13- 2 Chapter Thirteen Linear Regression and Correlation GOALS When you have completed this chapter, you will be able to: ONE Draw a scatter diagram. TWO Understand and interpret the terms dependent variable and independent variable. THREE Calculate and interpret the coefficient of correlation, the coefficient of determination, and the standard error of estimate. FOUR Conduct a test of hypothesis to determine if the population coefficient of correlation is different from zero. Goals
13- 3 Chapter Thirteen continued Linear Regression and Correlation GOALS When you have completed this chapter, you will be able to: FIVE Calculate the least squares regression line and interpret the slope and intercept values. SIX Construct and interpret a confidence interval and prediction interval for the dependent variable. SEVEN Set up and interpret an ANOVA table. Goals
13- 4 Correlation Analysis Independent Variable The Independent Variable provides the basis for estimation. It is the predictor variable. Correlation Analysis Correlation Analysis is a group of statistical techniques to measure the association between two variables. Scatter Diagram A Scatter Diagram is a chart that portrays the relationship between two variables. Dependent Variable The Dependent Variable is the variable being predicted or estimated.
13- 5 The Coefficient of Correlation, r Negative values indicate an inverse relationship and positive values indicate a direct relationship. Coefficient of Correlation The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. Also called Pearson’s r and Pearson’s product moment correlation coefficient. It requires interval or ratio- scaled data. It can range from -1.00 to 1.00. Values of -1.00 or 1.00 indicate perfect and strong correlation. Values close to 0.0 indicate weak correlation.
13- 10 Formula for r We calculate the coefficient of correlation from the following formula. r= (X – X)(Y – Y) (n-1)s x s y
13- 11 Coefficient of Determination It is the square of the coefficient of correlation. It ranges from 0 to 1. It does not give any information on the direction of the relationship between the variables. coefficient of determination The coefficient of determination (r 2 ) is the proportion of the total variation in the dependent variable (Y) that is explained or accounted for by the variation in the independent variable (X).
13- 12 Example 1 Dan Ireland, the student body president at Toledo State University, is concerned about the cost to students of textbooks. He believes there is a relationship between the number of pages in the text and the selling price of the book. To provide insight into the problem he selects a sample of eight textbooks currently on sale in the bookstore. Draw a scatter diagram. Compute the correlation coefficient.
13- 13 Example 1 continued BookPage Price($) Introduction to History500 84 Basic Algebra700 75 Introduction to Psychology800 99 Introduction to Sociology600 72 Business Management400 69 Introduction to Biology500 81 Fundamentals of Jazz600 63 Principles of Nursing800 93
13- 17 (X – X)(Y – Y) (n-1)s x s y r = = 7800 7(138.87)(12.21) =.657 r 2 =.657 2 =.432 The correlation between the number of pages and the selling price of the book is 0.657. This indicates a moderate association between the variable. Example 1 continued
13- 18 Did a computed r come from a population of paired observations with zero correlation? t test for the coefficient of correlation t = r n- 2 1- r 2 With n-2 d.f. H o : = 0 (The correlation in the population is zero.) H 1 : ≠ 0 (The correlation in the population is different from zero. T-test of significance of r
13- 19 Step 4 H 0 is rejected if t>3.143 or if t<-3.143 or if p <.02. There are 6 degrees of freedom, found by n – 2 = 8 – 2 = 6. Step 3 The statistic to use follows the t distribution. Step 1 H 0 : the correlation in the population is zero. H 1 :The correlation in the population is not zero. Computed r =.657. Test the hypothesis that there is no correlation in the population. Use a.02 significance level. Step 2 Significance level is.02.
13- 20 Example 1 continued H 0 is not rejected. We cannot reject the hypothesis that there is no correlation in the population. The amount of association could be due to chance. t = r n- 2 1- r 2 =.657 8 – 2 1 -.657 2 = 2.135 p(t > 2.135) =.077 Step 5 Find the value of the test statistic.
