Quality Quiz from Professor Cleary
Congratulations:
"B" is correct.
Click here for a more complete video explanation
Polly responded boldly, but not so wisely. While she had the right idea of coding the day’s production by type of excavator used, her assigning 5 and 10 to the two machines was not only arbitrary, but was not based on any reasonable assumptions.
The proper approach to this analysis is to make “type of excavator” a dummy variable (no relation to a dummy decision like Polly’s). This is sometimes referred to as an “indicator variable.” Its purpose is to indicate whether the new or old system is utilized for each data point. The coding will have a value of 1 when the new excavator is used, and 0 for the old excavator. The data is as follows:
Cost |
Mined |
Excavator |
3 |
1 |
0 |
4 |
3 |
0 |
6 |
4 |
0 |
7 |
5 |
0 |
6 |
6 |
0 |
2 |
1 |
1 |
2 |
3 |
1 |
3 |
4 |
1 |
4 |
5 |
1 |
5 |
6 |
1 |
Regression statistics:
Multiple R |
0.930494143 |
R Square |
0.865819351 |
Observations |
10 |
ANOVA
|
df |
SS |
MS |
Regression |
2 |
23.29054054 |
11.64537027 |
Residual |
7 |
3.609459459 |
0.515637066 |
Total |
9 |
26.9 |
|
|
Coefficients |
Standard Error |
t-stat |
P-value |
Intercept |
2.863513514 |
0.595545703 |
4.658579039 |
0.001947729 |
Mined |
0.614864865 |
0.131985495 |
4.658579039 |
0.002318374 |
Digger |
-2.2 |
0.454152867 |
-4.84418389 |
0.001868755 |
Where:
  cost of mining coal
 amount of coal mined
 type of excavator (0 = old excavator; 1 = new)
Three questions can be answered from this data:
- Are tons of coal mined and type of excavator used valid predictors of the cost of mining?
- Is “tons of coal mined” alone a valid predictor of the cost of mining?
- Is the type of excavator used alone a valid predictor of the cost of mining?
Examining a scatter diagram of the data will reveal a clear lower-left-upper-right relationship between the cost of coal mined and amount of coal mined. Furthermore, the scatter diagram shows that the relationship between cost of coal mined and amount of coal mined is different for each of the two excavating machines.
To answer the three questions:
With respect to question #3,(Is the type of excavator used a valid predictor of the cost of mining?), hypothetical testing can answer this question.
Step #1:
Interpretation: The null hypothesis is that  (type of excavator) has no effect on the cost of mining. The alternative is the opposite of that hypothesis.
Step #2:
Interpretation: We are willing to accept a 5 percent chance of committing a Type I error (rejecting the null when it is actually true).
Step #3
Calculate the t value. From the above printout, one can see that the calculated t value is -4.84. Compare this calculated t value of 2.36462 (highlighted below) to the tabular t value in Step #4 to determine whether to accept or reject the null hypothesis.
Step #4:
First determine the appropriate tabular t value. In this case, we have 7 degrees of freedom, since there are 10 samples (n) and 2 independent variables (m). Degrees of freedom equal n-(m+1) or 10 – 2+1), or 7. The tabular t value is therefore 2.36462. From Step #3, we have a calculated t of -4.894 (note sign is not relevant), which is larger than the tabular t, so we are in rejection region.
The conclusion of this analysis is that type of excavator is indeed a predictor of cost. We can go further by examining the value of  in the regression equation:
 
The value of  is -2.2. This means that the new excavator saves 2.2 in cost over the old excavator.
Polly should be advising Hyde N. Sikh to purchase the new excavating equipment, if the investment costs are reasonable.
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