Foundations of Business Statistics - Professor Fluture Materials from Professor Friedman HOMEWORK - REGRESSION REGRESSION In simple regression (one X variable and one Y variable), there are three ways to test for significance. All three tests are equivalent and give the same results. You can test the regression for significance by examining the F-value. Or, you can test the correlation coefficient for significance using a t-test. Or, you can test the slope term (b1) for significance using a t-test. You should always do one test to make sure that you are not looking at a chance relationship that is meaningless. Even when a relationship is statistically significant, it may be of little practical importance if the correlation coefficient is low, say, below .30. SOLVE THE FOLLOWING PROBLEMS IN TWO WAYS: (a) USING CALCULATOR (by hand) OR EXCEL BASIC COMPUTATIONS (b) USING MS EXCEL REGRESSION ANALYSIS Write out the regression equation. What is the correlation coefficient? What is r-square (coefficient of determination)? Draw the conclusion (very important) Your answers should follow the format of problem 1 solution. NOTE: the above request is for a very good reason. I want you to get familiar with the used equations to get an understanding of them. By using excel only, you just follow a sequence of steps; some of you have no idea what is actually going on. Please do not disregard my request. In the final you will not be asked to cover all the formulas for regression and correlation. It will be your choice on how you will obtain them, using calculator, using excel regression output, using scatter plots. BUT for the homework please follow closely the solution example and cover ALL the steps. Problem (with solution): A researcher is interested in determining whether there is a relationship between number of packs of cigarettes smoked per day and longevity (in years). n=10. # packs of cigarettes smoked Longevity (X) (Y) 0 80 0 70 1 72 1 70 2 68 2 65 3 69 3 60 Regression p. 1 Foundations of Business Statistics - Professor Fluture Materials from Professor Friedman 4 4 58 55 Answer: A: 1- You are given data for Xi (independent variable) and Yi (dependent variable). X = 20; Y = 667; XY = 1247; X2 = 60; Y2 = 44,983 (done with basic excel computations) 2- Calculate the correlation coefficient, r: r= r= nX i Yi (X i )(Yi ) n X 2 i X i 2 n Y i 2 Yi 2 -1 r 1 (10*1247-20*667)/[squareroot(10*60-400)(10*44983-444889)]= -870/squareroot(200*4941)= -870/994.08 = -.875 3- Calculate the coefficient of determination: r2 = ExplainedVariation = (r)2 = .766 TotalVaria tion 0 r2 1 This is the proportion of the variation in the dependent variable (Yi) explained by the independent variable (Xi) 4- Calculate the regression coefficient b1 (the slope): nX i Yi (X i )(Yi ) b1 = = -870/(10*60 - 400) = - 4.35 2 nX i2 X i Note that you have already calculated the numerator and the denominator for parts of r. Other than a single division operation, no new calculations are required. BTW, r and b1 are related. If a correlation is negative, the slope term must be negative; a positive slope means a positive correlation. 5- Calculate the regression coefficient b0 (the Y-intercept, or constant): b0 = Y b1 X = 66.7 - (-4.35*2) = 75.4 6- The regression equation (a straight line) is: Yi = b0 + b1Xi Longevity = 75.4 - 4.35 (Packs of cigarettes smoked) Regression p. 2 Foundations of Business Statistics - Professor Fluture Materials from Professor Friedman B. Excel regression analysis SUMMARY OUTPUT Regression Statistics Multiple R 0.875179 R Square 0.765938 Adjusted R Square 0.73668 Standard Error 3.802138 Observations 10 ANOVA df Regression Residual Total Intercept X Variable 1 1 8 9 SS MS F Significance F 378.45 378.45 26.17898833 0.000911066 115.65 14.45625 494.1 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% 75.4 2.082517 36.2062 3.71058E-10 70.59770833 80.20229167 70.59770833 80.20229167 -4.35 0.850184 -5.11654 0.000911066 -6.310527365 -2.389472635 -6.310527365 -2.389472635 Conclusion (VERY IMPORTANT) The regression is statistically significant (F-value = 26.18, p =.0009); Explain why; Check file \"regression Interpretation\" r = -.875; it shows a strong negative relationship r2 = 76.6%; 76.6% of the variation in the dependent variable Longevity explained by the independent variable number of cigarette packs. The regression equation is: Yi = b0 + b1Xi Longevity = 75.4 - 4.35* (Packs of cigarettes smoked) Every pack a person smokes per day results on average in a loss of 4.35 years of life. Someone who does not smoke is expected to live for 75.4 years. Theoretically if you smoke 17 packs a day you will live less than a year Regression p. 3 Foundations of Business Statistics - Professor Fluture Materials from Professor Friedman Each of the following problems is worth 20 points; Make sure you that for each problem you give a complete solution that follows the format of the solved problem. The conclusion is very important. Problem 1: A researcher is interested in determining whether there is a relationship between price and quantity demanded for her firm. Price(X) 2 3 4 5 6 7 8 9 10 11 12 Shelf Space in feet(X) 7.0 3.5 4.0 4.2 4.8 3.9 4.9 7.5 3.0 5.9 5.0 Q-demanded(Y) 95 90 84 80 74 69 62 60 63 50 44 Books Sold(Y) 280 140 170 200 215 190 240 295 125 265 200 Problem 2: A researcher is interested in determining whether there is a relationship between shelf space and number of books sold for her bookstore. Problem 3: A researcher is interested in determining whether there is a relationship between grades and hours studied for statistics. Hours studied(X) 1 2 4 Regression Grade on final(Y) 20 30 40 p. 4 Foundations of Business Statistics - Professor Fluture Materials from Professor Friedman 7 6 7 8 9 8 10 60 65 80 80 95 95 98 Problem 4: A researcher is interested in determining whether there is a relationship between high school average and job performance ((the higher the number, the better the performance) at a certain company. High school average (X) 60 78 98 66 87 77 61 90 91 79 88 99 88 85 81 Job performance (Y) 2 5 10 3 8 5 4 6 7 6 7 9 4 8 9 Problem 5: A researcher is interested in determining whether there is a relationship between education (in years) and the net income (in thousands of dollars). Education in Years 9 Regression Income (in thousands) 20 p. 5 Foundations of Business Statistics - Professor Fluture Materials from Professor Friedman 10 11 11 12 14 14 16 17 19 20 20 Regression 22 24 23 30 35 30 29 50 45 43 70 p. 6