13- 21 Regression Analysis The least squares criterion is used to determine the equation. That is the term (Y – Y’) 2 is minimized. Regression Analysis In Regression Analysis we use the independent variable (X) to estimate the dependent variable (Y). The relationship between the variables is linear. Both variables must be at least interval scale.
13- 22 Regression Analysis The regression equation is Y’= a + bX where Y’ is the average predicted value of Y for any X. a is the Y-intercept. It is the estimated Y value when X=0 b is the slope of the line, or the average change in Y’ for each change of one unit in X The least squares principle is used to obtain a and b.
13- 23 Regression Analysis b = r sysxsysx a = Y – bX The least squares principle is used to obtain a and b. The equations to determine a and b are:
13- 24 Example 1 revisited b = r sysxsysx = (.657) 12.21 138.87 =.0578 a = Y – bX = 79.5 -.0578*625 = 43.39 Develop a regression equation for the information given in example 1 that can be used to estimate the selling price based on the number of pages.
13- 25 Example 1 revisited The regression equation is: Y’ = 43.39 +.0578X The slope of the line is.0578. Each addition page costs about a nickel. The sign of the b value and the sign of r will always be the same. The equation crosses the Y-axis at $43.39. A book with no pages would cost $43.39.
13- 26 Example 1 revisited Price= $43.39 +.0578(Number of Pages) = $43.39 +.0578(800) = $89.61 We can use the regression equation to estimate values of Y. The estimated selling price of an 800 page book is $89.61, found by
13- 27 The Standard Error of Estimate The formula that is used to compute the standard error: Standard Error of Estimate The Standard Error of Estimate measures the scatter, or dispersion, of the observed values around the line of regression
13- 28 Find the standard error of estimate for the problem involving the number of pages in a book and the selling price. Example 1 revisited = 9.944
13- 29 The Y values are statistically independent. This means that in the selection of a sample, the Y values chosen for a particular X value do not depend on the Y values for any other X values. For each value of X, there is a group of Y values, and these Y values are normally distributed. Assumptions Underlying Linear Regression The means of these normal distributions of Y values all lie on the straight line of regression. The standard deviations of these normal distributions are the same.
13- 30 Confidence Interval Y’is the predicted value for any selected X value X is an selected value of X X is the mean of the Xs n is the number of observations S y.x is the standard error of the estimate t is the value of t at n-2 degrees of freedom The confidence interval for the mean value of Y for a given value of X is given by:
13- 31 For our earlier price estimate of $89.61, the confidence interval, assuming a desired 95% confidence, is calculated as follows. Example 1
13- 32 Example 1 revisited Y’ the predicted value, is $89.61 X is 800 pages X is 625, the mean of the pages n is 8, the number of observations S y.x is 9.944, the standard error of the estimate t is 2.447 at 8-2 degrees of freedom and 95% confidence
13- 33 Prediction Interval The prediction interval for an individual value of Y for a given value of X
13- 34 Summarizing The Results The estimated selling price for a book with 800 pages is $89.61. The standard error of estimate is $9.94. The 95 percent confidence interval for all books with 800 pages is $89.61 + $14.43. This means the limits are between $75.18 and $104.04. The 95 percent prediction interval for a particular book with 800 pages is $89.61+ $28.29. The means the limits are between $61.32 and $117.90. These results appear in the following Minitab and Excel outputs. Example 1 revisited
13- 35 Regression Analysis The regression equation is Price = 43.4 + 0.0578 No of Pages Predictor Coef StDev T P Constant 43.39 17.28 2.51 0.046 No of Pages 0.05778 0.02706 2.13 0.077 S = 9.944 R-Sq = 43.2% R-Sq(adj) = 33.7% Analysis of Variance Source DF SS MS F P Regression 1 450.67 450.67 4.56 0.077 Error 6 593.33 98.89 Total 7 1044.00 Fit StDev Fit 95.0% CI 95.0% PI 89.61 5.90 ( 75.17, 104.05) ( 61.31, 117.91) Example 1 revisited MINITABMINITAB