1. Chapter 11 Correlation and Simple Linear Regression Statistics for Business (Econ) 1

2. 2 Introduction In this chapter we employ Regression Analysis to examine the relationship among quantitative variables. The technique is used to predict the value of one variable (the dependent variable - y) based on the value of other variables (independent variables x 1, x 2,…x k.)

3. 3 Correlation is a statistical technique that is used to measure and describe a relationship between two variables. The correlation between two variables reflects the degree to which the variables are related. For example: A researcher interested in the relationship between nutrition and IQ could observe the dietary patterns for a group of children and then measure their IQ scores. A business analyst may wonder if there is any relationship between profit margin and return on capital for a group of public companies.

4. 4 A set of n= 6 pairs of scores (X and Y values) is shown in a table and in a scatterplot. The scatterplot allows you to see the relationship between X and Y.

5. 5 Positive correlation

6. 6 Negative correlation

7. 7 Non-linear relationship

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9. 9 A strong positive relationship, approximately +0.90; A relatively weak negative correlation, approximately -0.40

10. 10 A perfect negative correlation, -1.00 No linear trend, 0.00.

11. 11 A demonstration of how one extreme data point (an outrider) can influence the value of a correlation.

12. 12 A demonstration of how one extreme data point (an outrider) can influence the value of a correlation.

13. 13 Pearson correlation The most common measure of correlation is the Pearson Product Moment Correlation (called Pearson's correlation for short). = = correlation coefficient

14. 14 The value r 2 is called the coefficient of determination because it measures the proportion of variability in one variable that can be determined from the relationship with the other variable. A correlation of r =0.80 (or -0.80), for example, means that r 2 =0.64 (or 64%) of the variability in the Y scores can be predicted from the relationship with X.

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25. 25 Least square fit

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30. 30 The linear model y = dependent variable x = independent variable  0 = y-intercept  1 = slope of the line = error variable x y 00 Run Rise   = Rise/Run  0 and  1 are unknown, therefore, are estimated from the data.

31. 31 To calculate the estimates of the coefficients that minimize the differences between the data points and the line, use the formulas: The regression equation that estimates the equation of the first order linear model is:

32. 32 Example 12.1 Relationship between odometer reading and a used car’s selling price. – A car dealer wants to find the relationship between the odometer reading and the selling price of used cars. – A random sample of 100 cars is selected, and the data recorded. – Find the regression line. Independent variable x Dependent variable y

33. 33 Solution – Solving by hand To calculate b 0 and b 1 we need to calculate several statistics first; where n = 100.

34. 34 – Using the computer (see file Xm17-01.xls) Tools > Data analysis > Regression > [Shade the y range and the x range] > OK

35. 35 This is the slope of the line. For each additional mile on the odometer, the price decreases by an average of $0.0312 The intercept is b 0 = 6533. 6533 0 No data Do not interpret the intercept as the “Price of cars that have not been driven”