CAPITULO I PROBLEMÁTICA DE LA INVESTIGACIÓN
4. CAPITULO IV METODOLOGIA
4.13 Presentación y análisis de los resultados focus group
This appendix expands the discussion in the text to help you understand and interpret regression output. S TA N D A R D E R R O R S O F T H E C O E F F I C I E N T S A N D t - S TAT I S T I C S
The standard errors of the coefficients give an idea of the confidence we can have in the fixed and variable cost coefficients. The smaller the standard error relative to its coefficient, the more precise the estimate. (Such computational precision does not necessarily indicate that the esti- mating procedure is theoretically correct, however.)
The ratio between an estimated regression coefficient and its standard error is known as the t-value or t-statistic. If the absolute value of the t-statistic is approximately 2 or larger, we can be relatively confident that the actual coefficient differs from zero.7
If a variable cost coefficient has a small t-statistic, we may conclude that little, if any, relation exists between this particular activity (or independent variable) and changes in costs. If a fixed cost coefficient has a small t-statistic, we may conclude that these costs have little, if any, fixed cost component (which we would expect for operating room supplies or direct materials in manufacturing, for example).
7Statistics books provide t-tables that make the analysis of t-statistics more precise.
EXHIBIT 5.14 Average Labor Hours Required for X Units
Average Labor Hours 10 0 100 10 100 1,000 Units Y = 125X–.322
R2
The R2attempts to measure how well the line fits the data (that is, how closely the data points cluster about the fitted line). If all the data points were on the same straight line, the R2would be 1.00—a perfect fit. If the data points formed a circle or disk, the R2would be zero, indicating that no line passing through the center of the circle or disk fits the data better than any other. Technically, R2is a measure of the fraction of the total variance of the dependent variable about its mean that the fitted line explains. An R2of 1 means that the regression explains all of the variance; an R2of zero means that it explains none of the variance. R2is sometimes known as the ‘‘coefficient of determination.’’
Many users of statistical regression analysis believe that a low R2indicates a weak relation between total costs (dependent variable) and the activity base (independent variable). A low standard error (or high t-statistic) for the estimated variable cost coefficient signals whether or not the activity base performs well as an explanatory variable for total costs. With a large number of data observations, both low R2 and significant regression coefficients can occur. Exhibit 5.15 illustrates this possibility.
C A U T I O N S W H E N U S I N G R E G R E S S I O N
Computers easily perform statistical estimating techniques but often do not provide the necessary warnings. We conclude this section by providing several cautionary comments. A relation achieved in a regression analysis does not imply a causal relation; that is, a correlation between two variables does not imply that changes in one will cause changes in the other. An assertion of causality must be based on either a priori knowledge or some analysis other than a regression analysis.
Users of regression analysis should be wary of drawing too many inferences from the results unless they are familiar with such statistical estimation problems as multicollinearity, autocorrelation, and heteroscedasticity and how to deal with them. Statistics books deal with these statistical estimation problems.
Briefly, multicollinearity refers to the problem caused in multiple linear regression (more than one independent variable) when the independent variables are not independent of each other but are correlated. When severe multicollinearity occurs, the regression coefficients are unreliable. For example, direct labor hours worked during a month are likely to be highly correlated with direct labor costs during the month, even when wage rates change over time. If both direct labor hours and direct labor costs are used in a multiple linear regression, we would expect to have a problem of multicollinearity.
Autocorrelation problems arise when the data represent observations over time. Autocorre- lation occurs when a linear regression is fit to data where a nonlinear relation exists between the dependent and independent variables. In such a case, the deviation of one observation from the fitted line can be predicted from the deviation of the prior observation(s). For example, if demand
EXHIBIT 5.15 Relation between Statistical Significance
of Variable Cost Coefficient and R2
Total Cost
Statistically Significant Variable Cost Coefficient
but High R2 Total
Cost
Statistically Significant Variable Cost Coefficient
but Low R2
for a product is seasonal and production is also seasonal, a month of large total costs will more likely follow another month of large total costs than a month of small total costs. In such a case, we would have autocorrelation in the deviations of the data points from a fitted straight line.
Autocorrelation affects the estimates of standard errors of the regression estimates, and therefore it affects the t-statistics. If autocorrelation exists, the estimates of standard errors may be understated and the t-statistics may be overstated in the regression output.
