# Statistical Quality Control Questions and Answers – Time-Weighted – Cumulative Sum Control Chart – 4

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This set of Statistical Quality Control Multiple Choice Questions & Answers (MCQs) focuses on “Time-Weighted – Cumulative Sum Control Chart – 4”.

1. Which of these is a correct expression for the one sided upper cusum?
a) $$C_i^+=min⁡[0,x_i-(μ_0+K)+C_{i-1}^+]$$
b) $$C_i^+=max⁡[0,x_i-(μ_0+K)+C_{i-1}^+]$$
c) $$C_i^+=min⁡[0,(μ_0-K)-x_i+C_{i-1}^+ ]$$
d) $$C_i^+=max⁡[0,(μ_0-K)-x_i+C_{i-1}^+]$$

Explanation: The tabular cusum works by accumulating derivations from mean that are above target with one statistic Ci+, which is called one sided upper cusum. It is expressed as,
$$C_i^+=max⁡[0,x_i-(μ_0+K)+C_{i-1}^+]$$

2. What is the value of one sided lower cusum?
a) $$C_i^-=min⁡[0,x_i-(μ_0+K)+C_{i-1}^-]$$
b) $$C_i^-=max⁡[0,x_i-(μ_0+K)+C_{i-1}^-]$$
c) $$C_i^-=min⁡[0,(μ_0-K)-x_i+C_{i-1}^-]$$
d) $$C_i^-=max⁡[0,(μ_0-K)-x_i+C_{i-1}^-]$$

Explanation: The tabular cusum works also by accumulating derivations from the mean that are below the target with one statistic Ci. The value of this value is called the one sided lower cusum and its expressed as,
$$C_i^-=max⁡[0,(μ_0-K)-x_i+C_{i-1}^-]$$

3. What is K called in the expressions of the one-sided Cusums?
a) Regarded value
b) Related value
c) Resultant value
d) Reference value

Explanation: The values of the statistics used in the construction of a tabular cusum are called one sided upper and lower Cusums. K is used in their expressions which is called Reference value.

4. Which of these is another name of the reference value?
a) Regarding value
b) Stoppage value
c) Stack value
d) Assignable value

Explanation: The constant K in the expression of the one-sided Cusums, is called the reference value. It is also called slack value, or allowance.

5. The value of the reference value is chosen ____________
a) 3/4 ways between mean and the out-of-control mean towards the mean
b) 1/2 ways between mean and the out-of-control mean
c) 3/4 ways between mean and the out-of-control mean towards the out-of-control mean
d) 1/4 ways between mean and the out-of-control mean towards the mean

Explanation: The value of the reference value or K is chosen such that, it stay halfway between the target mean and the out-of-control value of the mean, that we are interested in finding out quickly.

6. What is the value δ is called, when used in the expression of K?
a) The shift of mean in standard deviation units
b) The shift of variance in standard deviation units
c) The shift of standard deviation in mean units
d) The shift of mean in variance units

Explanation: The value δ is used in the expression of K or the reference value.
K=$$\frac{\delta\sigma}{2}$$
Here this value δ is called the shift of mean in standard deviation units.

7. What is the value of μ1(out-of-control mean) in the terms of the actual target mean μ0, and the shift?
a) μ10+δσ
b) μ10+2δσ
c) μ10+$$\frac{δ}{2}$$σ
d) μ10-δσ

Explanation: The value μ1 is called the out-of-control mean. It is the value of mean when process changes its state to out-of-control. Its value is,
μ10+δσ.

8. What is the value of K in the terms of out-of-control mean and the target mean?
a) $$K=\frac{|μ_0-μ_1|}{2}$$
b) $$K=\frac{|μ_1-μ_0|}{2}$$
c) $$K=\frac{|μ_1-μ_0|}{4}$$
d) $$K=3\frac{|μ_1-μ_0|}{4}$$

Explanation: The value K is called the allowance. This is used in the one-sided upper and lower Cusums expressions. It is expressed as,
$$K=\frac{|μ_1-μ_0|}{2}$$

9. What is the starting value of one-sided upper cusum?
a) 1
b) 6
c) 5
d) 0

Explanation: The one-sided upper cusum is expressed as follows,
$$C_i^+=max⁡[0,x_i-(μ_0+K)+C_{i-1}^+]$$
The starting value of this would be C0+ which is always taken as 0.

10. What is the starting value of the one-sided lower cusum?
a) 1.5
b) -1.5
c) 0
d) .3

Explanation: The lower one sided cusum is expressed as follows,
$$C_i^-=max⁡[0,(μ_0-K)-x_i+C_{i-1}^-]$$
This value is called the starting value of lower one-sided cusum when the value of i=0. The value of C0=0.

11. If the value of the quantity ”$$(μ_0-K)-x_i+C_{i-1}^-$$” becomes negative, what will be the value of the value Ci?
a) Negative
b) Positive
c) Zero
d) Can be both, ±1

Explanation: We know that,
$$C_i^-=max⁡[0,(μ_0-K)-x_i+C_{i-1}^-]$$
So, as the value of quantity $$(μ_0-K)-x_i+C_{i-1}^-$$ becomes negative, it becomes lesser than 0. So we take the maximum value between zero and it, i.e. Ci=0

12. Which of these is always correct?
a) $$C_i^-≥0$$
b) $$C_i^-≤0$$
c) $$C_i^->1.323$$
d) $$C_i^-<-1.323$$

Explanation: As the value of Ci is chosen as the maximum between zero and quantity $$(μ_0-K)-x_i+C_{i-1}^-$$, we always have Ci≥0.

13. After the value of Ci increasing than the value of _____ the process is said to be out-of-control.
a) Control interval
b) Decision interval
c) Distribution interval
d) Calculation interval

Explanation: There is a certain value decided for both Ci and Ci+, after increasing from which, the process is said to be out-of-control. It is called decision interval.

14. If the value of μ0 > μ1, K will have a negative value.
a) True
b) False

Explanation: We know that,
$$K=\frac{|μ_1-μ_0|}{2}$$
So even if μ0 > μ1, K will have its value greater than 0.

15. The generally used value of K is the only value, which substantially impact the performance of the cusum.
a) True
b) False 