q+D H 0 :b 1 =0H A :b 1 0_1117102536F% % Ole CompObjjObjInfoEquation Native G_1446451872F% % Ole CompObj iTS:F obs =MSRMSE=SSR/kSSE/(n-k-1)RR:F obs F a,k,n-k-1 FMicrosoft ʽ 3.0 İ DS Equation Equation.39qObjInfoOlePres000 4Equation Native _1446353757O F% %    .  `@& & MathTypep`Times New RomanT2w@5wx!f,-2 1y Times New RomanT2w@5wx!f,-2 5by2 /22 ,22 222 12Times New RomanT2w@5wx!f,-2 ~s22 :b2 Symbolwb! U4T2w@5wx!f,-2 -a2 SymbolwZ 3 1T2w@5wx!f,-2 -2Symbolwb! U4T2w@5wx!f,-2 2 Times New RomanT2w@5wx!f,-2 ~n2Times New RomanT2w@5wx!f,-2 t2 & "System,x!f, 8l-NANIh30 b 1 t n"2,/2 s b 1  FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj fObjInfoOlePres000+&4 D .  "&`" & MathTypePTimes New RomanT2w@5wf- 2 `feet)2 `B e 2 `(square2 ` s 2 `0.109772 `J . 2 `98.248332 `B  8 2 `T price82 `  r 2 `house82 `. o 2 `: EstimatedSymbolw& `c/T2w@5wf-2 `+o2 ` =o & "Systemf 8l-NANIX.+ Estimated house price =98.24833 +0.10977 (square feet)Equation Native _1446353857F% % Ole CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39qE Z .  & & MathType-d8ObjInfoOlePres000Equation Native _1446353966&F% % Times New RomanT2w@5wf- 2 `0.58082 2   32600.5000 2 n 18934.9348 2 SST8 2 nSSR82 `:RS Times New RomanT2w@5wf-2 c2SSymbolw /BmT2w@5wf-2 `=S2 `=S2 `R=S & "Systemf 8l-NANIý li R 2 =SSRSST=18934.934832600.5000=0.58082 FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoOlePres000B {  .  @& & MathType`Times New RomanT2w@5wf- 2 41.330322 .s1 Times New RomanT2w@5wf-2 1Symbolw$ ,(0T2w@5wf-2 =1 & "Systemf 8l-NANI?0- s  =41.33032 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native [_1446354182F% % Ole CompObj!fObjInfoOlePres000 "REquation Native _1446354388)%F%        !"#$%'(+./0123456789;<=>ADEFGIJKMNOPQRSUVWXYZ[\]^_`abce:   .   & B & MathType-  Times New RomanT2w@5wy$f- 2  329e2  .22 `322 s22 022 4b22 @t2`Times New RomanT2w@5wy$f-2 D512 Times New RomanT2w@5wy$f-2 b22 122 02Symbolw XdT2w@5wy$f-2 422 -22 =2 & "Systemy$f 8l-NANIz0 t 0 =b 1 "0s b 1  H"3.329 FMicrosoft Equation 3.0 DS Equation Equation.39q   .  `@& & MathTypep`Ole CompObj$'fObjInfoOlePres000&(4Times New Roman82w@5wf-2 1y Times New Roman82w@5wf-2 5by2 /22 ,22 222 12Times New Roman82w@5wf-2 ~s22 :b2 Symbolw>& 82w@5wf-2 -a2 Symbolws 82w@5wf-2 -2Symbolw>& 82w@5wf-2 2 Times New Roman82w@5wf-2 ~n2Times New Roman82w@5wf-2 t2 & "Systemf 8P-NANIh0 b 1 t n"2,/2 s b 1 Equation Native &_1446354543+F  Ole )CompObj*-*f FMicrosoft Equation 3.0 DS Equation Equation.39q* l .  &F & MathTypepObjInfo,OlePres000,.-Equation Native :5_14463522933 F  Times New RomanT2w@5w"fX- 2 317.852 ,0) 2 N 0.1098(2002   )2   ) 2 98.255 2 `(sq.ft.)2 `J s 2 `B 0.10982 `  . 2 `98.2582 ` 8 2 `price82 ` r 2 `:house8Symbolw cT2w@5w"fX-2 N=o2 . +o2 N=o2 `. +o2 `N=o & "SystemX"fX 8l-NANIt house price =98.25 +0.1098 (sq.ft.)=98.25 + 0.1098(2000)=317.85Ole ?EPRINT02P-CompObj@oObjInfo15Bl) EMFP-l \KhCF, EMF+@,,F\PEMF+"@ @ $@ 0@?!@ @     !" !" !  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'%     Ld 5!??Ld8!??Rp@ ArialV0YMZ$@f1wvXMZ/V ,_/V<B/V@f1wLO/V$H/TxxI/h&!0 J"& J#/@BuT WAu<q c.dv%    T 2AEA LhSUMMARY OUTPUT## !Rp@ ArialV0YMZ$@f1w=vXMZ/=V =,_/V=<B/V@f1wLO/V$H/TxxI/h&!0# s"&# s#/@BuT WAu<q c.dv% ToAEAoLxRegression Statistics     % T AEA L`Multiple R#   T|AEAL\0.560079 T| AEA L\R Square T|AEAL\0.313688 % % " !% %    ;  T 9AEA LpAdjusted R Square    % % " !% %     T| 9AEA L\0.301855 % % " !% %   ?o  T ?mAEA ?LhStandard Error  % % " !% %     T|?mAEA?L\0.063123 % % " !% %   s  T sAEA r LdObservations! % % " !% %   s  TXpsAEAprLP60% % " !% %   % % " !% %     Tl  AEA LXANOVA!% % " !% %   %   TX*QDAEA*LPdf TX(DAEALPSSTXDAEALPMS#TTDAEALPF'k% % " !% %   E  TDAEALhSignificance F   % % " !% %   % % % " !% %   Iy  T IwAEA I L`Regression % % " !% %     TTIwAEAILP1TxIowAEAIL\0.10563 TxIAwAEAIL\0.10563 T|gIwAEAgIL\26.50969 T|=IwAEA=IL\3.27E-06 T| }AEA }L\Residual  TXp}AEAp}LP58T|}oAEA}L\0.231105 T|}AAEA}L\0.003985 % % " !% %     Tl gAEA LXTotal@  TXpAEApLP59T|oAEAL\0.336734 % % " !% %   % % % " !% %   O  TNAEA LdCoefficients   % % " !% %   vO  TuNAEALhStandard Error  % % " !% %     TpNAEALXt Stat   TxkNAEAkL\P-value. % T SAEA S L`Intercept   T|SAEASL\0.012818 T|SoAEASL\0.008223 T|SAAEASL\1.558903 Tx~SAEA~SL\0.12446 % % " !% %     T` [AEA LTTSET|AEAL\0.887691 T|oAEAL\0.172409 T|AAEAL\5.148756 T|kAEAkL\3.27E-06 % % " !% %   % % " !