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 under 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 under 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 calculate the proportion of failures for each instructor:

- Prof. A: 13/100 = 0.13
- Prof. B: 0/100 = 0
- Prof. C: 11/100 = 0.11
- Prof. D: 16/100 = 0.16

Next, we calculate the overall proportion of failures:

- Total failures = 13 + 0 + 11 + 16 = 40
- Total students = 4 x 100 = 400
- Overall proportion of failures = 40/400 = 0.1

Using a 5% risk of Type I error and assuming a binomial distribution, we can calculate the control limits:

- Upper control limit = p̂ + 1.96√((p̂(1-p̂))/n) = 0.1 + 1.96√((0.1(0.9))/100) = 0.164
- Lower control limit = p̂ - 1.96√((p̂(1-p̂))/n) = 0.1 - 1.96√((0.1(0.9))/100) = 0.036

If the proportion of failures for any instructor falls outside of these limits, we would conclude that they are out of control.

- Prof. A: 0.13 is within the control limits, so we conclude that they are in control.
- Prof. B: 0 is within the control limits, so we conclude that they are in control.
- Prof. C: 0.11 is within the control limits, so we conclude that they are in control.
- Prof. D: 0.16 is above the upper control limit, so we conclude that they are out of control.

Therefore, the answer is D: both Prof. B and Prof. D are out of control.

The Chair of the Operations Management Department at Quality University