The chair of the operations management department at Quality University wants to construct a p-chart for determining whether the four faculty teaching the basic P/OM course are in control with regard to the number of students who fail the course. Accordingly, he sampled 100 final grades from last year for each instructor, with the following results: $$\begin{array} { l c }
\text { Instructor } & \text { Number of Failures } \
\hline \text { Prof. A } & 13 \
\text { Prof. B } & 0 \
\text { Prof. C } & 11 \
\text { Prof. D } & 16
\end{array}$$
Using .95 control limits (5 percent risk of Type I error), which instructor(s), if any, should he conclude is (are) out of control?
A) none
B) Prof. B
C) Prof. D
D) both Prof. B and Prof. D
E) all

Question

The chair of the operations management department at Quality University wants to construct a p-chart for determining whether the four faculty teaching the basic P/OM course are in control with regard to the number of students who fail the course. Accordingly, he sampled 100 final grades from last year for each instructor, with the following results: $$\begin{array} { l c } 
\text { Instructor } & \text { Number of Failures } \
\hline \text { Prof. A } & 13 \
\text { Prof. B } & 0 \
\text { Prof. C } & 11 \
\text { Prof. D } & 16
\end{array}$$
Using .95 control limits (5 percent risk of Type I error), which instructor(s), if any, should he conclude is (are) out of control?
A) none
B) Prof. B
C) Prof. D
D) both Prof. B and Prof. D
E) all

Quizplus · Accepted Answer

To construct a p-chart, we need to first calculate the proportion of failures for each instructor:

$$\begin{array} { l c c c } 
\text { Instructor } & \text { Number of Failures } & \text{Sample Size} & \text{Proportion of Failures} \
\hline \text { Prof. A } & 13 & 100 & 0.13 \
\text { Prof. B } & 0 & 100 & 0 \
\text { Prof. C } & 11 & 100 & 0.11 \
\text { Prof. D } & 16 & 100 & 0.16
\end{array}$$

Next, we need to calculate the overall proportion of failures as the sum of all failures divided by the sum of all sample sizes:

$$p = \frac{13 + 0 + 11 + 16}{4\times 100} = 0.10$$

The control limits for the p-chart can then be calculated as:

$$ \begin{aligned} UCL &= p + 3\sqrt{\frac{p(1-p)}{n}} = 0.10 + 3\sqrt{\frac{0.10(0.90)}{100}} = 0.22 \
LCL &= p - 3\sqrt{\frac{p(1-p)}{n}} = 0.10 - 3\sqrt{\frac{0.10(0.90)}{100}} = 0.00 \end{aligned}$$

Any proportion of failures outside these control limits would be considered out of control.

For Prof. A, C, and D, their proportions of failures are within the control limits, so they are not out of control.

However, for Prof. B, the proportion of failures is 0, which is below the lower control limit of 0.00. This suggests that something unusual is happening in the course taught by Prof. B, either in terms of the quality of instruction or the difficulty of the grading standard. As a result, we should conclude that only Prof. B is out of control.

Therefore, the correct answer is D) both Prof. B and Prof. D.

The Chair of the Operations Management Department at Quality University