# Statistical Quality Control Questions and Answers – Time-Weighted – Cumulative Sum Control Chart – 3

This set of Advanced Statistical Quality Control Questions and Answers focuses on “Time-Weighted – Cumulative Sum Control Chart – 3”.

1. How many ways are there to represent the Cusum on the Cusum charts?
a) 3
b) 4
c) 2
d) 5

Explanation: There are two ways invented for the representation of the Cusums on the Cusum charts; namely Tabular Cusum, and V-mask form of the Cusum.

2. Which of these is a name of any way to represent the Cusum charts?
a) Logarithmic
b) Algorithmic
c) Exponential
d) Normal

Explanation: There are two ways of representing the Cusums. The first one is the tabular (or Algorithmic) Cusum, and the second one is the V mask form of Cusum.

3. The tabular Cusums are made to monitor the ____________ of the process.
a) Variance
b) Mean
c) Standard deviation
d) Capability

Explanation: The tabular Cusum is generally made to monitor the changes occurring in the mean of the process with respect to the time. This is preferred over the V-mask type.

4. Which of these is not one of the assumptions made for the construction of Tabular Cusum charts for the individual measurements?
a) The selected variable varies according to a lognormal distribution
b) The selected variable varies according to a normal distribution
c) The selected variable varies according to a binomial distribution
d) The selected variable varies according to an exponential distribution

Explanation: For the construction of the tabular cusum charts, it is assumed that the variable for which the Cusum charts are to be plotted, varies on a normal distribution.

5. Which of these is not an assumption made for the construction of the tabular Cusum charts?
a) The selected variable varies on a normal distribution
b) The mean of the normal distribution is available
c) The estimate or the exact value of the standard deviation of the variable
d) The assumptions are very inconsistent with phase II application of SPC

Explanation: Tabular Cusum charts are made with the assumptions that, the selected variable varies on normal distribution with a known mean and standard deviation. The standard deviation estimate may also be used.

6. If the process standard deviation increases, how will the cusum chart for monitoring process variability indicate the out-of-control state?
a) The value of Si+ will increase
b) The value of Si+ will decrease
c) The value of Si will decrease with decrease in the value of Si+
d) The value of Si will increase

Explanation: If the process standard deviation increases, Si+ value will be increased consequently. It will eventually exceed the decision interval h and we will get an out-of-control signal.

7. Which of these is an indication of out-of-control process with low standard deviation?
a) Increase in the value of Si+
b) Decrease in the value of Si+ with increase in the value of Si
c) Increase in the value of Si
d) Decrease in the value of Si with decrease in the value of Si

Explanation: If the process becomes out-of-control with low value of the process standard deviation, the value of Si will increase and eventually exceed the decision interval, h.

8. Who was the first person to recommend plotting the cusum charts for mean and standard deviation on same graph?
a) Hawkins
b) Roberts
c) Atkinson
d) Crowder

Explanation: Although it is customary to plot the cusum charts for mean and cusum charts for standard deviation on different graphs, Hawkins (1993a) was the first person who recommended to plot them on same graph.

9. If the deployment of the cusum is extended to the case of averages of the rational subgroups where sample size n>1, what will be done?
a) The value of xi be replaced by xi
b) The value of xi be replaced by xi
c) The value of Ci+ be replaced by value of Ci
d) The value of Ci be replaced by the value of Ci+

Explanation: When the rational subgrouping procedure is adopted with sample size n>1, the cusum charts are built by simply replacing xi by xi and σ by σ/√n.

10. A cusum for normal variance is having the value of upper cusum equal to __________
a) $$C_i^+=max⁡(0,C_{i-1}^+ + S^2 – k)$$
b) $$C_i^+=max⁡(0,C_{i-1}^+ + S^2 + k)$$
c) $$C_i^+=max⁡(0,C_{i-1}^+ – σ^2 + k)$$
d) $$C_i^+=max⁡(0,C_{i-1}^+ + σ^2 + k)$$

Explanation: The value of the upper cusum for the cusum for normal variance has the value of σ2 replaced by the sample variance Si2. It is expressed as,
$$C_i^+=max⁡(0,C_{i-1}^+ + S^2 + k)$$

11. A cusum for a normal variance has the value of lower cusum equal to ___________
a) $$C_i^+=min⁡(0,C_{i-1}^- + S^2 + k)$$
b) $$C_i^-=max⁡(0,C_{i-1}^- + S^2 + k)$$
c) $$C_i^+=max⁡(0,C_{i-1}^- + S^2 + k)$$
d) $$C_i^-=min⁡(0,C_{i-1}^- + S^2 + k)$$

Explanation: The value of the lower cusum for the normal variance is denoted by $$C_i^-$$. It is expressed by the following formula,
$$C_i^-=min⁡(0,C_{i-1}^- + S^2 + k)$$

12. If the value of Ci-1+S2+k is lesser than (-1), what will be the value of Ci equal to?
a) Lesser than -1
b) Higher than 1
c) Lower than 0
d) 0

Explanation: We know that
$$C_i^+=min⁡(0,C_{i-1}^- + S^2 + k)$$
Also the cusum values, whether it is the upper cusum, or lower cusum, are either zero or positive. So even if the value of Ci-1+S2+k is lesser than (-1), the value of Ci will be zero.

13. The V-mask procedure was proposed by ___________
a) Crowder
b) Roberts
c) Brinson
d) Barnard

Explanation: There are two procedures to plot the cusum charts. One is called the tabular cusum, and the other is called the V-mask procedure. The V-mask procedure was first introduced by Barnard (1959).

14. Only the value of lower cusum can be negative.
a) True
b) False

Explanation: We know that the value of lower cusum is denoted by,
$$C_i^-=max(0,C_{i-1}^- + S^2 + k)$$
So we get to know that the value of the lower cusum is always greater than or equal to zero.

15. Tabular Cusums can be constructed for both, individual measurements, and the averages of the rational subgroups.
a) True
b) False

Explanation: It is a big advantage of the tabular Cusum charts that they can be constructed for both cases; for individual measurements, and the averages of the rational subgroups.

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