You are given the following data on sales (in tens of thousands of dollars) and advertising expenditure (in thousands of dollars): Sales, Y 31 40 25 30 20 26 Advertising expenditures, X 5 11 3 4 3 5 a. Calculate the regression equation of sales on advertising expenditures. BA 578: Fall 2012 Page 2 of 20 b. Construct a 95% confidence interval for the mean value of Y if X =9. c. Use a 5% level of significance and test BA 578: Fall 2012 Page 3 of 20 d. Predict sales if $6,000 is spent on advertising. e. Calculate a 95% confidence interval for the individual value of Y if X = 6. 2. Suppose a statistician obtained the following estimated regression model where is a dummy variable indicating the gender of the ith individual and =1 if the individual is a male. Suppose another statistician estimated the model using the same data but defined if the individual is female. a. What would happen to the constant term? BA 578: Fall 2012 Page 4 of 20 b. What would happen to the coefficient of ? c. What would happen to the coefficient of ? 3. Fill in the missing values in following ANOVA table Source df SS MS F Factor 5 205.5 Error 637 Total 25 Answer a. In the above ANOVA table, is the factor significant at 5% level of significant? 4. Fill in the missing values in following ANOVA table Source df SS MS F Factor 2 3.24 Error 17 Total 40.98 Answer a. In the above ANOVA table, is the factor significant at 5% level of significant? BA 578: Fall 2012 Page 5 of 20 5. Fill in the missing values in following ANOVA table Source df SS MS F Factor 346.2 115.4 20.79 Error 16 Total Answer a. In the above ANOVA table, is the factor significant at 5% level of significant? 6. A regression model relating xs, number of sales persons at branch offices, to y, annual sales at the office ($1000s), has been developed. The computer output from a regression analysis of the data follows. The regression equation is ^ Predictor Coef Stdev t-ratio Constant 80.0 11.333 7.06 50.0 5.482 9.12 30.0 2.68 11.19 Analysis of Variance SOURCE DF SS MS Regression 2 6828.6 3414.3 Error 27 2298.8 85.1407 Total 29 9127.4 a. Write the estimated regression equation. b. How many branch offices were involved in the study? BA 578: Fall 2012 Page 6 of 20 c. Compute the F statistic and test the significance of the relationship at a .05 level of significance. 7. Answer the following questions: a) If r2 = 0.95, n = 11 and the ? ¯ = 100, what is 2 e S or 2 y \ x S ? b) If r2 = 1, then 2 e S in (1a) can be shown to have what type of relationship with SST and SSR? c) What relationship exists between y ˆ and Y where r2 = 1? d) What relationship exists between y ˆ and Y where r2 = 0? BA 578: Fall 2012 Page 7 of 20 e) Given the following information: r2 = 0.95, n = 10 and k =2 what is the t-value for b1? f) In (e), you calculated t-value at the 0.05 level of significance; what statistical decision can you make about b1? 8. Calculate r2 for the following sets of data: a) ? ¯ and ? ^ ¯ = 140 b) ? ¯ and ? ^ = 10 c) ? ^ ¯ and ? ^ = 25 BA 578: Fall 2012 Page 8 of 20 9. A metropolitan bus system sampler’s rider counts on one of its express commuter routes for a week. Use the following data to establish whether rider ship is evenly balanced by day of the week. Let ? = 0.05. Day Monday Tuesday Wednesday Thursday Friday Rider Count 10 34 21 57 44 1. Is the ? 2 value significant at 5% level of significant? 2. Write the conclusion for this question 10. The Acme corp. wished to predict maintenance costs per year on its factory equipment by knowledge of the equipment’s age. The following data was collected which yielded the regression equation: (see the printout) Age Maintenance Costs 6 920 7 1810 1 230 3 400 6 1260 a) What is the point estimate of maintenance cost for a machine that is 3 years old? BA 578: Fall 2012 Page 9 of 20 b) Determine the 4 missing values in the printout given below. 11. For the following ANOVA table are the results from treating four cultures with six observations for each culture. The enzymes are contained in test tubes with differing levels of enzymes applied. Source of Variation df SS MS F Treatment 180 Error 60 Total a. What are the null and alternate hypotheses? SUMMARY OUTPUT Regression Statistics Multiple R ( ) R Square 0.843041 Adjusted R Square 0.790722 Standard Error ( ) Observations 5 ANOVA df SS MS F Significance F Regression 1 ( ) 1394491 16.1133 0.027751 Residual 3 259628.6 86542.86 Total 4 1654120 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -158.095 299.9618 -0.52705 0.634646 -1112.71 796.5172 -1112.71 796.5172 X Variable 1 235.2381 58.60239 ( ) 0.027751 48.73913 421.7371 48.73913 421.7371 BA 578: Fall 2012 Page 10 of 20 b. What is your decision rule, Use c. Indicate your statistical decision. d. Is there a difference among the means? 12. Under what conditions can the interpretation of the intercept or constant term be meaningful? BA 578: Fall 2012 Page 11 of 20 13. What does a constant term in an equation reflect? 