CHAPTER DISCUSSION QUESTIONS Q3-1 Q3-2 Q3-3 Q3-4 The total dollar amount of a fixed cost is constant at different levels of activity within the relevant range, but fixed cost per unit of activity varies In contrast, the total amount of a variable cost varies at different levels of activity, but the variable cost per unit remains constant within the relevant range A semivariable cost contains both fixed and variable elements Consequently, both total semivariable cost and semivariable cost per unit vary with changes in activity The relevant range is the range of activity over which a fixed cost remains constant in total or a variable cost remains constant per unit of activity The underlying assumptions about the relationship of the activity and the incurrence of cost change outside the relevant range of activity Consequently, the amount of fixed cost or the variable cost rate must be recomputed for activity above or below the relevant range The fixed and variable components of a semivariable cost should be segregated in order to plan, analyze, control, measure, and evaluate costs at different levels of activity Separation of the fixed and variable components of semivariable cost is necessary to: (a) compute predetermined factory overhead rates and analyze variances; (b) prepare flexible budgets and analyze variances; (c) analyze direct cost and the contribution margin; (d) determine the break-even point and analyze the effect of volume on cost and profit; (e) compute differential cost and make comparative cost analyses; (f) maximize short-run profits and minimize short-run costs; (g) budget capital expenditures; (h) analyze marketing profitability by territories, products, and customers The obvious advantage to using managerial judgement to separate fixed and variable 3-1 Q3-5 Q3-6 Q3-7 costs is expediency, i.e., it requires less time and is, therefore, less costly than the use of any of the three computational methods The disadvantage is that the use of managerial judgment to separate fixed and variable costs often results in unreliable estimates of cost Cost behavior is not always readily apparent from casual observation As a consequence, managers often err in determining whether a cost is fixed or variable and frequently ignore the possibility that some costs are semivariable The three computational methods available for separating the fixed and variable components of semivariable costs are: (1) the high and low points method; (2) the statistical scattergraph method; and (3) the method of least squares The high and low points method has the advantage of being simple to compute, but it has the disadvantage of using only two data points in the computation, thereby resulting in a significant potential for bias and inaccuracy in cost estimates The scattergraph has the advantage of using all of the available data, but it has the disadvantage of determining the fixed and variable components on the basis of a line drawn by visual inspection through a plot of the data, thereby resulting in bias and inaccuracy in cost estimates The method of least squares has the advantage of accurately describing a line through all the available data, thereby resulting in unbiased estimates of the fixed and variable elements of cost, but it has the increased disadvantage of computational complexity The $200 in the equation, referred to as the y intercept, is an estimate of the fixed portion of indirect supplies cost The $4 in the equation, referred to as the slope of the regression equation, is an estimate of the variable cost associated with a unit change in machine hours These estimates may not be perfectly accurate because they were derived from a sample of data that may not be entirely 3-2 Q3-8 Q3-9 Chapter representative of the universe population, and because activities not included in the regression equation may have some influence on the cost being predicted The coefficient of correlation, denoted r, is a measure of the extent to which two variables are related linearly It is a measure of the covariation of the dependent and independent variables, and its sign indicates whether the independent variable has a positive or negative relationship to the dependent variable The coefficient of determination is the square of the coefficient of correlation and is denoted r The coefficient of determination is a more easily interpreted measure of the covariation than is the coefficient of correlation, because it represents the percentage of variation in the dependent variable explained by the independent