Heteroscedasticity refers to the phenomenon that occurs when the average deviation of the dependent variable from the best-fitting linear relation is systematically larger in one part of the range of independent variable(s) than in others. For example, if the firm uses less-reliable equipment and employs less-skilled labor in months of large total production, variation in total costs during months of large total production is likely to be greater than in months of small total production. Heteroscedasticity affects the reliability of the estimates of standard errors of the regression coefficients (and therefore affects the reliability of the t-statistics).
Que stions , Exercis e s , and Problems
R E V I E W Q U E S T I O N S
1. Review the meaning of the concepts or terms given in Key Terms and Concepts.
2. Which method of cost estimation does not rely primarily on historical cost data? What are the drawbacks of this method?
3. Name three methods of cost estimation.
4. The simplifying assumptions on which cost estimations are based include which of the following?
a. Cost behavior depends on one activity variable (except multiple regression). b. Cost behavior patterns are not linear within the relevant range.
c. Costs are only fixed. d. All of the above. 5. (See Appendix 5.2.) R2
a. measures how well the line fits the data. b. is a perfect fit when its value is 1.0. c. is the standard error of the coefficient. d. a. and b.
6. Multiple regression
a. has one dependent variable.
b. has more than one independent variable. c. has only one independent variable. d. a. and b.
C R I T I C A L A N A LY S I S A N D D I S C U S S I O N Q U E S T I O N S
7. ‘‘The concepts of short-run costs and long-run costs are relative—short run could mean a day, a month, a year, or even 10 years, depending on what you are looking at.’’ Comment. 8. ‘‘My variable costs are $2 per unit. If I want to increase production from 100,000 units to
150,000 units, my total costs should go up by only $100,000.’’ Comment.
9. What methods of cost estimation rely primarily on historical data? Discuss the problems an unwary user may encounter with the use of historical cost data.
10. What steps would you take to deal with a supervisor who asks you to falsify the results of your cost estimation?
11. In what cultures might analysts be more willing to reveal errors they made in estimating costs?
12. When estimating fixed and variable costs, it is possible to have an equation with a negative intercept. Does this mean that at zero production the company has negative fixed costs?
13. Refer to the Managerial Application ‘‘United Airlines Uses Regression to Estimate Cost Behavior.’’ What independent variables did the analysts use in the regression? What did the analysts who developed the regression conclude? Based on your experience, do you agree with their conclusions?
14. (See Appendix 5.2.) How is regression used to identify what cost drivers might be used in activity-based costing?
15. (See Appendix 5.1.) Describe the phenomenon that gives rise to learning curves. To what type of costs do learning curves apply?
16. ‘‘Simplification of all costs into only fixed and variable costs distorts the actual cost behavior pattern of a firm. Yet businesses rely on this method of cost classification.’’ Comment. 17. ‘‘The account analysis method uses subjective judgment. So we cannot really consider it a
valid method of cost estimation.’’ Comment.
18. Suggest ways that one can compensate for the effects of inflation when preparing cost estimates.
E X E R C I S E S
Solutions to even-numbered exercises are at the end of the chapter.
19. Graphs of cost relations. Sketch cost graphs for the following situations:
a. A 30 percent increase in fixed costs will enable Donelan Company to produce up to 75 percent more. Variable costs per unit will remain unchanged.
b. Refer to part a. What if Donelan Company’s variable costs per unit triple for the addi- tional units it intends to produce?
c. Richmond’s variable marketing costs per unit decline as more units are sold.
d. Anderson Paper pays a flat fixed charge per month for electricity plus an additional rate of $0.20 per unit for all consumption over the first 2,000 units.
e. Indirect labor costs at KMD Bank consist only of supervisors’ salaries. The bank needs one supervisor for every 20 clerks.
f. National Plastics currently operates close to capacity. A short-run increase in production would result in increasing unit costs for every additional unit produced.
20. Cost behavior in event of capacity change. Slopeside Resort, a lodge located in a fast- growing ski resort, is planning to open its new wing this coming winter, increasing the number of beds by 40 percent. Although variable costs per guest-day will remain unchanged, total fixed costs will increase by 25 percent. Last year’s costs follow:
Variable Costs ... $50,000 Total Fixed Costs... 30,000
a. Sketch the cost function.
b. Calculate the additional fixed operating costs that Slopeside Resort will incur next year.
