# Statistical Quality Control Questions and Answers – Attribute Charts – Control Charts for Fraction Nonconforming – 3

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This set of Statistical Quality Control Multiple Choice Questions & Answers (MCQs) focuses on “Attribute Charts – Control Charts for Fraction Nonconforming – 3”.

1. For LCL for the p-chart to be higher than zero, what condition should the sample size fulfill?
a) $$n>\left[\frac{(1-p) L^2}{p}\right]$$
b) $$n>\left[\frac{(1-p) L^2}{2p}\right]$$
c) $$n<\left[\frac{(1-p) L^2}{2p}\right]$$
d) $$n<\left[\frac{(1-p) L^2}{p}\right]$$

Explanation: For the LCL of the p-chart to be higher than zero, the sample size n should fulfill the following condition.
$$n>\left[\frac{(1-p) L^2}{p}\right]$$

2. For what value of sample size, will the p-chart will have a LCL value higher than 0 when p=0.05 and 3 sigma limits are used?
a) n>121
b) n>100
c) n>125
d) n>171

Explanation: We know for a p chart having LCL value higher than zero, we have,
$$n>\frac{1-p}{p}*L^2$$
For p=0.05 and L=3, we get, n>171.

3. For narrower limits of control, which of this is true?
a) Chart becomes insensitive to small shift
b) Chart becomes more sensitive to small shift but gives false alarms
c) Chart becomes more sensitive to small shift but gives accurate alarms
d) Chart gives no alarms at all

Explanation: If the narrower limits like 1-sigma limits or 2 – sigma limits are chosen for p-chart, the chart becomes more sensitive to small shifts. But in addition, it also gives more numbers of false alarms.

4. If the probability of a unit being conforming or nonconforming, depends on the previous unit being conforming or nonconforming, can p-chart be applied on the process?
a) Yes
b) No
c) This depends on the sample size
d) This depends on the operator changes

Explanation: In processes where nonconforming units are clustered, i.e. the probability of one unit to be conforming or nonconforming depends on the previous unit being defective or not, p-chart can’t be applied to them, as p-charts are designed on the basis of binomial probability or independent trials.

5. The number nonconforming chart is also called ___________
a) np-chart
b) p-chart
c) c-chart
d) s-chart

Explanation: The number nonconforming chart is also called the np-chart. This uses the number of nonconforming items in a sample instead of the fraction nonconforming.

6. The center line of the np-control chart represents the value equal to ____________
a) p
b) p
c) np
d) 1-np

Explanation: The center line of the np chart or the number nonconforming chart represents the value equal to np or np.

7. If the p=0.2313 for a np chart, and the number of items in a sample are 50, what will be the center line value of the np chart?
a) 10.34
b) 11.56
c) 10.11
d) 13.21

Explanation: The center line of the np chart represents the value equal to,
CL=np
Here, n=50 and p=0.2313, putting these values in the CL value, we get, CL=11.565.

8. For a control chart data having the average sample fraction nonconforming= 0.2313 and the sample size=50, what will be the value of the UCL for the np control chart?
a) 2.62
b) 20.510
c) 11.56
d) 11.892

Explanation: We know that for a np chart,
UCL=$$n\bar{p}+3\sqrt{n\bar{p} (1-\bar{p})}$$
Putting the values, we get, UCL=20.51.

9. What is the expression to calculate the variable-width control limits for the p-chart?
a) $$\bar{p} \pm \frac{1}{2} \sqrt{\bar{p}(1-\bar{p})/n_i}$$
b) $$\bar{p} \pm \sqrt{\bar{p}(1-\bar{p})/n_i}$$
c) $$\bar{p} \pm 3\sqrt{\bar{p}(1-\bar{p})/n_i}$$
d) $$\bar{p} \pm \sqrt{\bar{p}(1-\bar{p})/n_i}$$

Explanation: The variable-width control limits of the p-chart, when the sample size is varying, are,
$$\bar{p} \pm 3\sqrt{\bar{p}(1-\bar{p})/n_i}$$

10. For a process, there are 25 samples taken from its output of variable sample size. There are total 234 defects in all the samples combined. If the total of all the sample sizes is 2450, what will be the value of the UCL in the case of variable-width control limits of p-chart?
a) $$0.096+3\sqrt{0.096*\frac{0.904}{n_i}}$$
b) $$0.087+3\sqrt{0.087*\frac{0.913}{n_i}}$$
c) $$0.096-3\sqrt{0.096*\frac{0.904}{n_i}}$$
d) $$0.087-3\sqrt{0.087*\frac{0.913}{n_i}}$$

Explanation: The UCL of variable-width control limits of p-chart is written as,
$$\bar{p} + 3\sqrt{\bar{p}(1-\bar{p})/n_i}$$
Here $$\bar{p} = \frac{234}{2450}$$=0.096. Putting this value in UCL formula, we get,
$$0.096+3\sqrt{0.096*\frac{0.904}{n_i}}$$

11. Which one of these is not a method to plot the variable sample size data on p-chart?
a) Variable-width control limits
b) Control limits based on average sample size
c) Tolerance diagram
d) Standardized Control Chart

Explanation: The three approaches to deal with variable sample size are: variable-width control limits, control limits based on average sample size, and standardized control chart.

12. In the case of average sample size based control limits, which of these states the correct expression for average sample size?
a) $$\bar{n}=\frac{∑_{i=0}^m n_i}{m}$$
b) $$\bar{n}=\frac{∑_{i=1}^m n_i}{m}$$
c) $$\bar{n}=\frac{∑_{i=1}^m n_i}{m-1}$$
d) $$\bar{n}=\frac{∑_{i=1}^m n_i}{1-m}$$

Explanation: The average sample size in the case of control limits based on average sample size, is expressed by,
$$\bar{n}=\frac{∑_{i=1}^m n_i}{m}$$

13. The approximate upper control limit calculated for control chart limits based on an average sample size, is expressed by _____
a) UCL=$$\bar{p} + 0.5\sqrt{\bar{p}(1-\bar{p})/\bar{n}}$$
b) UCL=$$\bar{p} + \sqrt{\bar{p}(1-\bar{p})/\bar{n}}$$
c) UCL=$$\bar{p} – 3\sqrt{\bar{p}(1-\bar{p})/\bar{n}}$$
d) UCL=$$\bar{p} + 3\sqrt{\bar{p}(1-\bar{p})/\bar{n}}$$

Explanation: The approximate upper control limit for control limits of p-chart based on an average sample size, is expressed by,
UCL=$$\bar{p} + 3\sqrt{\bar{p}(1-\bar{p})/\bar{n}}$$

14. Even if 100% inspection is done, control chart can have a variable sample size.
a) True
b) False

Explanation: In some applications of the p-chart, the sample is 100% inspection of process output over a time period. Since different number of units could be produced in each period, the control chart would then have a variable sample size.

15. The points plotting below the lower control limit of p-chart do not always represent the out-of-control process or an assignable cause.
a) True
b) False 