14. What is the role of the error term in an equation, and what does it reflect? BA 578: Fall 2012 Page 12 of 20 BA 578: Fall 2012 Page 13 of 20 15. Use the printout above answer the questions. a. Is the relationship between working capital and net sales statistically significant? b. What is the coefficient of determination? What do you conclude in terms of the variables? c. What is the correlation coefficient? What do you conclude in terms of the variables? BA 578: Fall 2012 Page 14 of 20 d. What is the standard error of estimate? e. What is the regression equation? Interpret the regression equation. f. If working capital equals $100,000 what is the estimate for net sales? g. What are 95% prediction and confidence intervals for net sales if working capital equals $ 100,000? BA 578: Fall 2012 Page 15 of 20 True / False _______ 16. The usual objective of regression analysis is to predict estimate the value of one variable when the value of another variable is known. _______ 17. Correlation analysis is concerned with measuring the strength of the relationship between two variables. _______ 18. The term ei in the simple linear regression model indicates the amount of change in Y for a unit change in X. _______ 19. In the sample regression equation y = a + bx, b is the slope of the regression line. _______ 20. The coefficient of determination can assume any value between -1 and +1. _______ 21. In the least squares model, the explained sum of squares is always smaller than the regression sum of squares. _______22. The sample correlation coefficient and the sample slope will always have the same sign. _______ 23. Given the sample regression equation y = -3 + 5x, we know that in the sample X and Y are inversely related. _______ 24. Given the sample regression equation y = 5 – 6x, we know that when X = 2, Y = 17. _______25. An important relationship in regression analysis is (Y Y) i ? = Y) Y ˆ Y Y) ( ˆ ( ? ? i ? . _______ 26. Regression analysis is concerned with the form of the relationship among variables, whereas correlation analysis is concerned with the strength of the relationship. _______ 27. The correlation coefficient indicates the amount of change in Y when X changes by one unit. _______ 28. In simple linear regression analysis, when the slope is equal to zero, the independent variable does not explain any of the variability in the dependent variable. _______29. One of the purposes of regression analysis is to estimate a mean of the independent variable for given values of the dependent variable. BA 578: Fall 2012 Page 16 of 20 _______ 30. The variable that can be manipulated by the investigator is called the independent variable. _______ 31. When b = 0, X and Y are not related. _______ 32. If zero is contained in the 95% confidence interval for b, we may reject Ho: b = 0 at the 0.05 level of significance. _______ 33. If in a regression analysis the explained sum of squares is 75 and the unexplained sum of square is 25, r2 = 0.33. _______ 34. In general, the smaller the dispersion of observed points about a fitted regression line, the larger the value of the coefficient of determination. _______ 35. When small values of Y tend to be paired with small values of X, the relationship between X and Y is said to be inverse. _______ 36. An alternative hypothesis (Ha) is a theory that contradicts the null hypothesis. The alternative hypothesis will be accepted when there is strong evidence leading us to reject the null hypothesis. _______ 37. The p-value of a test depends on the observed data, but the critical values of a test do not. _______ 38. Other things being equal, decreasing increases . _______ 39. The larger the p-value associated with a test of hypothesis, the stronger the support for the null hypothesis. _______ 40. The probability that the test statistic will fall in the critical region, given that H0 is true, represents the probability of making a type II error. 41. A multiple regression analysis based on n= 24 data points yielded the following fitted model: ^ = 130.0093 – 3.5017 + .0034 The error sum of squares was 465.1348. Each of the following statements is incorrect; your job is to correct each. a. For each one-unit change in , we expect Y to decrease by -3.5017 units. BA 578: Fall 2012 Page 17 of 20 b. The estimated standard error of the regression is 22.1493. c. There is a direct relation between and Y. d. The fitting error for the data point Y = 50, , and = 1024 is 52.1565. e. If we reduce the value of from 0.0034 to 0.0014, the error sum of squres will be less than 465.1348. BA 578: Fall 2012 Page 18 of 20 42. Find the cutoff F-value for the following testing situations: a. = = 0; n = 28; and = 0.05. b. = = = = 0; n = 35; and = 0.10. c. = 0; n = 15; and = 0.01. BA 578: Fall 2012 Page 19 of 20 43. Using the following model: = + + + ( ) + Where UN = unemployment rate, % V = job vacancy rate, % D = 1 for the period 1996 – 2000 = 0 for the period 2001 – 2011 t = time, measured in annuals Derive the above equation for the two periods BA 578: Fall 2012 Page 20 of 20 44. ( =0.05) a. Use the above data to estimate a multiple regression equation. b. Please include your computer printout and interpret your result. c. Are the estimates for 1946 through 1954 different from those of 1955 through 1963? Year Y X Dum DumX 1946 0.36 8.8 0 0 1947 0.21 9.4 0 0 1948 0.08 10 0 0 1949 0.2 10.6 0 0 1950 0.1 11 0 0 1951 0.12 11.9 0 0 1952 0.41 12.7 0 0 1953 0.5 13.5 0 0 1954 0.43 14.3 0 0 1955 0.59 15.5 1 15.5 1956 0.9 16.7 1 16.7 1957 0.95 17.7 1 17.7 1958 0.82 18.6 1 18.6 1959 1.04 19.7 1 19.7 1960 1.53 21.1 1 21.1 1961 1.94 22.8 1 22.8 1962 1.75 23.9 1 23.9 1963 1.99 25.2 1 25.2