Business statistics
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Correlation and Simple Linear Regression Analysis 1
Correlation and Simple Linear Regression
Analysis
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Correlation and Simple Linear Regression Analysis 2
Learning Objectives
Upon completion of this chapter, you will be able to:
Ø Use the simple linear regression equation Ø Compute the coefficient of correlation and understand its
interpretation. Ø Understand the concept of measures of variation, coefficient of
determination, and standard error of the estimate Ø Understand and use residual analysis for testing the
assumptions of regression Ø Measure autocorrelation by using the Durbin–Watson statistic Ø Understand statistical inference about slope, correlation
coefficient of the regression model, and testing the overall model
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Correlation and Simple Linear Regression Analysis 3
Measures of Association
Ø Measures of association are statistics for measuring the strength of relationship between two variables.
Ø Correlation measures the degree of association between two variables.
Ø Karl Pearson’s coefficient of correlation is a quantitative measure of the degree of relationship between two variables. Suppose these variables are x and y, then Karl Pearson’s coefficient of correlation is defined as
Ø The coefficient of correlation lies in between +1 and –1.
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Correlation and Simple Linear Regression Analysis 4
Figure 15.1: Interpretation of correlation coefficient
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Correlation and Simple Linear Regression Analysis 5
Table 15.2 shows the sales revenue and advertisement expenses of a company for the past 10 months. Find the coefficient of correlation between sales and advertisement.
Example 15.1
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Table 15.3 : Calculation of correlation coefficient between sales and advertisement
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Correlation and Simple Linear Regression Analysis 7
Figure 15.9: Five examples of correlation coefficient
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Using MS Excel, Minitab and SPSS for Computing Correlation Coefficient Ø Ch 15 Solved Examples\Excel\Ex 15.1.xls Ø Ch 15 Solved Examples\Minitab\Ex 15.1.MPJ Ø Ch 15 Solved Examples\SPSS\Ex 15.1.sav Ø Ch 15 Solved Examples\SPSS\Output Ex 15.1.spv
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Correlation and Simple Linear Regression Analysis 9
Introduction to Simple Linear Regression
Ø Regression analysis is the process of developing a statistical model, which is used to predict the value of a dependent variable by at least one independent variable.
Ø In simple linear regression analysis, there are two types of variables. The variable whose value is influenced or to be predicted is called dependent variable and the variable which influences the value or is used for prediction is called independent variable.
Ø In regression analysis, independent variable is also known as regressor or predictor, or explanatory while the dependent variable is also known as regressed or explained variable. In a simple linear regression analysis, only a straight line relationship between two variables is examined.
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Correlation and Simple Linear Regression Analysis 10
A Deterministic and Probabilistic Model
ε is the error of the regression line in fitting the points of the regression equation. If a point is on the regression line, the corresponding value of ε is equal to zero. If the point is not on the regression line, the value of ε measures the error. It can be noticed that in the deterministic model, all the points are assumed to be on the regression line and hence, in all the cases random error ε is equal to zero. Probabilistic model includes an error term which allows the value of y to vary for any given value of x.
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Correlation and Simple Linear Regression Analysis 11
Figure 15.10: Error in simple regression
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Correlation and Simple Linear Regression Analysis 12
Figure 15.11: Summary of the estimation process for simple linear regression.
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Correlation and Simple Linear Regression Analysis 13
A cable wire company has spent heavily on advertisements. The sales and advertisement expenses (in thousand rupees) for the 12 randomly selected months are given in Table 14.2. Develop a regression model to predict the impact of advertisement on sales.
Example 15.2
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Correlation and Simple Linear Regression Analysis 14
Using Ms Excel, Minitab, and Spss for Simple Linear Regression
Ø Ch 15 Solved Examples\Excel\Ex 15.2.xls Ø Ch 15 Solved Examples\Minitab\EX 15.2.MPJ Ø Ch 15 Solved Examples\SPSS\Ex 15.2.sav Ø Ch 15 Solved Examples\SPSS\Output Ex 15.2.spv