variable The standard error of the estimate is defined as the standard deviation about the regression line It is essentially a measure of the variability of the actual observations of the dependent variable from the points predicted on the regression line A small value for the standard error of the estimate indicates a good fit A standard error of zero would indicate a perfect fit, i.e., all actual observations would be on the regression fine Q3-10 Heteroscedasticity means that the distribution of observations around the regression line is not uniform for all values of the independent variable If heteroscedasticity is present, the standard error of the estimate and confidence interval estimates, based on the standard error, are unreliable measures Q3-11 Serial correlation means that rather than being random, the observations around the regression line are correlated with one another If serial correlation is present, the standard error of the estimate and confidence interval estimates, based on the standard error, are unreliable measures Q3-12 Multicollinearity means that two or more of the independent variables in a multiple regression analysis are correlated with one another When the degree of multicollinearity is high, the relationship between one or more of the correlated independent variables and the dependent variable may be obscured However, this circumstance would normally not affect the estimate of cost Chapter 3-3 EXERCISES E3-1 Activity Level High 2,600 hours Low 2,100 Difference 500 hours Cost $1,300 1,100 $ 200 Variable rate: $200 ÷ 500 machine hours = $.40 per machine hour High $1,300 Low $1,100 Total cost Variable cost: $.40 × 2,600 hours 1,040 $.40 × 2,100 hours Fixed cost $ 260 840 $ 260 E3-2 $1,000 $900 SUPPLIES COST $800 $700 $600 $500 $400 $300 $200 $100 $0 200 400 600 DIRECT LABOR HOURS Average cost ($7,575 total ÷ 10 months) Fixed cost per month $757.50 350.00 Average total variable cost $407.50 $407.50 average total variable cost 5,875 total direct labor hours ÷ 10 months = $.6936 variable cost per direct labor hour 800 3-4 Chapter E3-3 Σ( x i − x )(y i − y ) 87, 000 = = $60 Σ( x i − x )2 1, 450 a = y– – bx– = $10,000 – ($60 × 125) = $2,500 Travel and entertainment expense for 200 sales calls would be: yi = a + bxi = $2,500 + ($60 × 200 calls) = $14,500 b= E3-4 (1) y Electricity Cost Month January $1,600 February 1,510 March 1,500 April 1,450 May 1,460 June 1,520 July 1,570 August 1,530 September 1,480 October 1,470 November 1,450 December 1,460 Total $18,000 (2) –) (y – y Cost Deviation 100 10 (50) (40) 20 70 30 (20) (30) (50) (40) (3) x Machine Hours 2,790 2,680 2,600 2,500 2,510 2,610 2,750 2,700 2,530 2,520 2,490 2,520 31,200 (4) (x – x–) Activity Deviation 190 80 (100) (90) 10 150 100 (70) (80) (110) (80) (5) (x – x–)2 (6) – –) (x – x )(y – y (4) Squared 36,100 6,400 10,000 8,100 100 22,500 10,000 4,900 6,400 12,100 6,400 123,000 (4) × (2) 19,000 800 5,000 3,600 200 10,500 3,000 1,400 2,400 5,500 3,200 54,600 – y = Σy = n = $18,000 ÷ 12 = $1,500 x– = Σx = n = 31,200 ÷ 12 = 2,600 54, 600 Σ( x − x )(y − y ) Column total = = $.44 = Σ( x − x ) Column total 123, 000 – – bx– Fixed cost (a) = y = $1,500 – ($.44)(2,600) = $356 Variable rate (b ) = Chapter 3-5 E3-5 r= Σ( x i − x )(y i − y ) Σ( x i − x )2 Σ(y i − y )2 = 1, 564 (850) (3, 400) = 92 r = (.92)2 − 8464 E3-6 (1) y Shipping Month Expense January $ 560 February 600 March 600 April 580 May 570 June 550 July 590 August 610 September 650 October 620 November 630 December 640 Total $7,200 (2) –) (y – y Expense Deviations (40) 0 (20) (30) (50) (10) 10 50 20 30 40 (3) x Sales Revenue $26,500 30,000 29,000 28,000 27,000 25,500 30,000 33,000 35,000 32,000 30,500 33,500 $360,000 (4) (x – x–) Activity Deviations (3,500) (1,000) (2,000) (3,000) (4,500) 3,000 5,000 2,000 500 3,500 (5) (x – x–)2 (6) –) (x – x–)(y – y (7) –)2 (y – y (4) Squared 12,250,000 1,000,000 4,000,000 9,000,000 20,250,000 9,000,000 25,000,000 4,000,000 250,000 12,250,000 (4) × (2) 140,000 0 40,000 90,000 225,000 30,000 250,000 40,000 15,000 140,000 (2) Squared 1,600 0 400 900 2,500 100 100 2,500 400 900 1,600 97,000,000 970,000 y = Σy ÷ n = $7, 200 ÷ 12 = $600 x = Σx ÷ n = $360, 000 ÷ 12 = $30, 000 Σ( x − x )(y − y ) 970, 000 r= = = 939 2 (97, 000, 000) (11, 000) Σ( x i − x ) Σ(y i − y ) 39)2 = 882 r = (.93 11,000 3-6 Chapter E3-7 (1) r= Σ( x i − x )(y i − y ) = Σ( x i − x )2 Σ(y i − y )2 2, 400 = 96 (6, 250) (1, 000) r = (.96)2 = 9216 (2) (3) b= Σ( x i − x )(y i − y ) 2, 400 variable maintenance = = $.384 cost per machine ho our Σ( x i − x ) 6, 250 y = Σy i ÷ n = $50, 000 ÷ 10 = $5, 000 x = Σx i ÷ n = 40, 000 ÷ 10 = 4, 000 hours Since y = a + bx , then : a = y − bx a = $5, 000 − ($.384 )(4, 000) a = $5, 000 − $1, 536 a = $3, 464 E3-8 (1) For electricity cost and direct labor hours: r= Σ( x i − x )(y i − y ) Σ( x i − x ) Σ(y i − y ) 2 = 5, 700 5, 700 = 9497 = (28, 500) (1, 264 ) 6, 002 r = (.9497 )2 = 9019 (2) For electricity cost and machine hours: r= Σ( x i − x )(y i − y ) Σ( x i − x )2 Σ(y i − y )2 = 7, 000 7, 000 = 8805 = (50, 000) (1, 264 ) 7, 950 r = (.8805)2 = 7753 (3) In this case, direct labor hours should be chosen as the appropriate activity measure to be used in predicting electricity cost because the coefficient of determination (r2 = 9019) is higher than that for machine hours (r2 = 7753) Chapter 3-7 E3-8 (Concluded) (4) b= Σ( x i − x )(y i − y ) 5, 700 variable ellectricity = = $.20 cost rate Σ( x i − x ) 28, 500 Since y = a + bx and y = Σy i ÷ n and x = Σx i ÷ n, then : a = ( Σy i ÷ n ) − b ( Σx i ÷ n ) 42, 000 ÷ 20) − (.20)(180, 000 ÷ 20) a = (4 a = 2, 100 − (.20)(9, 000) a = 2, 100 − 1, 800 = $300 fixed electricity cost E3-9 (1) xi Month January February March April May June July August September October November December Total s′= Labor Hours 2,650 3,000 2,900 2,800 2,700 2,550 3,000 3,300 3,500 3,200 3,050 3,350 36,000 (2) yi Actual Utility Cost $ 3,600 4,000 4,000 3,800 3,700 3,500 3,900 4,100 4,500 4,200 4,300 4,400 $48,000 (3) (4) (5) (y′′i = a + bxi) (yi – y′′i) (yi – y′′i)2 Predicted Prediction Utility Error (4) Cost (2) – (3) Squared $ 3,650 (50) $2,500 4,000 0 3,900 100 10,000 3,800 0 3,700 0 3,550 (50) 2,500 4,000 (100) 10,000 4,300 (200) 40,000 4,500 0 4,200 0 4,050 250 62,500 4,350 50 2,500 $48,000 $130,000 Σ(y i − y i′ )2 Column total $130, 000 = = = $114.018 12 − 10 n −2 3-8 Chapter E3-10 s′= Σ(y i − y ′i )2 $49, 972 = = 3, 844 = $62 15 − n −2 The 90 percent confidence interval estimate at the 1,500-hour level of activity would be: y ′i ± t 90% s′ 1+ ( x i − x )2 + n Σ( x i − x )2 $500 ± (1.771)($62) 1+ (1, 500 − 1, 300)2 + 150, 000 15 $500 ± (1.771)($62) 1.3333 $500 ± (1.771)($62) (1 1.1547 ) $500 ± $126.79 Chapter 3-9 PROBLEMS P3-1 (1) Coefficient of correlation and coefficient of determination between: (a) Travel expenses and the number of calls made: (1) y Travel Expense Month January February March April May June July August September October November December Total r= r= r= (2) –) (y – y Expense Deviations (3) x Calls Made $ 3,000 3,200 2,800 3,400 3,100 3,200 2,900 3,300 3,500 3,400 3,200 3,400 (200) (400) 200 (100) (300) 100 300 200 200 410 420 380 460 430 450 390 470 480 490 440 460 $38,400 5,280 Σ( x i − x )(y i − y ) Σ( x i − x )2 Σ(y i − y )2 = (4) (x – x–) Activity Deviations (30) (20) (60) 20 (10) 10 (50) 30 40 50 20 r = 8957 (6) – –) (x – x )(y – y (7) –)2 (y – y (4) Squared (4) × (2) (2) Squared 900 400 3,600 400 100 100 2,500 900 1,600 2,500 400 6,000 24,000 4,000 1,000 15,000 3,000 12,000 10,000 4,000 40,000 160,000 40,000 10,000 90,000 10,000 90,000 40,000 40,000 13,400 79,000 520,000 Column total (Column total) (Column total) 79, 000 79, 000 = 00 (13, 400)(520, 000) 6, 968, 000, 00 79, 000 = 9464 83, 475 (5) (x – x–)2 3-10 Chapter P3-1 (Concluded) (b) Travel expenses and orders received: Month January February March April May June July August September October November December (1) y Travel Expense $ 3,000 3,200 2,800 3,400 3,100 3,200 2,900 3,300 3,500 3,400 3,200 3,400 Total $38,400 r= = = (2) (3) (4) – (y – y ) x (x – x–) Expense Orders Activity Deviations Received Deviations (200) $53,000 (13,000) 65,000 (1,000) (400) 48,000 (18,000) 200 73,000 7,000 (100) 62,000 (4,000) 67,000 1,000 (300) 60,000 (6,000) 100 76,000 10,000 300 82,000 16,000 200 62,000 (4,000) 64,000 (2,000) 200 80,000 14,000 $792,000 Σ( x i − x )(y i − y ) Σ( x i − x ) Σ(y i − y ) 2 = (5) (x – x–)2 (4) Squared 169,000,000 1,000,000 324,000,000 49,000,000 16,000,000 1,000,000 36,000,000 100,000,000 256,000,000 16,000,000 4,000,000 196,000,000 1,168,000,000 (6) (7) – – –)2 (x – x )(y – y ) (y – y (4) × (2) (2) Squared 2,600,000 40,000 0 7,200,000 160,000 1,400,000 40,000 400,000 10,000 0 1,800,000 90,000 1,000,000 10,000 4,800,000 90,000 (800,000) 40,000 0 2,800,000 40,000 21,200,000 520,000 Column total (Column total) (Column total) 21, 200, 000 21, 200, 000 = 607, 360, 000, 000, 000 (1, 168, 000, 000)(520, 000) 21, 000, 000 = 8602 24, 644, 675 r = 7399 (2) Perfect direct correlation would be evidenced by a correlation coefficient of one The coefficient of 9464 revealed in (1)(a) is closer to