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SOLUTION
(30 min.) Multiple regression (continuation of 10-42).
1. Solution Exhibit 10-43 presents the regression output for medical supplies costs using both
number of procedures and number of patient-hours as independent variables (cost drivers).
SOLUTION EXHIBIT 10-43
Regression Output for Multiple Regression for Medical Supplies Costs Using Both Number of
Procedures and Number of Patient-Hours as Independent Variables (Cost Drivers)
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.9180632
7
0.8428401
ANOVA
df SS MS F
Significance
F
Coefficient
s
Standard
Error t Stat P-value Lower 95%
Upper
95%
2.
Economic
plausibility
A positive relationship between medical supplies costs and each of the
independent variables (number of procedures and number of patient-hours)
is economically plausible.
10-1
3. Multicollinearity is an issue that can arise with multiple regression but not simple regression
analysis. Multicollinearity means that the independent variables are highly correlated.
4. The simple regression model using the number of patient-hours as the independent variable
achieves a comparable r2 to the multiple regression model. However, the multiple regression
10-44 Cost estimation. Hankuk Electronics started production on a sophisticated new
smartphone running the Android operating system in January 2017. Given the razor-thin margins in
the consumer electronics industry, Hankuk’s success depends heavily on being able to produce the
phone as economically as possible.
At the end of the first year of production, Hankuk’s controller, Inbee Kim, gathered data on
its monthly levels of output, as well as monthly consumption of direct labor-hours (DLH). Inbee
views labor-hours as the key driver of Hankuk’s direct and overhead costs. The information
collected by Inbee is provided below:
10-2
Required:
1. Inbee is keen to examine the relationship between direct labor consumption and output
levels. She decides to estimate this relationship using a simple linear regression based on the
monthly data. Verify that the following is the result obtained by Inbee:
Regression 1: Direct labor-hours = a + (b Output units)
Variable Coefficient Standard Error t-Value
Constant 345.24 589.07 0.59
Independent variable: Output units 0.71 0.93 0.76
r2 = 0.054; Durbin-Watson statistic = 0.50
2. Plot the data and regression line for the above estimation. Evaluate the regression using the
criteria of economic plausibility, goodness of fit, and slope of the regression line.
3. Inbee estimates that Hankuk has a variable cost of $17.50 per direct labor-hour. She expects
that Hankuk will produce 650 units in the next month, January 2018. What should she budget
as the expected variable cost? How confident is she of her estimate?
SOLUTION
(30 min.) Cost estimation.
1. Here is the summary output for the monthly regression of Direct Labor Hours on Output
Units for Hankuk Electronics:
SUMMARY OUTPUT
Regression Statistics
10-3
ANOVA
df SS MS F
Significanc
e F
Coefficient
s
Standar
d Error t Stat P-value Lower 95%
Upper
95%
2. The plot and regression line for monthly direct labor hours on monthly output for Hankuk
Electronics are given below:
450 500 550 600 650 700 750
400
600
800
1,000
1,200
1,400
1,600
f(x) = 0.71x + 345.24
R² = 0.05
Hankuk Electronics
Output (Units)
Direct Labor Hours
Economic
plausibility
A positive relationship between direct labor hours and monthly output is
economically plausible since increased levels of production should lead to
the consumption of greater amounts of direct labor.
Significance of
Independent
The t-value of 0.76 for output units is not significant at the 0.05 level.
10-4
Variables
3. Given Inbee’s expectation that Hankuk will produce 650 units in January 2018, her best
estimate given the linear regression above is that Hankuk will use:
At an estimated variable cost of $17.50 per direct labor-hour, this implies that Inbee should
budget
for direct labor costs for January 2018.
Note that 650 units is in the range of output values that were used to find the regression equation,
10-45 Cost estimation, learning curves (continuation of 10-44). Inbee is
concerned that she still does not understand the relationship between output and labor
consumption. She consults with Jim Park, the head of engineering, and shares the results of her
regression estimation. Jim indicates that the production of new smartphone models exhibits
significant learning effects—as Hankuk gains experience with production, it can produce
additional units using less time. He suggests that it is more appropriate to specify the following
relationship:
y = axb
where x is cumulative production in units, y is the cumulative average direct labor-hours per unit
(i.e., cumulative DLH divided by cumulative production), and a and b are parameters of the
learning effect.
To estimate this, Inbee and Jim use the original data to calculate the cumulative output and
cumulative average labor-hours per unit for each month. They then take natural logarithms of
these variables in order to be able to estimate a regression equation. Here is the transformed data:
10-5
Required:
1. Estimate the relationship between the cumulative average direct labor-hours per unit and
cumulative output (both in logarithms). Verify that the following is the result obtained by
Inbee and Jim:
Regression 1: Ln (Cumulative avg DLH per unit) = a + [b Ln (Cumulative Output)]
Variable Coefficient Standard Error t-Value
Constant 2.087 0.024 85.44
Independent variable: Ln (Cum Output) –0.208 0.003 –69.046
r2 = 0.998; Durbin-Watson statistic = 2.66
2. Plot the data and regression line for the above estimation. Evaluate the regression using the
criteria of economic plausibility, goodness of fit, and slope of the regression line.
