1. Chapter 12 Simple Regression Statistika

2.  Analisis regresi adalah analisis hubungan linear antar 2 variabel random yang mempunyai hub linear,  Variabel bebas (variabel pengaruh)  Variabel respon (variabel terpengaruh)  Regression analysis is used to:  Predict the value of a dependent variable (Y) based on the value of at least one independent variable (X)  Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain Independent variable: the variable used to explain the dependent variable Introduction to Regression Analysis

3.  The relationship between X and Y is described by a linear function  Changes in Y are assumed to be caused by changes in X  Linear regression population equation model  Where  0 and  1 are the population model coefficients and will be estimated from data  and  is a random error term. Linear Regression Model

4. Linear component Simple Linear Regression Model The population regression model: Population Y intercept Population Slope Coefficient Random Error term Dependent Variable Independent Variable Random Error component

5. Random Error for this X i value Y X Observed Value of Y for X i Predicted Value of Y for X i XiXi Slope = β 1 Intercept = β 0 εiεi Simple Linear Regression Model

6.  b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared differences between y and : Least Squares Estimators Differential calculus is used to obtain the coefficient estimators b 0 and b 1 that minimize SSE

7.  The slope coefficient estimator is  And the constant or y-intercept is  The regression line always goes through the mean x, y Least Squares Estimators

8. The simple linear regression equation provides an estimate of the population regression line Simple Linear Regression Equation Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) y value for observation i Value of x for observation i The individual random error terms e i have a mean of zero

9.  The coefficients b 0 and b 1, and other regression results in this chapter, will be found using a computer  Hand calculations are tedious (boring)  Statistical routines are built into Excel  Other statistical analysis software can be used Finding the Least Squares Equation

10.  b 0 is the estimated average value of y when the value of x is zero (if x = 0 is in the range of observed x values)  b 1 is the estimated change in the average value of y as a result of a one- unit change in x Interpretation of the Slope and the Intercept

11.  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 Simple Linear Regression Example

12. Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X) 2451400 3121600 2791700 3081875 1991100 2191550 4052350 3242450 3191425 2551700

13. Graphical Presentation  House price model: scatter plot  From the scatter plot there is a linear trend

14.  Tools / Data Analysis / Regression Regression Using Excel

15. Excel Output Regression Statistics Multiple R0.76211 R Square0.58082 Adjusted R Square0.52842 Standard Error41.33032 Observations10 ANOVA dfSSMSFSignificance F Regression118934.9348 11.08480.01039 Residual813665.56521708.1957 Total932600.5000 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept98.2483358.033481.692960.12892-35.57720232.07386 Square Feet0.109770.032973.329380.010390.033740.18580 The regression equation is:

16.  House price model: scatter plot and regression line Graphical Presentation Slope = 0.10977 Intercept = 98.248

17.  b 0 is the estimated average value of Y when the value of X is zero (if X = 0 is in the range of observed X values)  Here, no houses had 0 square feet, so b 0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet Interpretation of the Intercept, b 0 House price = 98.24833 + 0.10977*(squarefeet)

18.  b 1 measures the estimated change in the average value of Y as a result of a one-unit change in X  Here, b 1 =.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 Interpretation of the Slope Coefficient, b 1 House price = 98.24833 + 0.10977*(squarefeet)