21. Cost behavior when costs are semivariable. Data from the shipping department of Brawn Company for the past two months follow:
Number of Packages Shipped Shipping Department Costs November... 6,000 $12,000 December ... 9,000 15,000
a. Sketch a line describing these costs as a function of the number of packages shipped. b. What is the apparent variable cost per package shipped?
c. The line should indicate that these shipping costs are semivariable. What is the apparent fixed cost per month of running the shipping department during November and December?
22. Identifying cost behavior. Data from the shipping department of Wanda’s Gourmet Foods for the past four months follow:
Number of Packages Shipped Shipping Department Costs May... 0 $1,500 June... 2,000 3,500 July ... 2,500 4,500 August ... 1,500 3,000
Are these costs fixed, semifixed, variable, or semivariable?
23. Cost estimation using regression analysis. Dali’s Financial Services prepares tax returns for small businesses. Data on the company’s total costs and output for the past six months appear in the table that follows. The results of regression analysis are also provided. a. Plot the data and the regression line on a graph. (See Problem 5.4 for Self-Study.) b. Estimate total monthly costs for a month when 330 tax returns are prepared, using the
estimates from the regression output.
Month Tax Returns Prepared Total Costs January... 200 $160,000 February ... 280 192,000 March ... 300 198,000 April... 260 180,000 May... 260 186,000 June... 240 170,000 Regression output: TC ¼ $78,045 þ ($401 Number of tax returns).
24. Learning curve. Phantom Co. makes technical products for mysterious customers. To make Product J24, the company recently recorded the following costs, which decline subject to an 85 percent cumulative learning curve.
Cumulative Number of Units Produced
Average Labor Costs per Unit 1 ... $5,000 2 ... 4,250 4 ... ? 8 ... ? 16 ... ?
Complete the chart by filling in the cost amounts for volumes of 4, 8, and 16 units. 25. Average cost calculations. Abs Health Club has the following cost equation:
Total Costs¼ $40,000 þ $80n, where n¼ number of memberships.
a. Calculate Abs’s average fixed cost per membership when there are 1,000 memberships. b. What is the average variable cost per membership when there are 1,000 memberships? c. Calculate the average cost per membership when there are 1,000 memberships. 26. Repair cost behavior. The Shilling Company analyzed repair costs by month using linear
regression analysis. The equation fit took the following form: Total
Repair Costs
¼Fixed Costsþ
Variable Repair Costs per Machine Hour Used
during Month Machine Hours Actually Used during Month 0 @ 1 A TRC¼ a þ bx:
The results were
TRC¼ $20,000 $0:75x
Average monthly repair costs have been $18,800, and machine hours used have averaged 1,600 hours per month. Management worries about the ability of the analyst who carried out this work because of the negative coefficient for variable cost.
What is your evaluation of these results?
27. Interpreting regression results. The output of a regression of overhead costs on direct labor costs per month follows:
Regression Results Equation: Intercept ... $38,000 Slope ... 2.20 Statistical Data: R2... 0.85 Smalltime Consulting Services plans to operate at a level that would call for direct labor costs of $20,000 per month for the coming year.
a. Use the regression output to write the overhead cost equation.
b. Based on the cost equation, compute the estimated overhead cost per month for the coming year.
c. (See Appendix 5.2.) How well does this regression explain the relation between direct labor and overhead?
28. Interpreting regression data. A marketing manager of a company used a pocket calculator to estimate the relation between sales dollars for the past three years and monthly advertising expenditures (the independent variable). The regression results indicated the following equation:
Sales Dollars¼ $97,000 ð1:45 Advertising DollarsÞ
Do these results imply that advertising hurts sales? Why would there appear to be a negative relation between advertising expenditures and sales?
29. Cost estimation using regression analysis. Milky Chocolates has observed the following overhead costs for the past 12 months:
Month Overhead Costs Boxes of Output January... $11,400 4,500 February ... 15,600 11,000 March ... 16,800 12,000 April ... 12,000 5,500 May... 14,100 9,000 June ... 15,600 10,500 July ... 13,200 7,500 August ... 12,300 5,000 September... 15,600 11,500 October... 12,900 6,000 November... 14,400 8,500 December ... 15,000 10,000
The results of the regression analysis are:
TC¼ $8,781 þ ð$0:63 Number of BoxesÞ
a. Plot the data and the regression line (like Problem 5.4 for Self-Study).
b. Estimate total monthly costs for a month when 10,200 boxes of chocolate are produced.