one than the coefficient of 8602 in (1)(b) This means that the variable portion of travel expense varies more directly with movements in the number of calls made than with the value of orders received To explain this further, the relative coefficients of determination are obtained by squaring the coefficients of correlation and expressing the answer as a percentage in each case The coefficients of determination are 89.57% for calls made and only 73.99% for orders received This means that approximately 90% of the movements in the variable portion of travel expense are related to fluctuations in the number of calls made, and the remaining 10% of the movements are related to other factors 3-22 Chapter P3-6 (Continued) (c) Month January February March April May June July August September October November December Total (1) y Electricity Cost $ 455 450 435 485 470 475 400 450 435 500 495 470 (2) (3) (4) – (y – y ) x (x – x–) Cost Number Activity Deviations of Billets Deviations (5) 2,000 (10) 1,800 (200) (25) 1,900 (100) 25 2,200 200 10 2,100 100 15 2,000 (60) 1,400 (600) (10) 1,900 (100) (25) 1,800 (200) 40 2,400 400 35 2,300 300 10 2,200 200 $5,520 24,000 (5) (x – x–)2 (4) Squared 40,000 10,000 40,000 10,000 360,000 10,000 40,000 160,000 90,000 40,000 800,000 (6) (7) – – –)2 (x – x )(y – y ) (y – y (2) (4) × (2) Squared 25 2,000 100 2,500 625 5,000 625 1,000 100 225 36,000 3,600 1,000 100 5,000 625 16,000 1,600 10,500 1,225 2,000 100 81,000 8,950 – = Σy ÷ n = $5,520 ÷ 12 = $460 y x– = Σx ÷ n = 24,000 ÷ 12 = 2,000 81, 000 Σ( x − x )(y − y ) Column total = = $.10125 = Σ( x − x ) Column total 800, 000 = y − bx Variable rate (b ) = Fixed cost (a ) = $460 − ($.10125)(2, 000) = $257.50 (2) The coefficient of correlation (r) and the coefficient of determination (r2), using data from the answer in requirement (1)(c) follow: r= Σ( x i − x )(y i − y ) Σ( x i − x )2 Σ(y i − y )2 r = (.957 )2 = 916 = 81, 000 = 957 (800, 000) (8, 950) Chapter 3-23 P3-6 (Concluded) (3) Month January February March April May June July August September October November December Total (1) y Actual Electricity Cost $ 455 450 435 485 470 475 400 450 435 500 495 470 $5,520 (2) x Number of Billets 2,000 1,800 1,900 2,200 2,100 2,000 1,400 1,900 1,800 2,400 2,300 2,200 24,000 (3) (y′′= a + bx) Estimated Electricity Cost $ 460 440 450 480 470 460 399 450 440 501 490 480 $5,520 (4) (y – y′′ ) (5) (y – y′′ )2 (1) – (3) (5) 10 (15) 15 (5) (1) (10) (4) Squared 25 100 225 25 225 25 25 100 752 Σ(y − y ′ )2 Column total $752 s′= = = = $8.672 12 − 10 n −2 The 95% confidence interval for electricity costs at the 2,200 Billets level of activity would be determined as follows: a + bx ± t 95% s′ 1+ ( x − x )2 + n Σ( x − x ) 28)($8.672) 1+ $257.50 + ($.10125)(2,200) ± (2.22 (2,200 − 2,000)2 + 12 800,000 $480.25 ± (2.228)($8.672)(1.065) $480.25 ± $20.58 or between a low of $459.67 and a high of $500.83 3-24 Chapter P3-7 (1) (1) yi Factory Overhead Cost (2) –) (yi – y Difference from Average of $7,900 Direct Labor Hours $8,500 9,900 8,950 9,000 8,150 7,550 7,050 6,450 6,900 7,500 7,150 7,800 600 2,000 1,050 1,100 250 (350) (850) (1,450) (1,000) (400) (750) (100) 2,000 2,400 2,200 2,300 2,000 1,900 1,400 1,000 1,200 1,700 1,600 1,900 200 600 400 500 200 100 (400) (800) (600) (100) (200) 100 8,700 9,300 9,300 8,700 8,000 7,650 6,750 7,100 7,350 7,250 7,100 7,500 800 1,400 1,400 800 100 (250) (1,150) (800) (550) (650) (800) (400) 2,100 2,300 2,200 2,200 2,000 1,800 1,200 1,300 1,500 1,700 1,500 1,800 Jan 8,600 Feb 9,300 Mar 9,400 Apr 8,700 May 8,100 June 7,600 July 7,000 Aug 6,900 Sep 7,100 Oct 7,500 Nov 7,000 Dec 7,600 Total $284,400 700 1,400 1,500 800 200 (300) (900) (1,000) (800) (400) (900) (300) 2,000 2,300 2,300 2,200 2,000 1,800 1,300 1,200 1,300 1,800 1,500 1,900 64,800 Month 20A Jan Feb Mar Apr May June July Aug Sep Oct Nov Dec (3) xi (4) (5) (xi – x–) (xi – x–)2 Difference from Average of 1,800 (4) Hours Squared (6) (7) –) (y – y–)2 (xi – x–)(yi – y i (4) × (2) (2) Squared 40,000 360,000 160,000 250,000 40,000 10,000 160,000 640,000 360,000 10,000 40,000 10,000 120,000 1,200,000 420,000 550,000 50,000 (35,000) 340,000 1,160,000 600,000 40,000 150,000 (10,000) 360,000 4,000,000 1,102,500 1,210,000 62,500 122,500 722,500 2,102,500 1,000,000 160,000 562,500 10,000 300 500 400 400 200 (600) (500) (300) (100) (300) 90,000 250,000 160,000 160,000 40,000 360,000 250,000 90,000 10,000 90,000 240,000 700,000 560,000 320,000 20,000 690,000 400,000 165,000 65,000 240,000 640,000 1,960,000 1,960,000 640,000 10,000 62,500 1,322,500 640,000 