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Chapter 6 (cont.) Regression Estimation

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Simple Linear Regression: review of least squares procedure 2

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Introduction n We will examine the relationship between quantitative variables x and y via a mathematical equation. n x: explanatory variable n y: response variable n Data: 3

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The Model 4 House size House Cost Most lots sell for $25,000 Building a house costs about $75 per square foot. House cost = 25000 + 75(Size) The model has a deterministic and a probabilistic component

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The Model 5 House cost = 25000 + 75(Size) House size House Cost Most lots sell for $25,000 However, house costs vary even among same size houses! Since cost behave unpredictably, we add a random component.

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The Model n The first order linear model y = response variable x = explanatory variable 0 = y-intercept 1 = slope of the line = error variable 6 x y 00 Run Rise = Rise/Run 0 and 1 are unknown population parameters, therefore are estimated from the data.

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Estimating the Coefficients n The estimates are determined by –drawing a sample from the population of interest, –calculating sample statistics. –producing a straight line that cuts into the data. 7 Question: What should be considered a good line? x y

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The Least Squares (Regression) Line 8

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9 3 3 4 1 1 4 (1,2) 2 2 (2,4) (3,1.5) Sum of squared differences =(2 - 1) 2 +(4 - 2) 2 +(1.5 - 3) 2 + (4,3.2) (3.2 - 4) 2 = 6.89 Sum of squared differences =(2 -2.5) 2 +(4 - 2.5) 2 +(1.5 - 2.5) 2 +(3.2 - 2.5) 2 = 3.99 2.5 Let us compare two lines The second line is horizontal The smaller the sum of squared differences the better the fit of the line to the data.

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The Estimated Coefficients 10 To calculate the estimates of the slope and intercept of the least squares line, use the formulas: The least squares prediction equation that estimates the mean value of y for a particular value of x is:

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Example: Consumer’s Union recently evaluated 26 brands of frozen pizza based on taste (y) We will examine the taste scores (y) and the corresponding fat content (x). 11 Simple Linear Regression

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The Simple Linear Regression Line (example, cont.) 12 Solution –Solving by hand: Calculate a number of statistics where n = 26.

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The Simple Linear Regression Line (example, cont.) 13 Solution – continued –Using the computer 1. Scatterplot 2. Trend function 3. Data tab > Data Analysis > Regression

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Regression Statistics Multiple R0.723546339 R Square0.523519305 Adjusted R Square0.503665943 Standard Error8.785081398 Observations26 ANOVA dfSSMSFSignificance F Regression12035.1208912035.12126.36932.95293E-05 Residual241852.26372477.17766 Total253887.384615 CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept39.002083225.5610982197.0133782.99E-0727.524540650.47962583 Fat1.726028940.3361234075.1351052.95E-051.0323043242.419753555 The Simple Linear Regression Line (example, cont.) 14

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The Simple Linear Regression Line (example, cont.) 15

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Regression estimator of a population mean y

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