Question

The Bendrix Company manufactures various types of parts for automobiles. The manager of the factory wants to get a better understanding of overhead costs. These overhead costs include supervision, indirect labor, supplies, payroll taxes, overtime premiums, depreciation, and a number of miscellaneous items such as insurance, utilities, and janitorial and maintenance expenses. Some of the overhead costs are “fixed” in the sense they do not vary appreciably with the volume of work being done, whereas others are “variable” and do vary directly with the volume of work being done. It is not easy to draw a clear line between the fixed and variable overhead components

The Bendrix manager has tracked total overhead costs for 36 months). To help explain these, he also collected data on two variables that are related to the amount of work done at the factory. These variables are:

14. MachHrs: number of machine hours used during the month
14. ProdRuns: the number of separate production runs during the month. To understand this variable we must know that Bendrix manufactures parts in fairly large batches called production runs. Between each run there is a downtime.

The manager believes both of these variables might be responsible for variations in overhead costs. He implemented the regression model and found the following output.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.930819542
R Square	0.866425021
Adjusted R Square	0.858329567
Standard Error	4108.99309
Observations	36
ANOVA
	df	SS	MS	F	Significance F
Regression	2	3614020661	1.81E+09	107.0261	3.75374E-15
Residual	33	557166199.1	16883824
Total	35	4171186860
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	3996.678209	6603.650932	0.605223	0.549171	-9438.56128	17431.92
MachHrs	43.53639812	3.5894837	12.12887	1.05E-13	36.23353283	50.83926
ProdRuns	883.6179252	82.25140753	10.74289	2.61E-12	716.2760457	1050.96

Please help the manager summarize all relevant information presented in the output and write an executive report.

Answer 1

The regression model to fit 'overhead costs' (y) on the independent variable 'MachHrs' (x1) and 'ProdRuns' (x2) is

$y = \beta_{0} +\beta_{1} \ast x_{1} + \beta_{2} \ast x_{2} + \epsilon$

From the given output, the sample regression equation is

3996.678209 + 43.53639S12 * .r1 + 883.6179252 * .r2 y

To test the overall significance of the model, i.e., the assumed model fits the data better than the intercept-only model we consider Significance F (p-value) of F-test.

The Null Hypothesis

Но : 31 = 32 = 0

versus

Alternative Hypothesis

$H_{1}: \beta_{j} \neq 0$ for at least one j =1,2

From the given output considering 'ANOVA' table, it is observed that the p-value (< 0.0000) < 1% as well as 5% significance level, inferring that we reject the null hypothesis $H_{0}$ and infer that the model fits the data better than the intercept-only model i.e., at least one of the coefficients of independent variables is statistically significantly different from '0'

Considering R Square which quantifies the proportion of variation in the dependent variable explained by the explanatory variables (independent variables) is 86.64% (= 0.8664 * 100) i.e., 86.64% of the total variation in the 'overhead cost' is explained by the explanatory variables 'MachHrs' and 'ProdRuns'

Consider the parameter estimates of the explanatory variables:

'MachHrs' -

1. For a unit increase in the number of machine hours used during the month keeping 'ProdRuns' constant, the average increase in the 'overhead costs' is $43.54 (rounded to nearest 2 digits)

2. The p-value (< 0.0000) is < 1% as well as 5% significance level implying that we reject the null hypothesis and infer that the parameter estimate is statistically significantly different from '0' and explain the variation in the dependent variable

3. Consider the 95% confidence intervals: With 95% confidence, we can infer that the parameter estimate of the variable 'MachHrs' lies within the interval 36.23353 and 50.8393

'ProdRuns' -

1. For a unit increase in the number of separate production runs during the month keeping 'MachHrs' constant, the average increase in the 'overhead costs' is $883.62 (rounded to nearest 2 digits)

2. The p-value (< 0.0000) is < 1% as well as 5% significance level implying that we reject the null hypothesis and infer that the parameter estimate is statistically significantly different from '0' and explain the variation in the dependent variable

3. Consider the 95% confidence intervals: With 95% confidence, we can infer that the parameter estimate of the variable 'ProdRuns' lies within the interval 716.2760 and 1050.96