302,500 422,500 640,000 160,000 200 500 500 400 200 (500) (600) (500) (300) 100 40,000 250,000 250,000 160,000 40,000 250,000 360,000 250,000 90,000 10,000 5,280,000 20B Jan Feb Mar Apr May June July Aug Sep Oct Nov Dec 20C 140,000 490,000 700,000 1,960,000 750,000 2,250,000 320,000 640,000 40,000 40,000 90,000 450,000 810,000 600,000 1,000,000 400,000 640,000 160,000 270,000 810,000 (30,000) 90,000 11,625,000 29,155,000 Chapter 3-25 P3-7 (Continued) b= Σ( x i − x )(y i − y ) Column total 11, 625, 000 = = = $2.20 variable cost rate Σ( x i − x )2 Column total 5, 280, 000 Since y = a + bx and y = Σy i ÷ n and x = Σx i ÷ n, then : ($284, 400 ÷ 36) = a + ($2.20)(64, 800 ÷ 36) $7, 900 = a + $3,, 960 a = $3, 940 fixed overhead cost (2) The coefficient of correlation and the coefficient of determination, using data from the requirement (1) answer: r = r = Σ( x i − x )(y i − y ) Σ( x i − x )2 Σ(y i − y )2 = Column total (Column total) (Column total) 11, 625, 000 11, 625, 000 11, 625, 000 = = = 9370 153, 938, 400, 000, 000 12, 407, 191 (5, 280, 000)(29, 155, 000) r = (.9370)2 = 8780 3-26 Chapter P3-7 (Continued) (3) The standard error of the estimate: (1) xi Month 20A January February March April May June July August September October November December 20B January February March April May June July August September October November December 20C January February March April May June July August September October November December Total Direct Labor Hours (2) yi Actual Factory Overhead Cost (3) (4) (5) (y′′i = a + bxi) (y′′i – yi) (y′′i – yi)2 Predicted Prediction Factory Prediction Error Overhead Error Squared Cost (2) – (3) (4) Squared 2,000 2,400 2,200 2,300 2,000 1,900 1,400 1,000 1,200 1,700 1,600 1,900 $ 8,500 9,900 8,950 9,000 8,150 7,550 7,050 6,450 6,900 7,500 7,150 7,800 $ 8,340 9,220 8,780 9,000 8,340 8,120 7,020 6,140 6,580 7,680 7,460 8,120 $160 680 170 (190) (570) 30 310 320 (180) (310) (320) $ 25,600 462,400 28,900 36,100 324,900 900 96,100 102,400 32,400 96,100 102,400 2,100 2,300 2,200 2,200 2,000 1,800 1,200 1,300 1,500 1,700 1,500 1,800 8,700 9,300 9,300 8,700 8,000 7,650 6,750 7,100 7,350 7,250 7,100 7,500 8,560 9,000 8,780 8,780 8,340 7,900 6,580 6,800 7,240 7,680 7,240 7,900 140 300 520 (80) (340) (250) 170 300 110 (430) (140) (400) 19,600 90,000 270,400 6,400 115,600 62,500 28,900 90,000 12,100 184,900 19,600 160,000 2,000 2,300 2,300 2,200 2,000 1,800 1,300 1,200 1,300 1,800 1,500 1,900 64,800 8,600 9,300 9,400 8,700 8,100 7,600 7,000 6,900 7,100 7,500 7,000 7,600 $284,400 8,340 9,000 9,000 8,780 8,340 7,900 6,800 6,580 6,800 7,900 7,240 8,120 $284,400 260 300 400 (80) (240) (300) 200 320 300 (400) (240) (520) 67,600 90,000 160,000 6,400 57,600 90,000 40,000 102,400 90,000 160,000 57,600 270,400 $3,560,200 Chapter 3-27 P3-7 (Concluded) s′= (4) Σ(y i − y ′i )2 Column total $3, 560, 200 = = = $104, 712 = $324 36 − 34 n −2 Since a large sample is used in this problem, t95% = z95% and the confidence interval is: y ′i ± z 95% s′ ($3, 940 + ($2.20)(2, 200)) ± (1.960)($324 ) $8, 780 ± $635 3-28 Chapter P3-8 (1) (1) yi Maintenance Months Cost Jan., 20A $ 1,195 Feb., 20A 1,116 Mar., 20A 1,390 Apr., 20A 1,449 May, 20A 1,618 June, 20A 1,525 July, 20A 1,687 Aug., 20A 1,650 Sep., 20A 1,595 Oct., 20A 1,675 Nov., 20A 1,405 Dec., 20A 1,251 Jan., 20B 950 Feb., 20B 1,175 Mar., 20B 1,425 Apr., 20B 1,506 May, 20B 1,608 June, 20B 1,653 July, 20B 1,675 Aug., 20B 1,724 Sep., 20B 1,626 Oct., 20B 1,575 Nov., 20B 1,653 Dec., 20B 1,418 Total $35,544 (2) –) (yi – y Cost Deviation (286) (365) (91) (32) 137 44 206 169 114 194 (76) (230) (531) (306) (56) 25 127 172 194 243 145 94 172 (63) (3) xi Labor Hours 950 1,024 1,109 1,148 1,313 1,261 1,552 1,372 1,366 1,455 1,221 1,150 999 1,022 1,220 1,283 1,339 1,250 1,440 1,290 1,335 1,164 1,373 1,124 (4) (xi – x–) Activity Deviation (290) (216) (131) (92) 73 21 312 132 126 215 (19) (90) (241) (218) (20) 43 99 10 200 50 95 (76) 133 (116) 29,760 (5) (xi – x–)2 (4) Squared 84,100 46,656 17,161 8,464 5,329 441 97,344 17,424 15,876 46,225 361 8,100 58,081 47,524 400 1,849 9,801 100 40,000 2,500 9,025 5,776 17,689 13,456 553,682 (6) (7) – – –)2 (xi – x )(yi – y ) (yi – y (2) (4) × (2) Squared 82,940 81,796 78,840 133,225 11,921 8,281 2,944 1,024 10,001 18,769 924 1,936 64,272 42,436 22,308 28,561 14,364 12,996 41,710 37,636 1,444 5,776 20,700 52,900 127,971 281,961 66,708 93,636 1,120 3,136 1,075 625 12,573 16,129 1,720 29,584 38,800 37,636 12,150 59,049 13,775 21,025 (7,144) 8,836 22,876 29,584 7,308 3,969 651,300 1,010,506 y– = Σyi ÷ n = $35,544 ÷ 24 = $1,481 x– = Σxi ÷ n = 29,760 ÷ 24 = 1,240 r = = Σ( x i − x )(y i − y ) Σ( x i − x )2 Σ(y i − y )2 = Column total (Column total) (Column total) 651, 300 651, 300 651, 300 = = 870725 = (553, 682)(1, 010, 506) 559, 498,, 983, 092 747, 997 r = (.870725)2 = 758162 Chapter 3-29 P3-8 (Continued) (1) yi Maintenance Months Cost Jan., 20A $1,195 Feb., 20A 1,116 Mar., 20A 1,390 Apr., 20A 1,449 May, 20A 1,618 June, 20A 1,525 July, 20A 1,687 Aug., 20A 1,650 Sep., 20A 1,595 Oct., 20A 1,675 Nov., 20A 1,405 Dec., 20A 1,251 Jan., 20B 950 Feb., 20B 1,175 Mar., 20B 1,425 Apr., 20B 1,506 May, 20B 1,608 June, 20B 1,653 July, 20B 1,675 Aug., 20B 1,724 Sep., 20B 1,626 Oct., 20B 1,575 Nov., 20B 1,653 Dec., 20B 1,418 Total $35,544 (2) –) (yi – y Cost Deviation (286) (365) (91) (32) 137 44 206 169 114 194 (76) (230) (531) (306) (56) 25 127 172 194 243 145 94 172 (63) (3) (4) xi (xi – x–) Machine Activity Hours Deviation 809 (266) 744 (331) 987 (88) 987 (88) 1,186 111 1,154 79 1,291 216 1,238 163 1,186 111 1,246 171 997 (78) 841 (234) 502 (573) 733 (342) 1,090 15 1,135 60 1,174 99 1,246 171 1,264 189 1,323 248 1,230 155 1,165 90 1,237 162 1,035 (40) 25,800 (5) (xi – x–)2 (4) Squared 70,756 109,561 7,744 7,744 12,321 6,241 46,656 26,569 12,321 29,241 6,084 54,756 328,329 116,964 225 3,600 9,801 29,241 35,721 61,504 24,025 8,100 26,244 1,600 1,035,348 (6) (7) –) (y – y–)2 (x – x–)(y – y i (2) (4) × (2) Squared 76,076 81,796 120,815 133,225 8,008 8,281 2,816 1,024 15,207 18,769 3,476 1,936 44,496 42,436 27,547 28,561 12,654 12,996 33,174 37,636 5,928 5,776 53,820 52,900 304,263 281,961 104,652 93,636 (840) 3,136 1,500 625 12,573 16,129 29,412 29,584 36,666 37,636 60,264 59,049 22,475 21,025 8,460 8,836 27,864 29,584 2,520 3,969 1,013,826 1,010,506 y– = Σyi ÷ n = $35,544 ÷ 24 = $1,481 x– = Σxi ÷ n = 25,800 ÷ 24 = 1,075 r = = Σ( x i − x )(y i − y ) Σ( x i − x )2 Σ(y i − y )2 = Column total (Column total) (Column total) 1, 013, 826 1, 013, 826 1, 013, 826 = = = 991176 1,, 046, 225, 366, 088 1, 022, 852 (1, 035, 348)(1, 010, 506) r = (.991176)2 = 982430 3-30 Chapter P3-8 (Continued) (2) The activity measure used to predict maintenance expense should be machine hours, which will result in the following cost estimates: Σ( x − x )(y − y ) Column total 1, 013, 826 = $.979213 variable rate = = Σ( x − x )2 Column total 1, 035, 348 Since y = a + bx , then the estimated fixed cost is determined as follows: a = y − bx b= a = $1, 481− ($.979213)(1, 075) a = $1, 481− $1, 052.65 a = $428.35 (3) (1) xi Months Jan., 20A Feb., 20A Mar., 20A Apr., 20A May, 20A June, 20A July, 20A Aug., 20A Sep., 20A Oct., 20A Nov., 20A Dec., 20A Jan., 20B Feb., 20B Mar., 20B Apr., 20B May, 20B June, 20B July, 20B Aug., 20B Sep., 20B Oct., 20B Nov., 20B Dec., 20B Total *rounding error Machine Hours 809 744 987 987 1,186 1,154 1,291 1,238 1,186 1,246 997 841 502 733 1,090 1,135 1,174 1,246 1,264 1,323 1,230 1,165 1,237 1,035 25,800 (2) (3) (4) yi (y′′i = a + bxi) (yi – y′′i) Actual Predicted Prediction Maintenance Maintenance Error Cost Cost (2) – (3) $ 1,195 $ 1,221 $(26) 1,116 1,157 (41) 1,390 1,395 (5) 1,449 1,395 54 1,618 1,590 28 1,525 1,558 (33) 1,687 1,693 (6) 1,650 1,641 1,595 1,590 1,675 1,648 27 1,405 1,405 1,251 1,252 (1) 950 920 30 1,175 1,146 29 1,425 1,496 (71) 1,506 1,540 (34) 1,608 1,578 30 1,653 1,648 1,675 1,666 1,724 1,724 1,626 1,633 (7) 1,575 1,569 1,653 1,640 13 1,418 1,442 (24) $35,544 $35,547* $ (3)* Σ(y i − y ′ )2 Column total $17, 817 = = = $28.458103 s′= 24 − 22 n −2 (5) (yi – y′′i)2 (4) Squared $ 676 1,681 25 2,916 784 1,089 36 81 25 729 900 841 5,041 1,156 900 25 81 49 36 169 576 $17,817 Chapter 3-31 P3-8 (Concluded) (4) The 95% confidence interval for maintenance cost at the 1,100 machine hour level of activity is: y ′i ± t 95% s′ 1+ ( x i − x )2 + n Σ( x i − x )2 $428.35 + ($.979213)(1, 100) ± (2.074 )(28.458103) 1+ $1, 505.48 ± $60.26 (1, 100 − 1, 075)2 + 24 1, 035, 348 3-32 Chapter CASES C3-1 (1) W = = = = a + bS 5.062 + (.023) (1,200) 5.062 + 27.6 32.662 or about 33 total workers Total workers needed Less permanent workers 33 10 Number of temporary workers needed 23 (2) Regression appears to be better than Regression because: (a) Data outside the relevant range have been excluded, thereby removing any bias (b) The standard error of the estimate (s′) for Regression is smaller than the standard error of the estimate for Regression (.432 compared to 2.012) (c) The coefficient of determination (r 2) is higher for Regression than the coefficient of determination for Regression (.998 compared to 962) (3) Jim Locter