#6 Business Statistics with Computer Applications II Problems from book: Chapter 12 (page 523) 12.62, 12.63, 12.65-12.69 Problems not in book: Casting aluminum. In casting metal parts, molten metal flows through a \"gate\" into a die that shapes the part. The gate velocity (the speed at which metal is forced through the gate) plays a critical role in die casting. A firm that casts cylindrical aluminum pistons examined 12 types formed from the same alloy. How does the piston wall thickness (inches) influence the gate velocity (feet per second) chosen by the skilled workers who do the casting? If there is a clear pattern, it can be used to direct new workers or to automate the process. Analyze these data and report your findings.1 A scatterplot (you need not make one) shows a moderately strong positive linear relationship. The Figure below displays part of the Minitab regression output. Exercises 1 through 3 refer to this output. 1 Peter H. Chen, Neftali Herrera, and Darren Christiansen, \"Relationships between gate velocity and casting features among aluminum round castings,\" no date. Provided by Darren Christiansen. Homework #6 Business Statistics with Computer Applications II 1. Casting aluminum: is there a relationship? Figure 24.13 leaves out the t statistics and their P-values. Based on the information in the output, test the hypothesis that there is no straight-line relationship between thickness and gate velocity. State hypotheses, give a test statistic and its approximate P-value, and state your conclusion. 2. Casting aluminum: intervals. The output in Figure 24.13 includes prediction for piston wall thickness x* = 0.5 inch. Use the output to give 90% intervals for (a) the slope of the population regression line of gate velocity on piston thickness. (b) the average gate velocity for a type of piston with thickness 0.5 inch. 3. Casting aluminum: residuals. The output in Figure 24.13 includes a table of the x and y variables, the fitted values for each x, the residuals, and some related quantities. (a) Plot the residuals against thickness (the explanatory variable). Use vertical scale 200 to 200 so that the pattern is clearer. Add the \"residual = 0\" line. Does your plot show a systematically nonlinear relationship? Does it show systematic change in the spread about the regression line? (b) Make a histogram of the residuals. Minitab identifies the residual for Observation 9 as a suspected outlier. Does your histogram agree? (c) Redoing the regression without Observation 9 gives regression standard error s = 42.4725 and predicted mean velocity 216 feet per second (90% confidence interval 191.4 to 240.6) for piston walls 0.5 inch thick. Compare these values with those in Figure 24.13. Is Observation 9 influential for inference? Thick 0.248 0.359 0.366 0.4 0.524 0.552 0.628 0.697 0.697 0.752 0.806 0.821 Veloc 123.8 223.9 180.9 104.8 228.6 223.8 326.2 302.4 145.2 263.1 302.4 302.4 Supplementary Problems Chain Chick-Fil-A SONIC Drive-Ins Domino's Pizza Panera Bread Arby's Jack in the Box Sales ($ billions) Number of Units (1000) 4.1 3.7 3.4 3.3 3.0 3.0 1.6 3.5 4.9 1.5 3.5 2.2 12.60 Shown here are the labor force figures (in millions) published by IndexMundi for the country of Bangladesh over a 10-year period. Develop the equation of a trend line through these data and use the equation to predict the labor force of Bangladesh for the year 2015. Year Labor Force (million) 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 64.10 64.02 65.49 66.60 68.00 69.40 70.86 72.35 73.87 75.42 12.61 How strong is the correlation between the inflation rate and 30-year treasury yields? The following data published by Fuji Securities are given as pairs of inflation rates and treasury yields for selected years over a 35-year period. Inflation Rate 30-Year Treasury Yield 1.57% 2.23 2.17 4.53 7.25 9.25 5.00 4.62 3.05% 3.93 4.68 6.57 8.27 12.01 10.27 8.45 Compute the Pearson product-moment correlation coefficient to determine the strength of the correlation between these two variables. Comment on the strength and direction of the correlation. 