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**Chapter 12 Simple Linear Regression**

Simple Linear Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance Using the Estimated Regression Equation for Estimation and Prediction Computer Solution Residual Analysis: Validating Model Assumptions

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**Simple Linear Regression Model**

The equation that describes how y is related to x and an error term is called the regression model. The simple linear regression model is: y = b0 + b1x +e b0 and b1 are called parameters of the model. e is a random variable called the error term.

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**Simple Linear Regression Equation**

The simple linear regression equation is: E(y) = 0 + 1x Graph of the regression equation is a straight line. b0 is the y intercept of the regression line. b1 is the slope of the regression line. E(y) is the expected value of y for a given x value.

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**Simple Linear Regression Equation**

Positive Linear Relationship E(y) Regression line Intercept b0 Slope b1 is positive x

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**Simple Linear Regression Equation**

Negative Linear Relationship E(y) Intercept b0 Regression line Slope b1 is negative x

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**Simple Linear Regression Equation**

No Relationship E(y) Regression line Intercept b0 Slope b1 is 0 x

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**Estimated Simple Linear Regression Equation**

The estimated simple linear regression equation is: The graph is called the estimated regression line. b0 is the y intercept of the line. b1 is the slope of the line. is the estimated value of y for a given x value.

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**Estimation Process Regression Model y = b0 + b1x +e**

Regression Equation E(y) = b0 + b1x Unknown Parameters b0, b1 Sample Data: x y x y1 xn yn b0 and b1 provide estimates of Estimated Regression Equation Sample Statistics b0, b1

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**Least Squares Method Least Squares Criterion where:**

yi = observed value of the dependent variable for the ith observation yi = estimated value of the dependent variable ^

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**The Least Squares Method**

Slope for the Estimated Regression Equation

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**The Least Squares Method**

y-Intercept for the Estimated Regression Equation where: xi = value of independent variable for ith observation yi = value of dependent variable for ith observation x = mean value for independent variable y = mean value for dependent variable n = total number of observations _ _

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**Example: Reed Auto Sales**

Simple Linear Regression Reed Auto periodically has a special week-long sale. As part of the advertising campaign Reed runs one or more television commercials during the weekend preceding the sale. Data from a sample of 5 previous sales are shown on the next slide.

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**Example: Reed Auto Sales**

Simple Linear Regression Number of TV Ads Number of Cars Sold 1 14 3 24 2 18 1 17 3 27

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**Example: Reed Auto Sales**

Slope for the Estimated Regression Equation b1 = (10)(100)/5 = _____ 24 - (10)2/5 y-Intercept for the Estimated Regression Equation b0 = (2) = _____ Estimated Regression Equation y = x ^

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**Example: Reed Auto Sales**

Scatter Diagram ^

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**The Coefficient of Determination**

Relationship Among SST, SSR, SSE SST = SSR + SSE where: SST = total sum of squares SSR = sum of squares due to regression SSE = sum of squares due to error ^

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**The Coefficient of Determination**

The coefficient of determination is: r2 = SSR/SST where: SST = total sum of squares SSR = sum of squares due to regression

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**Example: Reed Auto Sales**

Coefficient of Determination r2 = SSR/SST = 100/114 = The regression relationship is very strong because 88% of the variation in number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.

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**The Correlation Coefficient**

Sample Correlation Coefficient where: b1 = the slope of the estimated regression equation

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**Example: Reed Auto Sales**

Sample Correlation Coefficient The sign of b1 in the equation is “+”. rxy =

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**Model Assumptions Assumptions About the Error Term **

The error is a random variable with mean of zero. The variance of , denoted by 2, is the same for all values of the independent variable. The values of are independent. The error is a normally distributed random variable.

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**Testing for Significance**

To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 is zero. Two tests are commonly used t Test F Test Both tests require an estimate of s 2, the variance of e in the regression model.

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**Testing for Significance**

An Estimate of s 2 The mean square error (MSE) provides the estimate of s 2, and the notation s2 is also used. s2 = MSE = SSE/(n-2) where:

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**Testing for Significance**

An Estimate of s To estimate s we take the square root of s 2. The resulting s is called the standard error of the estimate.

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**Testing for Significance: t Test**

Hypotheses H0: 1 = 0 Ha: 1 = 0 Test Statistic where

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**Testing for Significance: t Test**

Rejection Rule Reject H0 if t < -tor t > t where: t is based on a t distribution with n - 2 degrees of freedom

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**Example: Reed Auto Sales**

t Test Hypotheses H0: 1 = 0 Ha: 1 = 0 Rejection Rule For = .05 and d.f. = 3, t.025 = _____ Reject H0 if t > t.025 = _____

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**Example: Reed Auto Sales**

t Test Test Statistics t = _____/_____ = 4.63 Conclusions t = 4.63 > 3.182, so reject H0

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**Confidence Interval for 1**

We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test. H0 is rejected if the hypothesized value of 1 is not included in the confidence interval for 1.

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**Confidence Interval for 1**

The form of a confidence interval for 1 is: where b1 is the point estimate is the margin of error is the t value providing an area of a/2 in the upper tail of a t distribution with n - 2 degrees of freedom

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**Example: Reed Auto Sales**

Rejection Rule Reject H0 if 0 is not included in the confidence interval for 1. 95% Confidence Interval for 1 = 5 +/ (1.08) = 5 +/- 3.44 or ____ to ____ Conclusion 0 is not included in the confidence interval. Reject H0

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**Testing for Significance: F Test**

Hypotheses H0: 1 = 0 Ha: 1 = 0 Test Statistic F = MSR/MSE

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**Testing for Significance: F Test**

Rejection Rule Reject H0 if F > F where: F is based on an F distribution with 1 d.f. in the numerator and n - 2 d.f. in the denominator

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**Example: Reed Auto Sales**

F Test Hypotheses H0: 1 = 0 Ha: 1 = 0 Rejection Rule For = .05 and d.f. = 1, 3: F.05 = ______ Reject H0 if F > F.05 = ______.

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**Example: Reed Auto Sales**

F Test Test Statistic F = MSR/MSE = ____ / ______ = 21.43 Conclusion F = > 10.13, so we reject H0.

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**Some Cautions about the Interpretation of Significance Tests**

Rejecting H0: b1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y. Just because we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.

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**Using the Estimated Regression Equation for Estimation and Prediction**

Confidence Interval Estimate of E(yp) Prediction Interval Estimate of yp yp + t/2 sind where: confidence coefficient is 1 - and t/2 is based on a t distribution with n - 2 degrees of freedom

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**Example: Reed Auto Sales**

Point Estimation If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be: y = (3) = ______ cars ^

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**Example: Reed Auto Sales**

Confidence Interval for E(yp) 95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is: = ______ to _______ cars

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**Example: Reed Auto Sales**

Prediction Interval for yp 95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is: = _____ to ______ cars

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**Residual Analysis Residual for Observation i yi – yi**

Standardized Residual for Observation i where: and ^ ^ ^ ^

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**Example: Reed Auto Sales**

Residuals

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**Example: Reed Auto Sales**

Residual Plot

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Residual Analysis Residual Plot Good Pattern Residual x

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Residual Analysis Residual Plot Nonconstant Variance Residual x

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**Model Form Not Adequate**

Residual Analysis Residual Plot Model Form Not Adequate Residual x

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End of Chapter 12

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