can use the regression in his planning for temporary workers if the following conditions exist: (a) The forecasted daily shipments are greater than 300 and not deviate too much from the actual shipments (b) The amount of work to be done is dependent only on the number of shipments to be made and does not change from shipment to shipment (c) Worker productivity is expected to remain approximately the same as that experienced during the period used to develop the regression (d) A strong cause and effect relationship exists between the dependent variable and the independent variable (e) The time frame for a forecast is short-term (4) The regression could be improved by the following: (a) Redeveloping the regression using the number of hours worked as the dependent variable (b) Performing another analysis if rush orders or deviations of actual orders from forecasts occur with any degree of regularity (c) Investigating the historical data used as a basis for the regression to determine if there are any further unusual circumstances that should be removed from the data set (d) Redoing the regression after a period of time, such as four to six months, to discover if there have been any changes in the relationship between the dependent and the independent variables Chapter 3-33 C3-2 (1) (2) The increase in y associated with a unit increase in x is 1.2 Therefore, a 500unit increase in x will result in a 600-unit increase (1.2 × 500) in y (direct labor hours) (a) The equation may be unreliable if the correlation is spurious The assumption is that there is a logical relationship between output and the use of electric power and direct labor (b) The equation may be reliable under the conditions at the time of the study, but if conditions change, the results may be unreliable (c) Data used were limited to a range of 500–2,000 units (d) It is assumed that a straight-line assumption is valid (e) The coefficient of correlation is a measure of the extent to which two variables are related linearly It is a relative measure of goodness of fit More of the variation in y is explained by the regression equation for direct labor hours than for electric power, that is, the equation for direct labor hours is a better fit than the equation for electric power (f) The standard error of the estimate is a measure of variation from the regression line If the observations are normally distributed about the regression equation, the standard error can be interpreted in the same way as the standard deviation The standard error is greater in the case of direct labor hours than in the case of electric power CGA-Canada (adapted) Reprint with permission C3-3 (1) An advantage of Alternative A is that using time as an independent variable is a convenient way to take into consideration all possible factors that may be influencing the dependent variable during each period of time A disadvantage of Alternative A is that there is no logical relationship between years and rental expense An advantage of Alternative B is that this method is logical because as revenues increase, the stores increase, and, thus, rental expense increases A disadvantage of Alternative B is that an estimate of revenues is required An advantage of Alternative C is that the mathematical calculations are relatively easy and the method is easy to understand A disadvantage of Alternative C is that the arithmetic average is an oversimplification that does not recognize any relationship between variables 3-34 Chapter C3-3 (Concluded) (2) (3) Motorco Corporation should select Alternative B because the relationship between revenue and the rental expense is logical, the coefficient of correlation is high, and the standard error of the estimate is low A statistical technique is an appropriate method for estimating rental expense before Motorco actually contacts Alpha Auto Parts A statistical technique attempts to measure the covariation between the variables that are presumed to have a cause and effect relationship, and such a relationship appears to exist in this situation Of course, Motorco is assuming that any relationships that exist in the historical data will continue in the future without change Management may want to adjust the variables for changes that it expects will occur, and Motorco may wish to introduce other quantitative variables C3-4 (1) (2) (3) The phrase “regression provides a relational statement rather than a causal statement” means that regression analysis is used to determine a relationship, but not necessarily a cause-and-effect