Least Squares Criterion b and b 1 are obtained by finding the values

# Least squares criterion b and b 1 are obtained by

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Chap 14-24 Least Squares Criterion b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared residuals 2 1 0 2 2 x)) b (b (y ) y ˆ (y e + - = - =
. Chap 14-25 The Least Squares Equation The formulas for b 1 and b 0 are: algebraic equivalent for b 1 : ∑ ∑ - - = n ) x ( x n y x xy b 2 2 1 - - - = 2 1 ) x (x ) y )(y x (x b x b y b 1 0 - = and
. Chap 14-26 b 0 is the estimated average value of y when the value of x is zero 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
. Chap 14-27 Finding the Least Squares Equation The coefficients b 0 and b 1 will usually be found using computer software, such as Excel or Minitab Other regression measures will also be computed as part of computer-based regression analysis
. Chap 14-28 Simple Linear Regression Example 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
. Chap 14-29 Sample Data for House Price Model House Price in \$1000s (y) Square Feet (x) 245 1400 312 1600 279 1700 308 1875 199 1100 219 1550 405 2350 324 2450 319 1425 255 1700
. Chap 14-30 Regression Using Excel Data / Data Analysis / Regression
. Chap 14-31 Excel Output Regression Statistics Multiple R 0.76211 R Square 0.58082 Adjusted R Square 0.52842 Standard Error 41.33032 Observations 10 ANOVA df SS MS F Significance F Regression 1 18934.9348 18934.9348 11.0848 0.01039 Residual 8 13665.5652 1708.1957 Total 9 32600.5000 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386 Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580 The regression equation is: feet) (square 0.10977 98.24833 price house + =
. Chap 14-32 0 50 100 150 200 250 300 350 400 450 0 500 1000 1500 2000 2500 3000 Square Feet House Price (\$1000s) Graphical Presentation House price model: scatter plot and regression line feet) (square 0.10977 98.24833 price house + = Slope = 0.10977 Intercept = 98.248
. Chap 14-33 Interpretation of the Intercept, b 0 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 feet) (square 0.10977 98.24833 price house + =
. Chap 14-34 Interpretation of the Slope Coefficient, b 1 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 feet) (square 0.10977 98.24833 price house + =
. Chap 14-35 Least Squares Regression Properties The sum of the residuals from the least squares regression line is 0 ( ) The sum of the squared residuals is a minimum (minimized ) The simple regression line always passes through the mean of the y variable and the mean of the x variable The least squares coefficients are unbiased estimates of β 0 and β 1 0 ) y (y = - ˆ 2 ) y (y ˆ -
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