Chapter 11 Correlation and Simple Linear Regression Statistics for Business (Econ) 1
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 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 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 Positive correlation
6 Negative correlation
7 Non-linear relationship
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9 A strong positive relationship, approximately +0.90; A relatively weak negative correlation, approximately -0.40
10 A perfect negative correlation, No linear trend, 0.00.
11 A demonstration of how one extreme data point (an outrider) can influence the value of a correlation.
12 A demonstration of how one extreme data point (an outrider) can influence the value of a correlation.
13 Pearson correlation The most common measure of correlation is the Pearson Product Moment Correlation (called Pearson's correlation for short). = = correlation coefficient
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 Least square fit
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30 The linear model y = dependent variable x = independent variable 0 = y-intercept 1 = slope of the line = error variable x y 00 Run Rise = Rise/Run 0 and 1 are unknown, therefore, are estimated from the data.
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 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 Solution – Solving by hand To calculate b 0 and b 1 we need to calculate several statistics first; where n = 100.
34 – Using the computer (see file Xm17-01.xls) Tools > Data analysis > Regression > [Shade the y range and the x range] > OK
35 This is the slope of the line. For each additional mile on the odometer, the price decreases by an average of $ The intercept is b 0 = No data Do not interpret the intercept as the “Price of cars that have not been driven”