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

1. What is the value of τ in the expression of C_{pm} 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) \(τ=\sqrt{E[(x+T)^2]}\)

View Answer

Explanation: The value of τ in the expression of C

_{pm}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) σ^{2}-(μ+T)^{2}

b) σ^{2}+(μ-T)^{2}

c) σ^{2}-(μ-T)^{2}

d) σ^{2}+(μ+T)^{2}

View Answer

Explanation: The value of squared value of τ,

τ

^{2}=E[(x-T)

^{2}]=E[(x-μ)

^{2}]+(μ-T)

^{2}=σ

^{2}+(μ-T)

^{2}.

3. Which of these is having same value as C_{pm}?

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) \(\frac{USL+LSL}{6\sqrt{σ^2+(μ-T)^2}}\)

View Answer

Explanation: As we know,

τ

^{2}=σ

^{2}+(μ-T)

^{2}

We may write,

C

_{pm}= \(\frac{USL-LSL}{6\sqrt{σ^2+(μ-T)^2}}\)

4. What is value of ξ in the expression for C_{pm} Process Capability Ratio?

a) \(\frac{μ+T}{3σ}\)

b) \(\frac{μ-T}{6σ}\)

c) \(\frac{μ+T}{σ}\)

d) \(\frac{μ-T}{σ}\)

View Answer

Explanation: The value of ξ in the expression of C

_{pm}is given by,

ξ = \(\frac{μ-T}{σ}\)

5. If standard values are not given, the value of C_{pm} 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) \(\widehat{C_{pm}}=\frac{C_p}{\sqrt{1-V^2}}\)

View Answer

Explanation: The estimate of C

_{pm}, when the standard values of σ, μ are not given, is calculated by expression,

\(\widehat{C_{pm}}=\frac{C_p}{\sqrt{1+V^2}}\)

6. What is the value of V in the expression of the estimate of C_{pm} Process Capability Ratio?

a) \(\frac{\bar{x}-T}{s}\)

b) \(\frac{\bar{x}+T}{s}\)

c) \(\frac{\bar{x}-T}{s^2}\)

d) \(\frac{\bar{x}-T}{σ}\)

View Answer

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 C_{pm} when the value |μ-T|→∞ ?

a) 1

b) 0

c) ∞

d) -∞

View Answer

Explanation: The value of C

_{pm}approaches to zero asymptotically as the value |μ-T|→∞. This is because,

C

_{pm}= \(\frac{USL-LSL}{6\sqrt{σ^2+(μ-T)^2}}\)

8. What is the value of C_{pm} when the value μ=T?

a) C_{pm}=1

b) C_{pm}=0

c) C_{pm}=C_{p}

d) C_{pm}<C_{p}

View Answer

Explanation: As we know,

C

_{pm}= \(\frac{USL-LSL}{6\sqrt{σ^2+(μ-T)^2}}\)

When, μ=T we get C

_{pm}=C

_{p}.

9. What is the value of C_{pkm} 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) \(C_{pkm}=\frac{C_{pk}}{\sqrt{1-ξ^2}}\)

View Answer

Explanation: The value of PCR C

_{pkm}is given by

\(C_{pkm}=\frac{C_{pk}}{\sqrt{1+ξ^2}}\)

10. What is the value of ξ in the expression for C_{pkm} PCR?

a) \(\frac{μ-T}{6σ}\)

b) \(\frac{μ-T}{σ}\)

c) \(\frac{μ+2T}{3σ}\)

d) \(\frac{μ+T}{6σ}\)

View Answer

Explanation: The value of ξ in the expression of C

_{pkm}is given by,

ξ = \(\frac{μ-T}{σ}\)

11. C_{pkm} is also called _________ generation process capability ratio.

a) Third

b) Fourth

c) Second

d) First

View Answer

Explanation: C

_{pkm}is sometimes called third generation process capability ratio since it is constructed from the second generation PCR C

_{pm}and C

_{pk}.

12. C_{p} is called the _________ generation PCR.

a) First

b) Second

c) Third

d) Fourth

View Answer

Explanation: C

_{p}is called the first generation process capability ratio as it was designed before any other process capability ratio like C

_{pk}or C

_{pm}.

13. Which of these is one of the first generation process capability ratios?

a) C_{pk}

b) C_{pm}

c) C_{pc }

d) C_{pu}

View Answer

Explanation: As we know, the first Process capability ratios, that were designed, were C

_{p}, C

_{pu}, C

_{pl}. So they are called the first generation process capability ratios.

14. C_{pm} is the second generation process capability ratio.

a) True

b) False

View Answer

Explanation: C

_{pk}and C

_{pm}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

View Answer

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/d_{2} 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) d_{2}<1

View Answer

Explanation: When process is not in control, the value of s has a big difference with the value of R/d

_{2}. 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. P_{p} is a ___

a) Process capability ratio

b) Process stability ratio

c) Process prediction index

d) Process performance index

View Answer

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

_{p}is one of them.

18. What is the value of process performance index estimate (P_{p})?

a) \(\frac{USL-LSL}{4s}\)

b) \(\frac{USL+LSL}{4s}\)

c) \(\frac{USL-LSL}{6σ}\)

d) \(\frac{USL-LSL}{6s}\)

View Answer

Explanation: The estimate of process performance index P

_{p}has the value,

P

_{p}= \(\frac{USL-LSL}{6s}\)

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