% %   '% (   Ldipi!??&% 6% Ld!??Ld!??Ld!??% E6E% LdEHE!??Ld!??Ld !??% O6O% LdORO!??Ld!??% ( % " !%   % ( ( " F4(EMF+*@$??FEMF+@  F#Microsoft Office Excel 2003 Biff8Excel.Sheet.89q Oh+'0 `ht   zhuhongtaoMicrosoft Excel@2+nD@pR%WorkbookDOlePres00046 6=SummaryInformation(7C DocumentSummaryInformation8H     !"#$%&'()*+W./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUXY\]^_`abcdefghijklmnopqrstuvwxyz{|}~ \p zhuhongtao Ba==-$e<X@"1Arial1Arial1Arial1Arial1Arial1[SO1h>:[SO1,>:[SO1>:[SO1>:[SO1:[SO1:[SO1:[SO1>:[SO1?:[SO1 :[SO1 :[SO1 :[SO1 :[SO1:[SO1:[SO1 :[SO1:[SO"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)\$#,##0_);\(\$#,##0\)\$#,##0_);[Red]\(\$#,##0\) \$#,##0.00_);\(\$#,##0.00\)% \$#,##0.00_);[Red]\(\$#,##0.00\) 0.000000 0.00000 0.0000 0.000                                  + )            P  P     ff a8         , *      `    p  p "x@ &x@ "x  p   p ||Fc)c}A} .00\)_(*ef;_(@_) }A} .00\)_(*ef;_(@_) }A} .00\)_(*ef;_(@_) }A} .00\)_(*ef;_(@_) }A} .00\)_(*ef;_(@_) }A} .00\)_(*ef ;_(@_) }A} .00\)_(*L;_(@_) }A} .00\)_(*L;_(@_) }A} .00\)_(*L;_(@_) }A} .00\)_(*L;_(@_) }A} .00\)_(*L;_(@_) }A} .00\)_(*L ;_(@_) }A} .00\)_(*23;_(@_) }A} .00\)_(*23;_(@_) }A} .00\)_(*23;_(@_) }A} .00\)_(*23;_(@_) }A}  .00\)_(*23;_(@_) }A}! .00\)_(*23 ;_(@_) }A}$ a.00\)_(*;_(@_) }A}% .00\)_(*;_(@_) }A}& .00\)_(*;_(@_) }A}' .00\)_(*;_(@_) }A}( .00\)_(*;_(@_) }A}) .00\)_(*;_(@_) }A}* .00\)_(*;_(@_) }A}+ .00\)_(* ;_(@_) }-}, .00\)_(*}A}- .00\)_(*;_(@_) }A}. .00\)_(*?;_(@_) }A}/ .00\)_(*23;_(@_) }-}0 .00\)_(*}}1 .00\)_(*;_(@_) ??? ??? ??? ???}U}2 .00\)_(*;_(@_)  }x}3.00\)_(*;_( ??? ???}-}5 .00\)_(*}-}6 .00\)_(*}}7 }.00\)_(*;_(    }}: ??v.00\)_(*̙;_(    }}; ???.00\)_(*;_(??? ???  ??? ???}A}< e.00\)_(*;_(}A}= }.00\)_(*;_(}(}D.00\)_(*}(}E.00\)_(*!20% - :_eW[r 1O20% - :_eW[r 1 ef %!20% - :_eW[r 2O"20% - :_eW[r 2 ef %!20% - :_eW[r 3O&20% - :_eW[r 3 ef %!20% - :_eW[r 4O*20% - :_eW[r 4 ef %!20% - :_eW[r 5O.20% - :_eW[r 5 ef %!20% - :_eW[r 6O220% - :_eW[r 6  ef %!40% - :_eW[r 1O40% - :_eW[r 1 L %!40% - :_eW[r 2O#40% - :_eW[r 2 L湸 %!40% - :_eW[r 3O'40% - :_eW[r 3 L %!40% - :_eW[r 4O+40% - :_eW[r 4 L %!40% - :_eW[r 5O/40% - :_eW[r 5 L %!40% - :_eW[r 6O340% - :_eW[r 6  Lմ %!60% - :_eW[r 1O 60% - :_eW[r 1 23 %!60% - :_eW[r 2O$60% - :_eW[r 2 23ٗ %!60% - :_eW[r 3O(60% - :_eW[r 3 23֚ %!60% - :_eW[r 4O,60% - :_eW[r 4 23 %! 