Example 11.2 Explaining Overhead Costs at Bendrix Scatterplots: Graphing Relationships

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Objective To use scatterplots to examine the relationships between overhead, machine hours, and productions runs at Bendrix.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Background Information n The Bendrix Company manufactures various types of parts for automobiles. n The manager of the factory wants to get a better understanding of overhead costs. n 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.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Background Information -- continued n 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. n It is not easy to draw a clear line between the fixed and variable overhead components. n The Bendrix manager has tracked total overhead costs for 36 months.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Background Information -- continued n 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: –MachHrs: number of machine hours used during the month –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. n The manager believes both of these variables might be responsible for variations in overhead costs. Do scatterplots support his belief?

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b BENDRIX1.XLS n The data collected by the manager appears in this file. n Each observation (row) corresponds to a single month. n We want to investigate any possible relationship between the Overhead variable and the MachHrs and ProdRuns variables but because these are time series variables we should also look out for relationships between these variables and the Month variable.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b The Scatterplots n This data set illustrates, even with the modest number of variables, how the number of potentially useful scatterplots can grow quickly. n At the least, we need to look at the scatterplots between each potential explanatory variable (MacHrs and ProdRuns) and the response variable (Overhead). n These scatterplots are as follows:

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Scatterplot of Overhead versus Machine Hours

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Scatterplot of Overhead versus Production Runs

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b The Scatterplots -- continued n To check for possible time series patterns we can also create a time series plot for any of the variables. This is equivalent to a scatterplot of the variable versus the Month, with the points joined by lines. n One of these is the time series plot for Overhead. The plot is shown next and it shows a fairly random pattern through time, with no apparent upward trend or other obvious time series pattern. n We can check that the MachHrs and ProdRuns also indicate no obvious pattern.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Time Series Plot of Overhead versus Month

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Scatterplot of Machine Hours versus Production Runs

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b The Scatterplots -- continued n Finally, when multiple explanatory variables exist we can check for relationships between them. The scatterplot of MachHrs versus ProdRuns is a cloud of points that indicate no relationship worth pursuing.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b In Summary n The Bendrix manager should continue to explore the positive relationship between Overhead and each of the MachHrs and ProdRuns variables. n However, none of the variables appear to have any time series behavior, and the two potential explanatory variables do not appear to be related to each other.

Example 11.2a Explaining Overhead Costs at Bendrix Simple Linear Regression

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Objective To use scatterplots to examine the relationships between overhead, machine hours, and production runs at Bendrix.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Background Information n In Example 11.2 we created scatterplots for Bendrix. n We found that there was a positive relationship between Overhead and each of the MachHrs and ProdRuns variables. n However, none of the variables appear to have any time series behavior, and the two potential explanatory variables of MachHrs and ProdRuns do not appear to be related to each other.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b BENDRIX1.XLS n The data collected by the manager appears in this file. n The Bendrix manufacturing data set has two explanatory variables, MachHrs and ProdRuns. n Eventually we will estimate a regression equation with both of the variables included. n However, if we include only one at a time, what do they tell us about the overhead costs?

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Regression Output for Overhead versus MachHrs

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Regression Output for Overhead versus ProdRuns

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Least Squares Line Equations n The two least squares lines are therefore Predicted Overhead = 48, MacHrs and Predicated Overhead = 75, ProdRuns n Clearly these two equations are quite different, although each effectively breaks Overhead into a fixed component and a variable component. n The equations imply that expected overhead increases by about $35 for each extra machine hour and about $655 for each extra production run.

| 11.1a | 11.2a | 11.2b | 11.3 | 11.3a | 11.4 | 11.3b | 11.5 | a11.2a11.2b a b Least Squares Line Equations -- continued n The differences between these two lines can be attributed to neither one telling the whole story. n If the manager’s goal is to split overhead into a fixed and variable component, then the variable component should include both of the measures of work activity to give a more complete explanation of overhead. n We will see how this can be done using multiple regression.