P R O B L E M S
30. Multiple regression. The managers of Peterson’s Catering Company are analyzing the costs involved in providing catering services. Managers have selected the following cost drivers: units of meals produced, total deliveries, number of VIP services, number of new customers, and new products developed. Here are the cost data and levels of cost driver activity for the past 16 months.
Month Total Overhead Meals Produced Deliveries VIP Services New Customers New Products 1 ... $69,094 12,690 1,340 345 3 0 2 ... 64,927 11,980 1,180 310 4 0 3 ... 60,332 10,950 1,050 280 4 1 4 ... 57,953 10,280 930 245 5 0 5 ... 55,984 9,020 840 205 7 2 6 ... 53,119 8,130 780 185 8 2 7 ... 52,706 7,540 700 160 10 3 8 ... 53,874 6,980 630 144 12 4 9 ... 53,445 8,930 680 135 10 2 10 ... 54,869 9,800 760 120 9 0 11 ... 59,985 10,560 890 175 8 2 12 ... 61,121 11,560 1,070 200 7 0 13 ... 63,926 11,710 1,240 240 6 0 14 ... 66,602 12,460 1,390 285 4 0 15 ... 72,773 13,520 1,450 330 4 1 16 ... 71,391 13,620 1,510 315 2 0 Totals... $972,101 169,730 16,440 3,674 103 17
a. Using multiple regression, find the cost driver rates for each of the cost drivers. (Note: You must use a computer program such as Microsoft1
Excel to perform this step.) b. Management estimates the following levels of cost driver volumes for the next month for the budget. What is the estimated cost for the budget? (Don’t forget to include the intercept of the regression in your estimate.)
12,000 ... Meals produced 1,100 ... Deliveries
300... VIP services
5 ... New customers
2 ... New products
c. Peterson’s Catering Company is considering outsourcing deliveries. Compared to your answer in requirement b., how much would be saved per month by outsourcing the delivery service (before considering the cost of outsourcing)?
31. Account analysis. Refer to problem 30.
a. Indicate the information in addition to that provided in problem 30 required to perform account analysis.
b. Now assume that Peterson’s Catering Company had the following breakdown of costs for the 16 months reported in problem 30:
Total costs of meals produced... $334,368 Total costs for delivering... 164,400 Total costs of VIP services ... 140,352 Total costs of developing new customers... 154,904 Total costs of developing new products... 18,700 Total facilities-level costs (total for all 16 months)... 159,377 Total costs... $972,101
What are the cost driver rates for (1) meals produced, (2) deliveries, (3) VIP services, (4) new customers, and (5) new products developed using account analysis?
c. What are the estimated costs for a month assuming the following level of cost driver volumes? (Don’t forget to include the facilities-level costs in your estimate.)
12,000 ... Meals produced 1,100 ... Deliveries
300... VIP services 5 ... New customers
2 ... New products
d. Peterson’s Catering Company is considering outsourcing deliveries. Compared to your answer in requirement c., how much would the company save by outsourcing the delivery service (before considering the cost of outsourcing)?
32. Engineering method. Refer to problems 30 and 31. Peterson’s Catering Company hired an engineering consulting firm to perform an engineering estimate of its business costs. The consulting firm came up with the following monthly cost estimates based on information for the current period:
Facilities costs... $9,500
Meal production-level costs... $1.90 per meal produced
Delivery costs ... $11 per delivery VIP services costs... $38 per service New customer costs ... $1,250 per new customer New product costs... $1,000 per new product
a. Assuming the following level of cost driver volume for a month, what is the estimated cost using the engineering estimates? (Don’t forget to include the facilities costs in your estimate.) 12,000 ... Meals produced 1,100 ... Deliveries 300... VIP services 5 ... New customers 2 ... New products
b. Peterson’s Catering Company is considering outsourcing delivery. Compared to your answer in requirement a., how much would the company save by outsourcing the delivery service (before considering the cost of outsourcing)?
33. Multiple regression. Analysts for Brazil Brewery have selected the following cost drivers: volume of beer produced (in hectoliters, i.e., 1 hL ¼ 100 L), total amount of raw materials used (in kilograms), number of batches, volume of water used (in hL), number of cleaning procedures performed—cleanings in place (CIPs)—and number of new products. Here are the cost data and levels of cost driver activity for 18 months.
