# Statistical Quality Control Questions and Answers – Process Capability Ratios – 5

This set of Statistical Quality Control Problems focuses on “Process Capability Ratios – 5”.

1. What is the value of τ in the expression of Cpm process capability ratio?
a) $$τ=\sqrt{[E\big\{x-T)^2]}$$
b) $$τ=\sqrt{E[{x+T}^2]}$$
c) $$τ=\sqrt{E[(x-T)^2]}$$
d) $$τ=\sqrt{E[(x+T)^2]}$$

Explanation: The value of τ in the expression of Cpm is the square root of expected squared deviation from target T, i.e.
$$τ=\sqrt{[E\big\{x-T)^2]}$$

2. Which of these is equal to the squared value of τ?
a) σ2-(μ+T)2
b) σ2+(μ-T)2
c) σ2-(μ-T)2
d) σ2+(μ+T)2

Explanation: The value of squared value of τ,
τ2=E[(x-T)2]=E[(x-μ)2 ]+(μ-T)22+(μ-T)2.

3. Which of these is having same value as Cpm?
a) $$\frac{USL-LSL}{6\sqrt{σ^2-(μ-T)^2}}$$
b) $$\frac{USL+LSL}{6\sqrt{σ^2-(μ-T)^2}}$$
c) $$\frac{USL-LSL}{6\sqrt{σ^2+(μ-T)^2}}$$
d) $$\frac{USL+LSL}{6\sqrt{σ^2+(μ-T)^2}}$$

Explanation: As we know,
τ22+(μ-T)2
We may write,
Cpm = $$\frac{USL-LSL}{6\sqrt{σ^2+(μ-T)^2}}$$

4. What is value of ξ in the expression for Cpm Process Capability Ratio?
a) $$\frac{μ+T}{3σ}$$
b) $$\frac{μ-T}{6σ}$$
c) $$\frac{μ+T}{σ}$$
d) $$\frac{μ-T}{σ}$$

Explanation: The value of ξ in the expression of Cpm is given by,
ξ = $$\frac{μ-T}{σ}$$

5. If standard values are not given, the value of Cpm is estimated by expression _____
a) $$\widehat{C_{pm}}=\frac{C_p}{\sqrt{1+V^2}}$$
b) $$\widehat{C_{pm}}=\frac{C_p}{\sqrt{1-V}}$$
c) $$\widehat{C_{pm}}=\frac{C_p}{\sqrt{1+V}}$$
d) $$\widehat{C_{pm}}=\frac{C_p}{\sqrt{1-V^2}}$$

Explanation: The estimate of Cpm, when the standard values of σ, μ are not given, is calculated by expression,
$$\widehat{C_{pm}}=\frac{C_p}{\sqrt{1+V^2}}$$
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6. What is the value of V in the expression of the estimate of Cpm Process Capability Ratio?
a) $$\frac{\bar{x}-T}{s}$$
b) $$\frac{\bar{x}+T}{s}$$
c) $$\frac{\bar{x}-T}{s^2}$$
d) $$\frac{\bar{x}-T}{σ}$$

Explanation: The value of V is used at the place of ξ in the case when the standard values of σ and μ are not available. The value of V is equal to,
V = $$\frac{\bar{x}-T}{s}$$

7. What is the value of Cpm when the value |μ-T|→∞ ?
a) 1
b) 0
c) ∞
d) -∞

Explanation: The value of Cpm approaches to zero asymptotically as the value |μ-T|→∞. This is because,
Cpm = $$\frac{USL-LSL}{6\sqrt{σ^2+(μ-T)^2}}$$

8. What is the value of Cpm when the value μ=T?
a) Cpm=1
b) Cpm=0
c) Cpm=Cp
d) Cpm<Cp

Explanation: As we know,
Cpm = $$\frac{USL-LSL}{6\sqrt{σ^2+(μ-T)^2}}$$
When, μ=T we get Cpm=Cp.

9. What is the value of Cpkm process capability ratio?
a) $$C_{pkm}=\frac{C_{pk}}{\sqrt{1-ξ}}$$
b) $$C_{pkm}=\frac{C_{pk}}{\sqrt{1+ξ^2}}$$
c) $$C_{pkm}=\frac{C_{pk}}{\sqrt{1+ξ}}$$
d) $$C_{pkm}=\frac{C_{pk}}{\sqrt{1-ξ^2}}$$

Explanation: The value of PCR Cpkm is given by
$$C_{pkm}=\frac{C_{pk}}{\sqrt{1+ξ^2}}$$

10. What is the value of ξ in the expression for Cpkm PCR?
a) $$\frac{μ-T}{6σ}$$
b) $$\frac{μ-T}{σ}$$
c) $$\frac{μ+2T}{3σ}$$
d) $$\frac{μ+T}{6σ}$$

Explanation: The value of ξ in the expression of Cpkm is given by,
ξ = $$\frac{μ-T}{σ}$$

11. Cpkm is also called _________ generation process capability ratio.
a) Third
b) Fourth
c) Second
d) First

Explanation: Cpkm is sometimes called third generation process capability ratio since it is constructed from the second generation PCR Cpm and Cpk.

12. Cp is called the _________ generation PCR.
a) First
b) Second
c) Third
d) Fourth

Explanation: Cp is called the first generation process capability ratio as it was designed before any other process capability ratio like Cpk or Cpm.

13. Which of these is one of the first generation process capability ratios?
a) Cpk
b) Cpm
c) Cpc
d) Cpu

Explanation: As we know, the first Process capability ratios, that were designed, were Cp, Cpu, Cpl. So they are called the first generation process capability ratios.

14. Cpm is the second generation process capability ratio.
a) True
b) False

Explanation: Cpk and Cpm were designed after the first generation capability ratios, which were not able to consider the process centering fact into their evaluation. So they are called the second gen. process capability ratios.

15. Confidence intervals are not needed for $$\widehat{C_{pkm}}$$.
a) True
b) False

Explanation: All the estimates of the process capability ratios are point estimates. This means there is a strong possibility of error due to statistical fluctuation. So confidence interval is needed also for $$\widehat{C_{pkm}}$$.

16. What does usage of s instead of R/d2 in the confidence intervals indicate?
a) s=1
b) Process must be in control
c) Process may be in the out-of-control state
d) d2<1

Explanation: When process is not in control, the value of s has a big difference with the value of R/d2. So for the calculation of confidence interval s is used. This means, for a PCR to have a meaning, the process must be in control.

17. Pp is a ___
a) Process capability ratio
b) Process stability ratio
c) Process prediction index
d) Process performance index

Explanation: AIAG has recommended using the process performance indices to be used instead of the process capability ratios when process is not in control. Pp is one of them.

18. What is the value of process performance index estimate (Pp)?
a) $$\frac{USL-LSL}{4s}$$
b) $$\frac{USL+LSL}{4s}$$
c) $$\frac{USL-LSL}{6σ}$$
d) $$\frac{USL-LSL}{6s}$$

Explanation: The estimate of process performance index Pp has the value,
Pp = $$\frac{USL-LSL}{6s}$$

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