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) \(C_i^+=max[0,(μ_0-K)-x_i+C_{i-1}^+]\)

View Answer

Explanation: The tabular cusum works by accumulating derivations from mean that are above target with one statistic C

_{i}

^{+}, 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) \(C_i^-=max[0,(μ_0-K)-x_i+C_{i-1}^-]\)

View Answer

Explanation: The tabular cusum works also by accumulating derivations from the mean that are below the target with one statistic C

_{i}

^{–}. 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

View Answer

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

View Answer

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

View Answer

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

View Answer

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 μ_{1}(out-of-control mean) in the terms of the actual target mean μ_{0}, and the shift?

a) μ_{1}=μ_{0}+δσ

b) μ_{1}=μ_{0}+2δσ

c) μ_{1}=μ_{0}+\(\frac{δ}{2}\)σ

d) μ_{1}=μ_{0}-δσ

View Answer

Explanation: The value μ

_{1}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,

μ

_{1}=μ

_{0}+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) \(K=3\frac{|μ_1-μ_0|}{4}\)

View Answer

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

View Answer

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 C

_{0}

^{+}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

View Answer

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 C

_{0}

^{–}=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 C_{i}^{–}?

a) Negative

b) Positive

c) Zero

d) Can be both, ±1

View Answer

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. C

_{i}

^{–}=0

12. Which of these is always correct?

a) \(C_i^-≥0 \)

b) \(C_i^-≤0\)

c) \(C_i^->1.323\)

d) \(C_i^-<-1.323\)

View Answer

Explanation: As the value of C

_{i}

^{–}is chosen as the maximum between zero and quantity \((μ_0-K)-x_i+C_{i-1}^-\), we always have C

_{i}

^{–}≥0.

13. After the value of C_{i}^{–} 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

View Answer

Explanation: There is a certain value decided for both C

_{i}

^{–}and C

_{i}

^{+}, after increasing from which, the process is said to be out-of-control. It is called decision interval.

14. If the value of μ_{0} > μ_{1}, K will have a negative value.

a) True

b) False

View Answer

Explanation: We know that,

\(K=\frac{|μ_1-μ_0|}{2}\)

So even if μ

_{0}> μ

_{1}, 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

View Answer

Explanation: The cusum performance not only depends on the allowance or reference value K, but it also depends on the decision interval H. It is because H is related to the value of C

_{i}

^{+}or C

_{i}

^{–}above which the process becomes out-of-control.

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