relationship A specific value for a regression coefficient does not imply that the independent variable(s) causes a change in the dependent variable The meaning of each of the symbols in the basic formula for a regression equation follows: y′′i = estimated value of the i th observation of the dependent variable a = the y-axis intercept or constant term (e.g., the fixed portion of a semivariable expense) b = the regression coefficient corresponding to the independent variable x (e.g., the variable cost element associated with a one unit change in activity x) xi = the i th observation of the first independent variable c = the regression coefficient corresponding to the independent variable z (e.g., the variable cost element associated with a one unit change in activity z) zi = the i th observation of the second independent variable ei = the error term associated with the i th observation Statistical factors used to test a regression equation for goodness of fit include: (a) The coefficient of determination, r 2, which indicates the portion of the variance in the dependent variable explained by the independent variables A coefficient of determination approaching indicates a good fit Chapter 3-35 C3-4 (Concluded) (b) (4) (a) (b) (c) (d) (e) The standard error of the estimate which measures the dispersion of the observed points about the regression line A standard error of the estimate approaching zero indicates a good fit The term “linearity within a relevant range” means that in a specific situation, a straight-line relationship between the dependent variable and the independent variables can be assumed only within the range of historically observed values The term “constant variance (homoscedasticlty)” means that the distribution of the observations about the regression line is uniform for all values of the independent variables within the observed range of values The term “serial correlation” refers to the lack of independence in a series of successive observations over time The deviation of a value from the regression line should be unrelated to the deviation of any other point from this line The term “normality” means that the joint probability distribution of the variables is normally distributed (multivariate normal) The frequency of the observations should approximate a normal curve The term “multicollinearity” refers to the correlation of independent variables When independent variables are highly correlated with each other, the relationship(s) between the independent variables may obscure the relationship between the independent variables(s) and the dependent variable C3-5 (1) (2) (a) D = (2.455 + (.188)(1,500,000 ÷ 100,000)) × 10,000 units = (2.455 + 2.82) × 10,000 units = 5.275 × 10,000 units = 52,750 units (b) D = (2.491 + (.44)(12,000,000 ữ 1,000,000)) ì 10,000 units = (2.491 + 5.28) × 10,000 units = 7.771 × 10,000 units = 77,710 units The 50% confidence interval for demand is calculated as follows: D = 104,160 units ± (.69)(.922 × 10,000 units) = 104,160 units ± 6,361.8 units or between 97,798 units and 110,522 units 3-36 Chapter C3-5 (Concluded) (3) (4) Equation is the best The coefficient of correlation and the coefficient of determination are the highest of the four equations The coefficient of determination indicates that 70.3% of the sample variance of automobile sales is explained by the regression For predictive purposes, the standard error of the estimate at 922 is also the lowest of the four models, giving the tightest (smallest) physical confidence interval of any of the equations Equation assumes that factory rebates (R) are dependent on advertising funds (A) The results of the analysis show that factory rebates and advertising funds are almost totally independent and, therefore, cannot be used to predict each other The results of Equation lend credibility to the use of A and R in Equation The independence of A and R reduces the possible negative aspects of collinearity ... DAYS PER MONTH Fixed cost per month determined by inspection $350 Average cost Less fixed cost Variable cost $700 350 $350 $350 3,500 average guest days = $.10 variable cost per guest day... electricity cost Machine hours explain more of the variance in electricity cost than labor hours (3) With machine hours as the basis for predicting electricity cost, the fixed cost and the variable cost. .. supplies cost Labor hours explain more of the variance in supplies cost than machine hours (3) With labor hours as the basis for predicting supplies cost, the fixed cost and the variable cost rate