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Chapter 13

Chapter 13. Simple Linear Regression Analysis. The Goal. This chapter talks about methods for Measuring linear correlation between two variables Describing a linear relationship between two variables with a linear equation Making predictions with linear regression model

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Chapter 13

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  1. Chapter 13 Simple Linear Regression Analysis

  2. The Goal This chapter talks about methods for • Measuring linear correlation between two variables • Describing a linear relationship between two variables with a linear equation • Making predictions with linear regression model • Describing the usefulness of a linear regression model

  3. Different Values of the CorrelationCoefficient

  4. Measure the linear Relationship: The Simple Correlation Coefficient The linear coefficient (or simple correlation coefficient) r is a numerical measure of the strength of the linear relationship between two variables representing quantitative data.

  5. Interpret the Correlation Coefficient r • If r>0 we say the two variables are positively correlated; if r<0 we say they are negatively correlated. • If the absolute value of r is ≥ 0.8, we say the linear relationship is strong; • If the absolute value of r is below 0.8 but ≥ 0.5, we say the linear relationship is moderate. • If the absolute value of r is below 0.5, we say the linear relationship is weak.

  6. Properties of the Correlation Coefficient r The value of r is always between -1 and +1. 2. The value for r does not change if all values of either variables are converted to a different scale. 3. The value of r is not affected by the choice of x or y. Interchange all x and y values and the value of r will not change. 4. r measures the strength of a linear relationship. It is not designed to measure the strength of a relationship that is not linear.

  7. The Simple Linear Regression Model and the Least Squares Point Estimates • The dependent (or response) variable is the variable we wish to understand or predict, denoted by Y • The independent (or predictor or explanatory ) variable is the variable we will use to understand or predict the dependent variable, denoted by X • Regression analysis is a statistical technique that uses observed data to relate the dependent variable to one or more independent variables

  8. Form of The Simple LinearRegression Model • Y = β0 + β1X + ε • β0 + β1X is the mean value of the dependent variable Y when the value of the independent variable is X. The mean is in the form of a linear function. The mean determines the overall trend of the relationship between X and Y. • β0 is the y-intercept, the mean of y when X is 0; β1 is the slope, the change in the mean of Y per unit change in X • εis an error term that describes the effect on Y of all factors other than X • ŷ = b0 + b1x, ŷ is the estimate of mean value of Y when X=x

  9. The Least Squares Estimation Method

  10. The Simple Linear Regression Model Illustrated

  11. The Least Squares Point Estimates • Estimation/prediction equationŷ = b0 + b1x • Least squares point estimate of the slope β1

  12. The Least Squares Point EstimatesContinued • Least squares point estimate of the y-intercept 0

  13. Testing the Significance of the Slope • A regression model is not likely to be useful unless there is a significant relationship between x and y • To test significance, we use the null hypothesis:H0: β1 = 0 • Versus the alternative hypothesis:Ha: β1 ≠ 0

  14. Testing the Significance of the Slope #2

  15. The Simple Coefficient ofDetermination and Correlation • How useful is a particular regression model? • One measure of usefulness is the simple coefficient of determination • It is represented by the symbol r2 , because it is actually equal to the square of (simple) Correlation Coefficient which is denoted by r. • It is interpreted as the percentage of variation in Y that could be explained by the linear regression line b0 + b1x

  16. Prediction • To estimate the mean value of Y for X= x0, one just need to plug x0 into the regression line formula and calculate the estimate of Y by b0 + b1x0 . We usually denote the estimated mean value of Y from the regression line by ŷ = b0 + b1x0 and call ŷ the fitted value for X= x0. • window

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