# Statistical Quality Control Questions and Answers – Modeling Process Quality – Discrete Distributions – 1

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This set of Statistical Quality Control Multiple Choice Questions & Answers (MCQs) focuses on “Modeling Process Quality – Discrete Distributions – 1”.

1. What will be the variance of the sample following hyper geometric distribution and having 5 items defective in 100 lot items, if 10 samples are taken without replacement?
a) 0.431818
b) 0.410023
c) 0.483838
d) 0.568898

Explanation: variance of hyper geometric distribution is given by,
$$\sigma^2 = \frac{nD}{N} (1-\frac{D}{n})(\frac{N-n}{N-1})$$ Putting, N=100, n=10, D=5; we get variance=0.431818.

2. For a Poisson distribution, parameter λ is having a value of 7. What is the variance for the distribution?
a) 2.64
b) 2.78
c) 7
d) 14

Explanation: For the Poisson probability distribution, the variance is equal to the parameter λ of the probability distribution. So for the above stated distribution, the variance will be equal to 7.

3. For 9 Bernoulli trials, what will be the mean of trials when the probability of failure is 0.56?
a) 5.04
b) 1.74
c) 3.96
d) 2.82

Explanation: For n=9 and probability of failure=0.56, we get probability of success;
p = 1 – 0.56 = 0.44.
Using this, we get;
mean= np=9*0.44=3.96.

4. In sample fraction defective, the ̂ symbol represents that ______
a) p ̂ is an estimate of the true value of binomial parameter p
b) It is the “raised to power” operator
c) It is having no significance
d) It is the true value of binomial parameter p

Explanation: The ̂ symbol represents the estimate of the true, unknown value of the binomial parameter p.p is the value of sample fraction defective or sample fraction nonconforming.

5. For a Pascal distribution, the probability of occurrence of a event x is given by _____
a) $$p(x)=\left(x-1 \atop r-1\right) p^x (1-p)^{x-r}$$
b) $$p(x)=\left(r-1 \atop x-1\right) p^x (1-p)^{x-r}$$
c) $$p(x)=\left(x-1 \atop r-1\right) p^x (1-p)^{1-r}$$
d) $$p(x)=\left(x-1 \atop r-1\right) p^x (1-p)^{x-1}$$

Explanation: The Pascal probability distribution is given by the following equation,
$$p(x)=\left(x-1 \atop r-1\right) p^x (1-p)^{x-r}$$
Where x=r, r+1, r+2… and r≥1 is an integer.

6. When is Pascal distribution, called negative binomial distribution?
a) When r>0 but not necessarily an integer
b) When r<0 but an integer
c) When r>3 and an integer
d) When r=0 and an integer

Explanation: When in a Pascal probability distribution, r is greater than 0 but not necessarily an integer, the Pascal distribution is called, negative binomial distribution. This is a special case of Pascal distribution.

7. When r=1 in a Pascal distribution, what is this case called?
a) Negative binomial distribution
b) Positive binomial distribution
c) Hyper geometric distribution
d) Geometric distribution

Explanation: The special case of Pascal distribution when r=1, we get geometric distribution. This represents the number of Bernoulli trials until the first success occurs.

8. Lognormal distributions are a part of discrete distributions.
a) True
b) False

Explanation: Lognormal distributions are an important part of continuous distributions, as they represent probability distribution of variables having continuous values. They are not counted in discrete distributions.

9. Which of the expression represents the mean of Pascal distribution?
a) r/p
b) np
c) nD/N
d) np(1-p)

Explanation: The mean of the Pascal distribution is given by the following equation,
μ=r/p
Where r > 1 and an integer and p is probability of success.

10. The equation of variance, i.e. σ2=$$\frac{r(1-p)}{p^2}$$ is true for _____
a) Hyper geometric distribution
b) Pascal Distribution
c) Normal distribution
d) Binomial distribution

Explanation: The expression $$\frac{r(1-p)}{p^2}$$ is useful for finding out variance of the Pascal probability distribution. We can identify it as it has “r” in the expression of variance.

11. Bernoulli trials are a part of _____ probability distribution.
a) Poisson
b) Binomial
c) Hyper geometric
d) Normal

Explanation: Bernoulli trials are the independent trials having a result either a “success” or a “failure”. They are an important part of binomial probability distribution. The probability distribution is based upon the Bernoulli trials.

12. $$\left(a \atop b\right)$$ is expanded as __________
a) $$\frac{a!}{b!}$$
b) $$\frac{a!}{b!(a-b)!}$$
c) $$\frac{a!}{b!(b-a)!}$$
d) $$\frac{b!}{a!}$$

Explanation: $$\left(a \atop b\right)$$ is generally acronym for the combination $$a \atop b$$C or $$\frac{a!}{b!(a-b)!}$$

13. For a Pascal distribution; r=3 and the probability of success is 0.7895. What will be the value of “σ”?
a) 1.0131
b) 3
c) 1.0065
d) 1.7320

Explanation: We know, for a Pascal distribution,
σ2=r(1-p)/p2
Evaluation of the above equation using the values r=3 and p=0.7895, we get, σ=1.0063.

14. For evaluation probability of measurement of length of a part being a certain value, discrete probability distributions can be used.
a) True
b) False 