12.62 Shown below are data on the total sales generated by the seafood industry and the corresponding jobs supported by the seafood industry in the top 15 states by seafood sales. The data are published by the National Marine Fisheries Service of the National Oceanic and Atmospheric Administration of the U.S. Department of Commerce. Develop a simple regression model to predict the number of jobs supported by the seafood industry for a state from the total sales generated by the seafood industry of a state. Construct a confidence interval for 523 the average y value for sales of $3,000 (million). Use the t statistic to test to determine whether the slope is significantly different from zero using a = .05. State Total Sales Generated by the Seafood Industry (in $ million) Jobs Supported by the Seafood Industry (1,000) 20,054 14,250 8,026 7,754 6,564 5,103 4,685 2,278 1,867 1,802 1,743 1,734 1,490 1,351 1,025 122.1 72.3 67.0 98.4 43.6 41.8 63.3 27.7 22.1 32.8 15.3 31.1 11.1 18.6 9.2 California Florida Washington Massachusetts New Jersey New York Alaska Texas Virginia Louisiana Maryland Maine Georgia Oregon Rhode Island 12.63 People in the aerospace industry believe the cost of a space project is a function of the weight of the major object being sent into space. Use the following data to develop a regression model to predict the cost of a space project by the weight of the space object. Determine r 2 and se . Weight (tons) Cost ($ millions) 1.897 3.019 0.453 0.988 1.058 2.100 2.387 $ 53.6 184.9 6.4 23.5 33.4 110.4 104.6 12.64 The following data represent a breakdown of state banks and all savings organizations in the United States every 5 years over a 60-year span, according to the Federal Reserve System. Time Period State Banks 1 2 3 4 5 6 7 8 9 10 11 12 13 1342 1864 1912 1847 1641 1405 1147 1046 997 1070 1009 1042 992 All Savings 2330 2667 3054 3764 4423 4837 4694 4407 4328 3626 2815 2030 1779 Chapter 12 Simple Regression Analysis and Correlation Develop a regression model to predict the total number of state banks by the number of all savings organizations. Comment on the strength of the model. Develop a time-series trend line for All Savings using the time periods given. Forecast All Savings for period 15 using this equation. 12.65 Is the amount of money spent by companies on advertising a function of the total revenue of the company? Shown are revenue and advertising cost data for nine companies published by Advertising Age. Company Procter & Gamble General Motors Verizon Comcast AT&T JPMorgan Chase Ford American Express L'Oreal Advertising ($ millions) Revenues ($ billions) 5.0 3.1 2.5 2.5 2.4 2.4 2.1 2.1 2.1 82.6 148.9 110.9 55.8 126.7 97.2 136.3 30.0 28.3 Use the data to develop a regression line to predict the amount of advertising by revenues. Compute se and r 2. Assuming a = .05, test the slope of the regression line. Comment on the strength of the regression model. 12.66 Can the consumption of water in a city be predicted by air temperature? The following data represent a sample of a day's water consumption and the high temperature for that day. Water Use (millions of gallons) Temperature (degrees Fahrenheit) 219 56 107 129 68 184 150 112 103 39 77 78 50 96 90 75 Develop a least squares regression line to predict the amount of water used in a day in a city by the high temperature for that day. What would be the predicted water usage for a temperature of 100? Evaluate the regression model by calculating se , by calculating r 2, and by testing the slope. Let a = .01. INTERPRETING THE OUTPUT 12.67 Study the following Minitab output from a regression analysis to predict y from x. a. What is the equation of the regression model? b. What is the meaning of the coefficient of x? c. What is the result of the test of the slope of the regression model? Let a = .10. Why is the t ratio negative? d. Comment on r 2 and the standard error of the estimate. e. Comment on the relationship of the F value to the t ratio for x. f. The correlation coefficient for these two variables is -.7918. Is this result surprising to you? Why or why not? Regression Analysis: Y versus X The regression equation is Y = 67.2 - 0.0565 X Predictor Coef Constant 67.231 X -0.05650 S = 10.32 SE Coef 5.046 0.01027 R-Sq = 62.7% T 13.32 -5.50 p 0.000 0.000 R-Sq(adj) = 60.6% Analysis of Variance Source DF SS Regression 1 3222.9 Residual Error 18 1918.0 Total 19 5141.0 MS F P 3222.9 30.25 0.000 106.6 12.68 Study the following Excel regression output for an analysis attempting to predict the number of union members in the United States by the size of the labor force for selected years over a 30-year period from data published by the U.S. Bureau of Labor Statistics. Analyze the computer output. Discuss the strength of the model in terms of proportion of variation accounted for, slope, and overall predictability. Using the equation of the regression line, attempt to predict the number of union members when the labor force is 110,000. Note that the model was developed with data already recorded in 1,000 units. Use the data in the model as is. