# Statistical Quality Control Questions and Answers – Gauge and Measurement System Capability Studies – 2

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This set of Basic Statistical Quality Control Questions and Answers focuses on “Gauge and Measurement System Capability Studies – 2”.

1. What of these can be used as a reasonable model for measurement system capability studies? Here y,x and ε denote the observed measurement, true measurement, and the measurement error respectively.
a) y=x-2ε
b) y=x+ε
c) y=x-ε
d) y=x+2ε

Explanation: The reasonable model, which can be used for measurement system capability is expressed as,
y=x+ε

2. The variance of the total observed measurement is expressed by __________
a) $$\sigma_{Total}^2=\sigma_P^2-\sigma_{Gauge}^2$$
b) $$\sigma_{Total}^2=\sigma_P+\sigma_{Gauge}^2$$
c) $$\sigma_{Total}^2=\sigma_P^2+\sigma_{Gauge}$$
d) $$\sigma_{Total}^2=\sigma_P^2+\sigma_{Gauge}^2$$

Explanation: The variance of the total observed measurement uses the value of the sum of the variances of product errors, and the gauge errors. It is calculated by,
$$\sigma_{Total}^2=\sigma_P^2+\sigma_{Gauge}^2$$

3. The P/T ratio stands for ___________
a) Probability to tolerance ratio
b) Precision to time ratio
c) Probability to total ratio
d) Precision to tolerance ratio

Explanation: The P/T ratio is used in the measurement system capability analysis. In P/T ratio, P/T refers to Precision to tolerance ratio.

4. What is the value of the P/T ratio?
a) $$\frac{P}{T}=\frac{1.5k\hat{σ_p}}{USL-LSL}$$
b) $$\frac{P}{T}=\frac{k\hat{σ}_{gauge}}{USL-LSL}$$
c) $$\frac{P}{T}=\frac{2k\hat{σ_p}}{USL+LSL}$$
d) $$\frac{P}{T}=\frac{k\hat{σ}_{gauge}}{USL+LSL}$$

Explanation: The P/T ratio is calculated for the evaluation of the gauge capability. It uses the value of the $$\hat{σ_{gauge}}$$ The value of P/T ratio is given by,
$$\frac{P}{T}=\frac{k\hat{σ}_{gauge}}{USL-LSL}$$

5. If the number of standard deviations between the usual natural tolerance limits of a normal distribution, what is the value used for k in the P/T ratio?
a) 5.15
b) 8
c) 6
d) 5.60

Explanation: The value k=6 in the P/T ratio corresponds to the number of standard deviations, between the usual natural tolerance limits for a normal distribution.

6. For a process, which has, USL and LSL equal to 60, and 5 respectively, and the value of $$\hat{σ}_{gauge}$$ = 0.887, what will be the value of P/T ratio when k=6?
a) 0.087
b) 0.077
c) 0.067
d) 0.097

Explanation: We know that,
$$\frac{P}{T}=\frac{k\hat{σ}_{gauge}}{USL-LSL}$$
Putting the values in the question, we get P/T=0.097.

7. Which of these indicate an adequate measurement system?
a) $$\frac{P}{T}≤0.1$$
b) $$\frac{P}{T}≤0.5$$
c) $$\frac{P}{T}≥0.1$$
d) $$\frac{P}{T}=0.3$$

Explanation: If the value of P/T ratio is lesser than or equal to 0.1, the measurement system used is predicted to be adequate for the selected process.

8. The options are the P/T ratios for different measurement systems. Which of these shows an adequate measurement system?
a) 0.21
b) 0.13
c) 0.18
d) 0.06

Explanation: A P/T ratio lesser than 0.1 indicates an appropriate measurement system for any process. So here, 0.06<0.1, so it is an example of acceptable measurement systems.

9. The cause of calling a measurement system adequate because it has P/T ratio lesser than 0.1, is ____________
a) A measurement device should be calibrated in units one-tenth large as the accuracy required in final measurement
b) A measurement device should be calibrated in units one-third large as the accuracy required in final measurement
c) A measurement device should be calibrated in units one-fourth large as the accuracy required in final measurement
d) A measurement device should be calibrated in units three-tenth large as the accuracy required in final measurement

Explanation: Values of estimated P/T ratio of 0.1 or less indicate adequate measurement system. It’s based upon the general rule, which requires the measurement device to be calibrated in units one-tenth large, as the accuracy required in the final measurement.

10. Which of these can be used as the estimate of standard deviation of total variability which is including both product variability, and the gauge variability?
a) The sample mean
b) The sample variance
c) The sample standard deviation
d) No of defects in the sample

Explanation: The sample variance can be used as the estimate of the standard deviation of the total variability, which includes both, the product variability, and the gauge variability.

11. If the sample variance of a process is, 10.05, and the gauge capability standard deviation is estimated to be 0.79. What will be the value of the estimate of the standard deviation of the product variability?
a) 9.26
b) 3.04
c) 2.03
d) 8.91

Explanation: As we know that,
$$\sigma_{Total}^2=\sigma_P^2+\sigma_{Gauge}^2$$
If the estimates are to be used, the same equation can be written for the corresponding estimate values. Putting the values as mentioned, we get, $$\hat{\sigma}_p^2$$=3.04.

12. Which of these show a correct expression for the ρp?
a) $$ρ_p=\frac{\sigma_p^2}{2σ_{Gauge}^2}$$
b) $$ρ_p=\frac{\sigma_{gauge}^2}{σ_{Total}^2}$$
c) $$ρ_p=\frac{\sigma_P^2}{σ_{Gauge}^2}$$
d) $$ρ_p=\frac{\sigma_{gauge}^2}{2σ_{total}^2}$$

Explanation: The gauge capability ratio ρp is the ratio of the variances of the product error and the gauge errors. It is expressed as,
$$ρ_p=\frac{\sigma_P^2}{σ_{Gauge}^2}$$

13. The gauge capability ratio ρM is expressed as ____________
a) $$ρ_p=\frac{\sigma_p^2}{2σ_{Gauge}^2}$$
b) $$ρ_p=\frac{\sigma_{gauge}^2}{σ_{Total}^2}$$
c) $$ρ_p=\frac{\sigma_P^2}{σ_{Gauge}^2}$$
d) $$ρ_p=\frac{\sigma_{gauge}^2}{2σ_{total}^2}$$

Explanation: The value of gauge capability ratio ρM is a ratio of the variances of the gauge errors and the total observed errors. It is expressed as,
$$ρ_p=\frac{\sigma_{gauge}^2}{σ_{Total}^2}$$

14. The general rule, that is used to define a measurement system adequate by using P/T ratio equal to or less than 0.1, can be used without caution.
a) True
b) False

Explanation: The caution should be used in accepting this general rule of thumb in all cases. A gauge must be capable to measure product accurately enough and precisely enough, for the analyst to make a correct decision. This may not necessarily require P/T <=0.1.

15. ρP=1-ρM.
a) True
b) False

Explanation: It is known that,
$$ρ_p=\frac{\sigma_{gauge}^2}{σ_{Total}^2}; ρ_p=\frac{\sigma_P^2}{σ_{Gauge}^2}; σ_{Total}^2=σ_P^2+σ_{Gauge}^2$$
Combining all three equations, we get,
ρP=1-ρM.

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