Month Total Overhead Beer Produced (hL) Raw Material (kg) Number of Batches Water (hL) CIPs New Products January... $ 57,266.65 890 13,573 54 6,005 67 0 February ... 61,020.23 980 15,013 58 6,588 72 1 March ... 64,622.52 1,094 16,781 65 7,336 81 0 April... 68,630.16 1,212 18,551 73 8,002 88 0 May... 70,652.68 1,262 19,370 75 8,435 93 0 June... 79,927.29 1,494 23,182 89 9,940 110 2 July ... 82,867.34 1,557 24,202 95 10,420 106 3 August ... 81,748.55 1,528 23,797 94 10,326 112 2 September... 68,819.71 1,215 18,537 72 8,284 87 0 October... 66,375.05 1,145 17,582 69 7,746 85 0 November... 63,767.19 1,072 16,369 64 7,168 76 0 December ... 62,254.68 1,032 15,628 62 6,933 77 0 January... 56,837.54 872 13,158 50 5,902 61 1 February ... 61,298.34 1,006 15,224 60 6,759 75 0 March ... 63,179.60 1,041 15,763 62 6,990 81 1 April... 66,107.60 1,139 17,246 68 7,629 85 0 May... 69,759.22 1,228 18,593 75 8,205 89 1 June... 76,402.53 1,397 21,571 84 9,304 100 2 Totals... $1,221,536.88 21,164 324,140 1,269 141,972 1,545 13 Required:
a. Using multiple regression, find the cost driver rates for each of the cost drivers. (Note: You must use a computer program such as Microsoft1
Excel to perform this step.) b. Assuming the following level of cost driver volume for the next month, what is the
estimated cost? (Don’t forget to include the intercept of the regression in your estimate.) 1,650... hL of beer produced
25,500 ... kg of raw materials consumed 100 ... Batches
10,800 ... hL of water consumed 120 ... CIPs
1 ... New product
c. Brazil Brewery is considering a target for water consumption of 5.0 hL water per hL of beer produced. How much would the company have saved in total over the previous 18 months if it had reached this target in the previous 18 months?
34. Account analysis. Refer to problem 33.
a. Indicate the information in addition to that provided in problem 33 required to perform account analysis.
b. Now assume that Brazil Brewery had the following breakdown of costs:
Total costs of beer produced... $ 292,429.28 Total costs of raw materials consumption... 236,168.40 Total batch-level costs... 69,173.19 Total costs of water consumption ... 141,935.36 Total costs of CIPs performed... 29,392.65 Total costs of developing new products... 6,204.64 Total facilities-level costs (total for all 18 months)... 446,233.36 Total costs... $1,221,536.88
c. What are the estimated costs for a month assuming the following level of cost driver volumes? (Don’t forget to include the facilities-level costs in your estimate.)
1,650... hL of beer produced
25,500 ... kg of raw materials consumed 100 ... Batches
10,800 ... hL of water consumed 120 ... CIPs
1 ... New product
35. Engineering method. Refer to problems 33 and 34. Brazil Brewery hired an engineering consulting firm to perform an engineering estimate of beer production costs. The consulting firm came up with the following monthly cost estimates based on information for the current period:
Facilities costs... $26,008.00
Beer production-level costs ... $13.5 per hL produced Raw materials costs... $0.70 per kg consumed Batch-level costs ... $60 per batch Water costs... $0.90 per hL consumed CIP costs ... $18.5 per CIP performed
New product costs... $500 per new product
Required:
Assuming the following level of cost driver volume for a month, what is the estimated cost using the engineering estimates? (Don’t forget to include the facilities costs in your estimate.)
1,650... hL of beer produced 25,500 ... kg of raw materials consumed
100 ... Batches
10,800 ... hL of water consumed
120 ... CIPs
1 ... New product
36. Interpreting regression results (adapted from an example by G. Benston, The Accounting Review 41, 657–672). The Philly Company manufactures widgets and digits. Philly assem- bles the widgets in batches but makes digits one at a time. Philly believes that the cost of producing widgets is independent of the number of digits produced in a week. The firm gathered cost data for 156 weeks. The following notation is used:
C ¼ Total manufacturing costs per week N ¼ Number of widgets produced during a week
B ¼ Average number of widgets in a batch during the week