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.798 0.636 0.612 258.632 17 ANOVA SS 1756035.529 1003354.471 2759390 Regression Residual Total df 1 15 16 Intercept Total Employment Coefficients 20704.3805 -0.0390 MS 1756036 66890.3 Standard Error 879.6067 0.0076 F 26.25 Significance F 0.00012 t Stat 23.54 -5.12 P-value 0.00000 0.00012 12.69 Study the following Minitab residual diagnostic graphs. Comment on any possible violations of regression assumptions. Normal Plot of Residuals 25 20 15 10 5 0 5 10 15 20 Residual 524 2 1 0 1 Normal Score 2 Case 7 6 5 4 3 2 1 0 Residuals Versus the Fitted Values 25 20 15 10 5 0 5 10 15 20 Residual Frequency Histogram of Residuals 525 20 10 0.0 10 Residual 20 ANALYZING THE DATABASES 1. Develop a regression model from the Consumer Food database to predict Annual Food Spending by Annual Household Income. Discuss the model and its strength on the basis of statistics presented in this chapter. Now develop a regression model to predict Non-Mortgage Household Debt by Annual Household Income. Discuss this model and its strengths. Compare the two models. Does it make sense that Annual Food Spending and NonMortgage Household Debt could each be predicted by Annual Household Income? Why or why not? 2. Using the Hospital database, develop a regression model to predict the number of Personnel by the number of Births. Now develop a regression model to predict number of Personnel by number of Beds. Examine the regression output. Which model is stronger in predicting number of Personnel? Explain why, using techniques presented in this chapter. Use the second regression model to predict the number of Personnel in a hospital that has 110 beds. 45 50 Fitted Value 55 see www.wiley.com/college/black and WileyPLUS Construct a 95% confidence interval around this prediction for the average value of y. 3. Analyze all the variables except Type in the Financial database by using a correlation matrix. The seven variables in this database are capable of producing 21 pairs of correlations. Which are most highly correlated? Select the variable that is most highly correlated with P/E ratio and use it as a predictor to develop a regression model to predict P/E ratio. How did the model do? 4. Construct a correlation matrix for the six U.S. and international stock indicators. Describe what you find. That is, what indicators seem to be most strongly related to other indicators? Now focus on the three international stock indicators. Which pair of these stock indicators is most correlated? Develop a regression model to predict the DJIA by the Nikkei 225. How strong is the model? Develop a regression model to predict the DJIA by the Hang Seng. How strong is the model? Develop a regression model to predict the DJIA by the Mexico IPC. How strong is the model? Compare the three models. CASE DELTA WIRE USES TRAINING AS A WEAPON The Delta Wire Corporation was founded in 1978 in Clarksdale, Mississippi. The company manufactures high-carbon specialty steel wire for global markets and at present employs about 100 people. For the past few years, sales increased each year. A few years ago, however, things did not look as bright for Delta Wire because it was caught in a potentially disastrous bind. With the dollar declining in value, foreign competition was becoming a growing threat to Delta's market position. In addition to the growing foreign competition, industry quality requirements were becoming tougher each year. Delta officials realized that some conditions, such as the value of the dollar, were beyond their control. However, one area that they could improve upon was employee education. The company worked with training programs developed by the state of Mississippi and a local community college to set up its own school. Delta employees were introduced to statistical process control and other quality assurance techniques. Delta reassured its customers that the company was working hard on improving quality and staying competitive. Customers were invited to sit in on the educational sessions. Because of this effort, Delta has been able to weather the storm and continues to sustain a leadership position in the highly competitive steel wire industry. Delta continued its training and education program. In the 1990s, Delta instituted a basic skills training program