# Statistical Quality Control Questions and Answers – Control Charting Techniques – Control Charts for Multiple-Stream Processes – 3

This set of Statistical Quality Control Question Paper focuses on “Control Charting Techniques – Control Charts for Multiple-Stream Processes – 3”.

1. Which of these is a suitable pair of s, r, while using the GCC?
a) 2, 6
b) 3, 7
c) 9, 2
d) 6, 3

Explanation: The ARL of the MSP depends on the value of the s and r of the process. So to choose r is in our hand. (3, 7) and (4, 6) are the most used pairs which give desired results. Others will give too many false alarms.

2. The expected number of trials until r consecutive largest or smallest means come from the same stream, is called _________ of the MSP.
a) ARL
b) ARL(1)0
c) ARL(2)0
d) Predicted ARL

Explanation: The two sided in-control ARL of any process or ARL(2)0 is defined as the expected number of trials until r consecutive largest, or r consecutive smallest means come from a particular stream of any MSP, while MSP is in control.

3. Who was the first person to find out the one sided in-control ARL for any event of consecutive largest values being obtained from a single process stream?
a) Bond
b) Nelson
c) Robertson
d) Clarkes

Explanation: The one sided in-control ARL for any GCC of any MSP was first explained by Nelson (1986). He expressed it as,
$$ARL(1)_0=\frac{s^r-1}{s-1}$$

4. Who gave the Markov chain approach to compute the ARL(2)0?
a) Brook and Evans
b) Mortell
c) Runger
d) Nelson

Explanation: The Markov Chain Approach was given by the pair of persons, Brook and Evans, in the year of 1972. This was used by Nelson and Stephenson (1996) to calculate the ARL(2)0.

5. Who was not one of the scientists who used the Markov chain approach to compute the two sided in-control ARL for any GCC of any MSP?
a) Mortell
b) Robertson
c) Runger
d) Stephenson

Explanation: Mortell and Runger (1995), and Nelson and Stephenson (1996) used the Markov chain approach of Brook and Evans (1972) to compute ARL(2)0.
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6. Which of these is a lower bound on the ARL(2)0?
a) $$ARL(2)_0=2\frac{s^r-1}{s-1}$$
b) $$ARL(2)_0=\frac{2(r-1)}{s^r-1}$$
c) $$ARL(2)_0=\frac{s^r-1}{s-1}$$
d) $$ARL(2)_0=\frac{s^r-1}{2(r-1)}$$

Explanation: The ARL(2)0 close form expression could not be evaluated. So Nelson and Stephenson give a lower bound on ARL(2)0 which is written as,
$$ARL(2)_0=\frac{s^r-1}{2(r-1)}$$

7. Which of these is one of the drawbacks of GCC?
a) Each stream does not need to be sampled at a time
b) There is no information about nonextreme streams at each trial. Thus no past value related EWMA can be constructed
c) It can’t be used for MSPs
d) It is very easy to construct

Explanation: In GCC, there is no information about nonextreme streams at each trial. So, we can’t utilize past values to form an E.W.M.A. or a cusum statistic to improve on GCC performance.

8. Dependent streams of an MSP are also called ____________
a) Accuracy streams
b) Cross Acceptance
c) Cross-correlated
d) Uncorrelated

Explanation: The Dependent streams of a multiple stream process are the streams that have their data related to each other. So these streams are also called cross-correlated streams.

9. Which of these is the correct model proposed for MSP by Mortell and Runger to accommodate the practical case of the dependent streams?
a) xtjk = μ – At + ϵtjk
b) xtjk = μ + At + ϵtjk
c) xtjk = μ – At – ϵtjk
d) xtjk = μ + At – ϵtjk

Explanation: Mortell and Runger (1995) proposed the following model for the MSP to accommodate the practical case of cross-correlated streams:
xtjk = μ + At + ϵtjk
Here cross-correlated streams have their data related to each other.

10. What does the term xtjk correspond to in the model proposed by Mortell and Runger?
a) k Th measurement on the jTh stream at time t
b) j Th measurement on the kTh stream at time t
c) t Th measurement on the jTh stream at time k
d) k Th measurement on the tTh stream at time j

Explanation: The model proposed by Mortell and Runger, to accommodate the case of cross-correlated stream data of MSP, is expressed as,
xtjk = μ + At + ϵtjk
Here the term xtjk corresponds to k Th measurement on the j Th stream at time t.

11. The model given by Mortell and Runger represents two types of variability, σa2 accounting for the variance over time ___________ and σ2 accounting for the variation between the streams at specific time t.
a) Common to all streams
b) Specific to one stream
c) Specific to 2 streams
d) Specific to a group of 3 streams

Explanation: The MSP representation given by Mortell and Runger, has the total variation allocated into two sources, σa2 accounting for the variation over time common to all streams, and σ2 accounting for the variation between the streams at specific time t.

12. What is the correct value of cross correlation given by the Mortell and Runger model?
a) $$ρ=\frac{2σ_a^2}{(σ_a^2+σ^2)}$$
b) $$ρ=\frac{4σ_a^2}{(σ_a^2+σ^2)}$$
c) $$ρ=\frac{σ_a^2}{(σ_a^2+σ^2)}$$
d) $$ρ=\frac{2σ_a^2}{(σ_a^2-σ^2)}$$

Explanation: The MSP model was given by Mortell and Runger in the year of 1995. According to it, the value of the cross-correlation given by the model was expressed as,
$$ρ=\frac{σ_a^2}{(σ_a^2+σ^2)}$$

13. Mortell and Runger proposed monitoring the average at time t of the means across all the streams with _________ control chart to detect an overall assignable cause.
a) A cusum
b) A EWMA
c) A p-chart
d) An individuals

Explanation: Mortell and Runger proposed the MSP model. They recommended using an Individuals Chart to compare the average at time t of the means across all streams, to find an assignable cause.

14. The Mortell and Runger model did not propose to monitor the range of stream’s means at time t.
a) True
b) False

Explanation: According to Mortell and Runger, it was quite necessary to monitor the range of the stream’s means at time t. This range was denoted by,
Rt = max⁡(xtj )-min⁡(xtj).

15. The proposed control charts on residuals are quite good than the GCC.
a) True
b) False

Explanation: The Mortell and Runger model given in 1995, recommended to monitor the range of stream’s means at time t or the maximum residual at time t,
max⁡(xij )-xi
This chart is better than the GCC, especially when the variation in the process means over time is greater than the between-stream variability.

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