# Statistical Quality Control Questions and Answers – Time-Weighted – Cumulative Sum Control Chart – 5

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This set of Statistical Quality Control Questions & Answers for Exams focuses on “Time-Weighted – Cumulative Sum Control Chart – 5”.

1. What are the graphical displays of tabular cusum called?
a) Cusum regular charts
b) Cusum regulated display
c) Cusum level display
d) Cusum status charts

Explanation: It is useful to present a graphical display for the tabular cusum. These charts are sometimes called the cusum status charts.

2. Which of these is a correct statement for cusum status charts?
a) A plot between Ci+ or Ci and the sample number
b) A plot between the Ci+ or Ci and the sample mean
c) A plot between the Ci+ or Ci and the sample variance
d) A plot between the Ci+ or Ci and the sample standard deviation

Explanation: Cusum status charts are constructed by plotting Ci+ or Ci versus the sample number. These are used to get an quick information about the process from the displayed chart.

3. Each vertical bar in cusum status chart represents __________
a) The value of Ci+ and Ci
b) The value of Ci
c) The value of Ci+
d) Neither the value of Ci+ nor Ci

Explanation: Cusum status charts are the charts made to represent the cusum data on a graphical display, where each vertical bar in the cusum status charts, represents the value of Ci+ and Ci.

4. What is analogous to the control limits as in Shewhart control charts, in the cusum status charts?
a) The reference value
b) The allowance value
c) The value of the decision intervals
d) The value of Ci+ and Ci

Explanation: The value of the decision interval H can be used as the control limits in the cusum charts as in the Shewhart charts. This gives us an indication about when the process is out-of-control.

5. In the Minitab version of the cusum control charts, which of these is used?
a) $$C_i^+=max⁡\left\{0,x_i-μ_0+k+C_{i-1}^-\right\}$$
b) $$C_i^-=min⁡\left\{0,x_i-μ_0+k+C_{i-1}^-\right\}$$
c) $$C_i^-=max⁡\left\{0,x_i-μ_0-k+C_{i-1}^-\right\}$$
d) $$C_i^-=min⁡\left\{0,x_i-μ_0-k-C_{i-1}^-\right\}$$

Explanation: The Minitab version of the tabular cusum charts uses the negative of Ci i.e. it uses the following expression for Ci,
$$C_i^-=min⁡\left\{0,x_i-μ_0+k+C_{i-1}^-\right\}$$.

6. Which of these is true for the Minitab version of tabular cusum charts?
a) $$C_i^-≥0$$
b) $$C_i^-≥9$$
c) $$C_i^-≤0$$
d) $$C_i^-≤5$$

Explanation: The Minitab version of the tabular cusum uses the negative expression for the Ci, so the value of Ci always remains less than or equal to zero.

7. Which of these steps, is not carried out when the process becomes out-of-control when using the cusum control charts?
a) Search for an assignable cause
b) Taking corrective action
c) Restarting the control chart from zero
d) Continuing the control chart

Explanation: When an out-of-control situation is encountered while using the cusum control charts, we use the same procedure, used as in the Shewhart control charts. First, identifying the assignable cause, then second, taking corrective action, and then restarting the control chart from zero.

8. Which of these control charts may accurately tell when the assignable cause has occurred?
a) Cusum control charts
b) p-charts
c) c-charts
d) x bar and s charts

Explanation: As the pattern changes in the cusums due to an assignable cause, the changed pattern gets recorded in all the next cusums. So, we can identify from which cusum, the pattern changed, and hence the assignable cause location can be known.

9. Which of these can be used to calculate the estimate of mean of the process in the case of the tabular cusums when Ci+ > H?
a) $$\hat{μ}=\left\{μ_0-K+\frac{C_i^+}{N^+}\right\}$$
b) $$\hat{μ}=\left\{μ_0+K+\frac{C_i^+}{N^+}\right\}$$
c) $$\hat{μ}=\left\{μ_0+K-\frac{C_i^+}{N^+}\right\}$$
d) $$\hat{μ}=\left\{μ_0+K+\frac{C_i^-}{N^-}\right\}$$

Explanation: In cases when a manipulative variable is required to take an out-of-control process back to target valueμ0, we use the new process mean estimate, which is written as,
$$\hat{μ}=\left\{μ_0+K+\frac{C_i^+}{N^+}\right\}$$

10. Which of these following expressions can be used to calculate the estimate of mean of the process in the case of the tabular cusums when Ci > H?
a) $$\hat{μ}=\left\{μ_0+K+\frac{C_i^-}{N^-}\right\}$$
b) $$\hat{μ}=\left\{μ_0-K-\frac{C_i^-}{N^-}\right\}$$
c) $$\hat{μ}=\left\{μ_0-K+\frac{C_i^-}{N^-}\right\}$$
d) $$\hat{μ}=\left\{μ_0+K-\frac{C_i^-}{N^-}\right\}$$

Explanation: The cases when there is a need to estimate the changed process mean, the expression of new process mean estimate is written as,
$$\hat{μ}=\left\{μ_0-K-\frac{C_i^-}{N^-}\right\}$$

11. What of these can be used as the decision interval for the tabular cusum charts?
a) 2σ
b) 1σ
c) 5σ
d) 4σ

Explanation: It is a generalized rule that, the decision intervals should always be five times of the standard deviations of the process.

12. What is the expression for the two sided cusum ARL?
a) $$\frac{1}{ARL}=\frac{1}{2ARL^-}+\frac{1}{2ARL^+}$$
b) $$\frac{1}{ARL}=\frac{1}{ARL^-}±\frac{1}{ARL^+}$$
c) $$\frac{1}{ARL}=\frac{1}{2ARL^-}-\frac{1}{2ARL^+}$$
d) $$\frac{1}{ARL}=\frac{1}{ARL^-}+\frac{1}{ARL^+}$$

Explanation: The expression for the ARL of the two sided cusum from the ARLs of the two one sided statistics, is written as,
$$\frac{1}{ARL}=\frac{1}{ARL^-}+\frac{1}{ARL^+}$$
It is used to estimate the average time till an out-of-control signal comes from the cusum chart.

13. Which of these is an expression for Siegmund’s approximation?
a) $$ARL=\frac{exp⁡(-2∆b)+2∆b-1}{2∆^2}$$
b) $$ARL=\frac{exp⁡(-2∆b)+2∆b-1}{2∆}$$
c) $$ARL=\frac{exp⁡(-2∆b)-2∆b+1}{2∆^2}$$
d) $$ARL=\frac{exp⁡(-2∆b)-2∆b-1}{2∆^2}$$

Explanation: The Siegmund’s approximation is used to evaluate the average run length for the cusum charts. It is expressed as,
$$ARL=\frac{exp⁡(-2∆b)+2∆b-1}{2∆^2}$$

14. The value of K and H should be determined according to the ARL required for the corresponding cusum chart.
a) True
b) False

Explanation: ARL value tells about the time until the next out-of-control signal. So it is also an estimate to predict the type I error possibility. Hence, K and H values should be determined according to the ARL required.

15. The desired ARL is obtained by using the Siegmund’s approximation.
a) True
b) False 