60% - :_eW[r 5O060% - :_eW[r 5 23 %!!60% - :_eW[r 6O460% - :_eW[r 6  23 %"CSMOR#$CSMOR[0]$}Y5}Y  a%%]5]  %8^ĉ&:_eW[r 1C:_eW[r 1 O %':_eW[r 2C!:_eW[r 2 PM %(:_eW[r 3C%:_eW[r 3 Y %):_eW[r 4C):_eW[r 4 d %*:_eW[r 5C-:_eW[r 5 K %+:_eW[r 6C1:_eW[r 6  F % ,h+h I}% -h 1=h 1 I}%O .h 2=h 2 I}%? /h 3=h 3 I}%23 0h 4/h 4 I}%1hgUSCQ=                                        > >  >0EH?>t_lW? >܁:э?>s`? BB >%>jGq? ??}2*? >S?>*)t EE@Ѵw? >XU>&Kz ??j}Q? >#ng_y?>s? ??,3B(? >AN?>dVp;? @~ @N@ > ?>%VF#  >I ٩ >5w     >LQԙ{ʿ >v=̿ A A A A A A   >ǚA >Fa ? ~ ?? ?.؊ ? ?.؊ ? ?O{:@ ?VXn>  ~ >@%@ >fd? ? ~ ?M@ ?sה? ?iRp? ??  >vp71$? >⟿ @ ~ @M@ @? @@@ >?>/hҩ? >;>_?A A A A ACCCC >A>B4-? ? ?]@??0̕ր??['D??x\3ܿ????? >I-LN휿>l# DDOIg?@7P~?@JS@@Xn>???? >dWZF=?>k)3? >,baL>.X x >%:,B>c&Q/4 >?>}"Oi? >O`Ü>bN&O? >nQfL2>A >󬤅?>_Z'? >x[鵱>+F<ٱ? >qN`:{?>8L >H'>V6z? >0o?>9? >.҈?>gE(?>Y-? >xb֋?>i? D lD@L\\\\X@Lth@x@@@@@@@@@@@@@ !"#$%&'()*+,-./0123456789:;< >LD? >~ IJ?!>V 3!>#fF?">V{ ?">>Vp#>MK砿#>XyV?$>lg?$>1 f?%>; ?%>HLP÷&>&^?&> ce'> LnX'>Ց#(>0B(>ӈ})>}E?)>D+g?*>sѐ(u*>:vPx+>P+>K{?,>WL,>ѫJC->]M->oH.>f/?.>EN唿/>Ndǚ?0>}r ?0>VI?1>ۥ o1>a{?2>岩?2>׉ D?3>}#g]?3>[ ?4>VF??4> # v5>ڪ$?5><|?6>h󊧞6>uoE7>ɭI%r?7>et?8>tH8>fbal9> GJ9>SYvQ:>l??:> ?;>c^'?;> b<>PR`L?<>"rl=C?>X0$$$$$$$$$$$$$$$$$$$$$$$$$$$$>@=C:C7ggD 'Y < z&" WMFC~ *l853+ EMF*8>`<F, EMF+@``F\PEMF+"@ @ $@ 0@?!@ @     !" !" !  " !  9" !  9" !  9" !  9'%     LdEVE!??Ld&8&!??Rp@ ArialV0YMZ$@f1wBvXMZ/B BԳ_/BB/@f1wO/ȵ/ #7I/&F$ e% h $ e#/@@}8L}TbQv~Tdv%    T|r BUU BLhSUMMARY OUTPUT   Rp@ ArialV0YMZ$@f1wBvXMZ/B BԳ_/BB/@f1wO/ȵ/ #7I/&F$ g% h $ g#/ @  dv% % % " !% %   $4  T$}3r BUU B$LxRegression Statisticsp  % % " !% %   9%   T5<Dr BUU B5 L`Multiple R  T|I5}Dr BUU BI5L\0.560079T|F8Ur BUU BFL\R Square  T|IF}Ur BUU BIFL\0.313688% % " !% %   W@g  TW?fr BUU BWLpAdjusted R Squarea   % % " !% %   9  T|IW}fr BUU BIWL\0.301855% % " !% %   h@x  Th?wr BUU BhLhStandard Error  % % " !% %   9  T|Ih}wr BUU BIhL\0.063123% % " !% %   y@  Ty?r BUU By LdObservations % % " !