Statistical Quality Control Questions and Answers – Modeling Process Quality – Continuous Distributions – 2

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This set of Statistical Quality Control test focuses on “Modeling Process Quality – Continuous Distributions – 2”.

1. The display of possible outcomes of an event with their corresponding probabilities is called ____________
a) Probability Plot
b) Contingency table
c) Bayesian Table
d) Frequency Plot

Explanation: The graphical representation of possible outcomes of an event with their corresponding probabilities; is called Probability plot.

2. Which one of these is not one of the conditions for the binomial distribution?
a) Only two outcomes must be there for an event
b) Independent trials must be there
c) Trials must be more than 4
d) The probability of failure must be constant

Explanation: For a binomial distribution, Bernoulli trials are necessary, i.e. the outcomes of an event must be either a success or a failure, and this means the probability of failure must be constant.

3. For mean=7 and standard deviation=2, what will be probability that a variable, which follows normal distribution, will have a value less than or equal to 10?
a) 0.93319
b) 0.94520
c) 0.96409
d) 0.91924

Explanation: Here μ=7, σ=2, we know that, P[x≤a] = P[z≤$$\frac{a-\mu}{σ}$$]
So, P[x≤10]=P[z≤$$\frac{10-7}{2}$$]= P[z≤1.5]=φ[1.5]=0.93319.

4. The variance of lognormal distribution is given by ____
a) σ2 = e2θ+w ($$e^{w^{2}}$$-1)
b) σ2 = e2θ+w2 (ew-1)
c) σ2 = e2θ+w2 ($$e^{w^{2}}$$-1)
d) σ2 = e2θ+w2 ($$e^{w^{2}}$$-2)

Explanation: The variance of a variable having a lognormal distribution is given by,
σ2= e2θ+w2 (ew2-1).

5. The relationship between the mean and variance of the exponential distribution is expressed by _____
a) μ = λσ2
b) μ = σ2
c) σ22 μ
d) σ2=μ/λ2

Explanation: For an exponential distribution, μ=1λ and σ2 = 1/λ2
So, for an exponential distribution, μ = λσ2.

6. (1-e-λa) is the cumulative ______ distribution.
a) Gamma
b) Normal
c) Lognormal
d) Exponential

Explanation: The cumulative distribution for standard exponential distribution is given by following equation,
F(a)=(1-e-λa).

7. The exponential distribution is used in reliability engineering as a model of the time to failure of a system. The parameter λ is called _____ in this application.
a) Failure rate
b) MTF
c) Hazard rate
d) MTBF

Explanation: The parameter λ in the reliability engineering application of the exponential distribution is called the failure rate of the system. The application uses it as a model of time to failure.

8. The mean of the exponential distribution used in reliability engineering is used as mean time to failure.
a) True
b) False

Explanation: In reliability engineering, the exponential distribution is used as a model of the time to failure of a system. The mean of the distribution is called MTF or mean time to failure.

9. The mean of the gamma distribution is given by _________
a) μ=1/λ
b) μ=r/λ
c) μ=r2
d) μ=1/r

Explanation: For a gamma distribution,
μ=r/λ
Where r and λ are the shape parameter and the scale parameter, respectively.

10. Exponential distribution is a special case of gamma distribution.
a) True
b) False

Explanation: When the shape factor r=1, in the case of a gamma distribution, gamma distribution reduces to an exponential distribution which has only the scale parameter λ.

11. The relationship between the mean and the standard deviation of the gamma distribution is given as _____
a) $$\frac{\mu}{\sigma} = \sqrt{r}$$
b) $$\frac{\mu}{\sigma} = r$$
c) $$\frac{\mu}{\sigma} = r^2$$
d) $$\frac{\mu}{\sigma} = \sqrt[3]{r}$$

Explanation: The mean of a gamma distribution = $$\frac{r}{λ}$$ and standard deviation σ = $$\frac{\sqrt{r}}{λ}$$; so,
$$\frac{\mu}{\sigma} = \sqrt{r}$$.

12. If the shape parameter of the gamma distribution is twice the scale parameter, what will be the value of the mean of the distribution?
a) 1.5
b) 2
c) 4
d) 1

Explanation: As for a gamma distribution, mean μ=r/λ; putting the value r=2λ from the question, we get,
μ=λ=2.

13. For an exponential distribution, what is the probability that the value of the exponential random variable with parameter λ=3.2; will be having a value higher than 0.2353?
a) 0.5290
b) 0.4710
c) 0.2213
d) 0.3452

Explanation: P[x>0.2352]=1-P[x≤0.2352]=1-{1-exp⁡(-3.2*0.2353)}=0.4710.

14. Which of these will not be distributed on a discrete distribution?
a) The number of houses in a colony
b) The number of bedrooms in a house
c) The number of scratches on a car hood
d) The diameter of a car tire

Explanation: The number of houses, bedrooms, and scratches, are all incremented by a discrete number, which is an integer, whereas, the diameter of a car tire will be increased on a continuous scale. Thus, it must be plotted on a continuous distribution.

15. For a variable distributed log-normally with θ=6, ω=1.2; what is the probability that the variable exceeds a value of 500?
a) 0.4290
b) 0.5710
c) 0.4990
d) 0.4937

Explanation: For a lognormal distribution,
P[x > a]=1-P[x≤a]=1-P[z≤$$\frac{ln⁡(a)-θ}{ω}$$]
Putting the values of θ and ω we get,
P[x>500]=0.4290.

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