TIM7100 DATA FILE 6 ASSIGNMENT 7 1. Baseball wisdom says if you can’t hit, you can’t win. But is the number of games won by a major-league baseball team in a season related to the team’s batting average? The table below shows the number of games won and the batting averages for 14 games in one American League season. Team Cleveland New York Boston Toronto Texas Detroit Minnesota Baltimore California Milwaukee Seattle Kansas City Oakland Chicago Games Won, y 99 92 85 74 90 53 78 88 70 80 85 75 78 85 Team Batting Average, x .293 .288 .283 .259 .284 .256 .288 .274 .276 .279 .287 .267 .265 .281 a. If you were to model the relationship between the mean (or expected) number of games won by a major-league team and the team’s batting average x, using a straight line, would you expect the slope of the line to be positive or negative? Explain. b. Construct a scattergram of the data. Does the pattern revealed by the scattergram agree with your answer to part a? Construct a simple linear regression of the data using SPSS. What is the equation of the least squares line? c. Graph the least squares line on your scattergram. Does your least squares line seem to fit the points on your scattergram? d. Does the mean or expected number of games won appear to be strongly related to a team’s batting average? Explain. e. Interpret the values of 𝛽̂0 and 𝛽̂1 in the words of the problem. 2. To improve the quality of the output of any production process, it is necessary first to understand the capabilities of the process. In a particular manufacturing process, the useful life of a cutting tool is related to the speed at which the tool is operated. It’s necessary to understand this relationship in order to predict when the tool should be replaced and how many spare tools should be available. The data in the table below were derived from life tests for the two different brands of cutting tools currently used in the production process. Cutting Speed (meters per minute) 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 Useful Life (Hours) Brand A Brand B 4.5 6.0 3.5 6.5 5.2 5.0 5.2 6.0 4.0 4.5 2.5 5.0 4.4 4.5 2.8 4.0 1.0 3.7 4.0 3.8 2.0 3.0 1.1 2.4 1.1 1.5 0.5 2.0 3.0 1.0 a. Construct a simple linear regression of the data using SPSS. What is the equation of the least squares line for Brand A and for Brand B? b. For a cutting speed of 70 meters per minute, find 𝑦̂ ± 2s for each least squares line. c. For which brand would you feel more confident in using the least squares line to predict useful life for a given cutting speed? Explain. 3. The expenses involved in a manufacturing operation may be categorized as being for raw material, direct labor, and overhead. Overhead refers to all expenses other than those for raw materials and direct labor that are involved with running the factory. A manufacturer of 10-speed racing bicycles is interested in estimating the relationship between its monthly factory overhead and the total number of bicycles produced per month. The estimate will be used to help develop the manufacturing budget for next year. The data in the table below have been collected for the previous 12 months. Month 1 2 3 4 5 6 7 8 9 10 11 12 Production Level (1,000s of units) 16.9 15.6 17.4 11.6 17.7 17.6 16.3 15.5 23.4 28.4 27.1 19.2 Overhead ($1,000s) 41.4 35.0 38.3 29.5 39.6 37.4 37.5 37.0 47.9 55.6 53.1 40.6 a. Construct a simple linear regression of the data using SPSS. What is the equation of the least squares line? b. Does the straight-line model contribute information for predicting overhead costs? Test at α = 0.05. c. Which of the four assumptions we make about the random error ϵ may be inappropriate in this problem? Explain. 4. Describe the slope of the least squares line if a. r = 0.7 b. r = -0.7 c. r = 0 d. r2 = 0.64 5. After the Minnesota Department of Transportation installed more than 20 years ago what was then a state-of-the-art weigh-in-motion scale in the concrete surface of the eastbound lanes of Interstate 94, a study was undertaken to determine whether the scale’s readings corresponded with the static weights of the vehicles being monitored. After some preliminary comparisons using a two-axle, six-tire truck carrying different loads, calibration adjustments were made in the scale’s software and the scales were reevaluated. The data are in the table below. Construct a simple linear regression of the data using SPSS. a. What is the correlation coefficient for both sets of data? b. How can the correlation coefficients be used to evaluate the weigh-in-motion scale? Trial Number Static Weight of Truck x (thousand pounds) 1 2 3 4 5 6 7 8 9 10 27.9 29.1 38.0 27.0 30.3 34.5 27.8 29.6 33.1 35.5 Weigh in Motion Prior to Calibration Adjustment y1 (thousand pounds) 26.0 29.0 39.5 25.1 31.6 36.2 25.1 31.0 35.6 40.2 Weigh in Motion After Calibration Adjustment y2 (thousand pounds) 27.8 29.1 37.8 27.1 30.6 34.3 26.9 29.6 33.0 35.0 6. Work standards specify time, cost, and efficiency norms for the performance of work tasks. They are typically used to monitor job performance. In one distribution center, data were collected to develop work standards for the time to assemble or fill customer orders. The table below contains data for a random sample of 9 orders. Time (mins.) 27 15 71 35 8 60 3 10 10 Order Size 36 34 255 103 4 555 6 60 96 a. Construct a simple linear regression of the data using SPSS. What is the equation of the least squares line? Use time as the dependent variable. b. In general, we would expect the mean time to fill an order to increase with the size of the order. Do the data support this theory? Test using α = .05. 7. Firms planning to build new plants or make additions to existing facilities have become very conscious of the energy efficiency of proposed new structures and are interested in the relation between annual energy consumption and the number of square feet of building shell. The table below lists the energy consumption in British thermal units (BTUs) for 22 buildings that were all subjected to the same climatic conditions. a. Construct a simple linear regression of the data using SPSS. What is the estimate of the intercept β0 and slope β1? b. Is annual energy consumption positively linearly related to the shell area of the building? Test using α = .10. c. What is the observed significance level of the test of part b? Interpret its value. d. What is the coefficient of determination? Interpret its meaning. e. A company wishes to build a new warehouse that will contain 8,000 sq. ft. of shell area. Find the predicted value of energy consumption. Calculate a 95% prediction interval. Comment on the usefulness of this interval. BTU/Year (thousands) 3,870,000 1,371,000 2,422,000 672,200 233,100 218,900 354,000 3,135,000 1,470,000 1,408,000 2,201,000 2,680,000 337,500 567,500 555,300 239,400 2,629,000 1,102,000 423,500 423,500 1,691,000 1,870,000 Shell Area (square feet) 30,001 13,530 26,060 6,355 4,576 24,680 2,621 23,350 18,770 12,220 25,490 23,680 5,650 8,001 6,147 2,660 19,240 10,700 9,125 6,510 13,530 18,860
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