3. Verify that the estimated slope coefficient corresponds to an 86.6% cumulative average-time
learning curve.
4. Based on this new estimation, how will Inbee revise her budget for Hankuk’s variable cost
for the expected output of 650 units in January 2018? How confident is she of this new cost
estimate?
SOLUTION
(30 min.) Cost estimation, learning curves (continuation of 10-44).
1. Here is the summary output for the monthly regression of the natural log of Cumulative
Average Direct Labor-Hours per Unit on the natural logarithm of Cumulative Output:
10-6
SUMMARY OUTPUT
Regression Statistics
ANOVA
df SS MS F
Significanc
e F
Coefficient
s
Standar
d Error t Stat P-value Lower 95%
Upper
95%
2. The plot of the data and the regression line estimated above are provided next.
6.000 7.000 8.000 9.000
0.200
0.300
0.400
0.500
0.600
0.700
0.800
f(x) = - 0.21x + 2.09
R² = 1
Hankuk Electronics
Log of Cumulative Output
Log of Cumulave Average DLH per unit
Economic
A negative relationship between cumulative average direct-labor hours per
10-7
3. The original learning curve specification, y = axb is mathematically identical to the following
log-linear specification:
4. With an additional 650 units in January 2018, Hankuk’s cumulative output will go from 7,527
at the end of December 2016 to 8,177 (7,527 + 650). As Ln (8,177) = 9.0091, the cumulative
average direct-labor hours in logarithmic terms are given by:
The cumulative direct-labor hours per unit therefore equals Exp (0.2146) = 1.2394. This implies
654 × $17.50 = $11,445
for direct labor costs for January 2018.
While 9.0091 is outside the range of cumulative output values (measured in logarithms) used to
find the regression equation, unless there has been a structural break in the experience curve
10-8
10-46 Interpreting regression results, matching time periods. Nandita
Summers works at Modus, a store that caters to fashion for young adults. Nandita is responsible
for the store’s online advertising and promotion budget. For the past year, she has studied search
engine optimization and has been purchasing keywords and display advertising on Google,
Facebook, and Twitter. In order to analyze the effectiveness of her efforts and to decide whether
to continue online advertising or move her advertising dollars back to traditional print media,
Nandita collects the following data:
Required:
1. Nandita performs a regression analysis, comparing each month’s online advertising expense
with that month’s revenue. Verify that she obtains the following result:
Revenue = $51,999.64 – (0.98 Online advertising expense)
Variable Coefficient Standard Error t-Value
Constant $51,999.64 7,988.68 6.51
Independent variable: Online advertising expense –0.98 1.99 –0.49
r2 = 0.02; Durbin-Watson statistic = 2.14
2. Plot the preceding data on a graph and draw the regression line. What does the cost formula
indicate about the relationship between monthly online advertising expense and monthly
revenues? Is the relationship economically plausible?
3. After further thought, Nandita realizes there may have been a flaw in her approach. In
particular, there may be a lag between the time customers click through to the Modus website
10-9
and peruse its social media content (which is when the online ad expense is incurred) and the
time they actually shop in the physical store. Nandita modifies her analysis by comparing
each month’s sales revenue to the advertising expense in the prior month. After discarding
September revenue and August advertising expense, show that the modified regression yields
the following:
Revenue = $28,361.37 + (5.38 Online advertising expense)
Variable Coefficient Standard Error t-Value
Constant $28,361.37 5,428.69 5.22
Independent variable: Previous month’s online
advertising expense r2 = 0.65; Durbin-Watson
statistic = 1.71
5.38 1.31 4.12
4. What does the revised formula indicate? Plot the revised data on a graph. Is this relationship
economically plausible?
5. Can Nandita conclude that there is a cause-and-effect relationship between online advertising
expense and sales revenue? Why or why not?
SOLUTION
(25 min.) Interpreting regression results, matching time periods
1. Here is the summary output for the monthly regression of Sales Revenue on Online
Advertising Expense:
SUMMARY OUTPUT
Regression Statistics
ANOVA
df SS MS F
Significanc
e F
33972689.7
3397269
0.24245
Coefficient
s
Standard
Error t Stat P-value
Lower
95%
Upper
95%
69799.5
10-10
2. SOLUTION EXHIBIT 10-46A presents the data plot for the initial analysis. The formula
of Sales Revenue = $52,000 – (0.98 × Online advertising expense) indicates that there is a fixed
amount of revenue each month of $52,000, which is reduced by 0.98 times that month’s online
10-11