% %   9  TXpy}r BUU BpyLP60Tl0r BUU BLXANOVA % TXZer BUU BZLPdfTXr BUU BLPSS TXr BUU BLPMS TT$r BUU BLPF% % " !% %   A  TAr BUU B6LhSignificance F % % " !% %   9% % % " !% %   @  T?r BUU B L`Regression % % " !% %   9  TTw}r BUU BwLP1Txr BUU BL\0.10563lTxr BUU BL\0.10563T| =r BUU B L\26.50969T|J}r BUU BJL\3.27E-06 T|4r BUU BL\Residual TXp}r BUU BpLP58T|r BUU BL\0.231105T|r BUU BL\0.003985Tlr BUU BLXTotalTXp}r BUU BpLP59T|r BUU BL\0.336734% % % " !% %     T=r BUU B= LdCoefficients % % " !% %     Tr BUU BvLhStandard Error  % % " !% %   9  Tpr BUU BLXt Stat Tx 4r BUU B L\P-value r& WMFC *% T4%r BUU B L`InterceptT|I}%r BUU BIL\0.012818T|%r BUU BL\0.008223T|%r BUU BL\1.558903Tx=%r BUU BL\0.12446.T`'6r BUU B'LTTSE T|I'}6r BUU BI'L\0.887691T|'6r BUU B'L\0.172409T|'6r BUU B'L\5.148756T| '=6r BUU B 'L\3.27E-06 % % " !% %   9'% (   Ld"#"!??&% 464% Ld444!??Ld!??Ld!??% 6% Ld!??Ld!??Ld@A!??% 6A% Ld@A!??Ld7@87A!??% ( % " !%   9" F4(EMF+*@$??FEMF+@   9  '' ' ,9' ,9' ,9' ,9-   @ !E @ !&@ Arial- 2 SUMMARY OUTPUT   @ Arial--"System???{?{?{?{?-'-- ,4$ '2 $Regression Statistics  --'-- ,9- 2 5 Multiple R  2 5I0.5600792 FR Square  2 FI0.313688--'-- ,g@W !2 WAdjusted R Squaret   --'-- ,9 2 WI0.301855--'-- ,x@h 2 hStandard Error  --'-- ,9 2 hI0.063123--'-- ,@y 2 y Observations --'-- ,9  2 yp602 ANOVA1 - 2 Zdf 2 SS 2 MS 2 F--'-- ,A 2 6Significance F --'-- ,9---'-- ,@ 2  Regression --'-- ,9  2 w12 0.1056332 0.1056332 26.509692 J3.27E-06 2 Residual  2 p582 0.2311052 0.0039852 Total9 2 p592 0.336734---'-- , 2 = Coefficients --'-- , 2 vStandard Error  --'-- ,9 2 t Stat 2  P-value4 -2  InterceptE2 I0.0128182 0.0082232 1.5589032 0.124463 2 'TSE2 2 'I0.8876912 '0.1724092 '5.1487562 ' 3.27E-06 --'-- ,9-  @ !"-44- @ !4 @ ! @ !-- @ ! @ ! @ !A-A- @ !A @ !A7--'- ,9'NANI՜.+,0HP X`hp x  Sheet1  Oh+'0  0< \ h t <Regression: Finding the equation of the line ofGD* $M$ J$P4H]{n  w?BQ)? # fdjLgyРV3Wd1,Wʪ γZ'c*z$0N˃ȮOG Ɣ"ܛh*ntݣ˾mϚoEυ}Rlon?,NE,$r!R`R^3xݘYD"H""k*E+î :RUrQN^OBPfُI|ʋ=+UGqvh5_-fLȓ| rvwA&jA`!bv1XA>& H0xUPJA;M zZJSF 6)rF8yB"Z ,R`aoz!Ѕw}f*@/0êȼBdZk{#3ZH|EXg5ؽpelvWk턗4(쿚\p{(YjRia@N!;z{plRWMe Dd b  c $A? ?3"`?2 T:i0J!Kx- `! 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