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

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

1. Which of these cannot be shown on the continuous distributions?
a) Length dimension measurement of a box
b) Volume measurement of the box
c) Area measurement of one face of the box
d) Number of defects on the surface of the box

Explanation: Continuous distributions are used to describe the variation in the values of variables which are continuous, i.e. which take values on continuous scale. Number of defects is discrete not continuous parameter.

2. Which of these is a continuous distribution?
a) Pascal distribution
b) Lognormal distribution
c) Binomial distribution
d) Hyper geometric distribution

Explanation: Pascal, binomial, and hyper geometric distributions are all part of discrete distribution which are used to describe variation of attributes. Lognormal distribution is a continuous distribution used to describe variation of the continuous variables.

3. Which of these distributions has an appearance of bell-shaped or unimodal curve?
a) Lognormal distributions
b) Normal distribution
c) Exponential distribution
d) Cumulative exponential distributions

Explanation: Out of all continuous distributions, Normal distributions are the only distributions, which have a shape of curve as a bell. The curves of them are mostly unimodal.

4. The rule of multiplication of probability is possible only ____
a) When events are independent
b) When events are mutually exclusive
c) When events are Bayesian
d) When events are Empirical

Explanation: The rule of multiplication of probability is possible only when the events are independent.
P(A and B)=P(A).P(B)

5. Which of these equations describe the normal continuous distribution?
a) $$f(x)=\frac{1}{\sigma \sqrt{2π}} e^{-0.5(\frac{x-μ}{σ})^2}, -\infty < x < -\infty$$
b) $$f(x)=\frac{1}{\sqrt{2π}} e^{-0.5(\frac{x-μ}{σ})^2}, -\infty < x < -\infty$$
c) $$f(x)=\frac{1}{\sigma \sqrt{π}} e^{-0.5(\frac{x-μ}{σ})^x}, -\infty < x < -\infty$$
d) $$f(x)=\frac{1}{\sigma \sqrt{2π}} e^{-0.5(\frac{x-μ}{σ})^x}, -\infty < x < -\infty$$

Explanation: Normal distribution is a part of continuous distributions, which is described by the following equation,
$$f(x)=\frac{1}{\sigma \sqrt{2π}} e^{-0.5(\frac{x-μ}{σ})^2}, -\infty < x < -\infty$$

6. “ϕ(∙)” is said to be cumulative distribution function of _________
a) Standard binomial distribution
b) Standard normal distribution
c) Standard exponential distribution
d) Standard gamma distribution

Explanation: For the standard normal distribution, “ϕ(∙)” is described as the cumulative distribution function, which has the value of mean equal to zero and the corresponding standard deviation equal to unity.

7. The probability that the normal random variable x is less than or equal to some value a is said to be the _____
a) Standard normal distribution
b) Lognormal distribution
c) Exponential distribution
d) Cumulative normal distribution

Explanation: The normal distribution has several important special cases, out of which, the cumulative normal distribution is defined as the probability that the normal random variable x≤a.

8. Which of these is true for the normal distributions?
a) $$μ_y = a_1 μ_1 + a_2 μ_2 +⋯+ a_n μ_n$$
b) $$μ_y = a_1 μ_1^2 + a_2 μ_2^2 +⋯+ a_n μ_n^2$$
c) $$μ_y^2 = a_1^2 μ_1^2 + a_2^2 μ_2^2 +⋯+ a_n^2 μ_n^2$$
d) $$σ_y^2 = a_1^2 σ_1^2 + a_2^2 σ_2^2 +⋯+ a_n^2 σ_n^2$$

Explanation: From property, for any normal distribution which has,
$$y = a_1^2 x_1 + a_2^2 x_2 +⋯+ a_n^2 x_n$$
We have,
$$μ_y = a_1 μ_1 + a_2 μ_2 +⋯+ a_n μ_n$$

9. The central limit theorem is true for _______ distribution.
a) Normal distribution
b) Lognormal distribution
c) Exponential distribution
d) Gamma distribution

Explanation: The central limit theorem uses an assumption that, the normal distribution is an appropriate distribution for a random variable. That’s why; the central limit theorem is true only for Normal distribution.

10. If a variable x is having its value equal to the exponential function of another variable w, i.e. “x=exp⁡(w)”, and w has normal distribution; the distribution of x is called _______
a) Normal distribution
b) Exponential distribution
c) Lognormal distribution
d) Gamma distribution

Explanation: For a random variable x, which has “x=exp⁡(w)” where w is having a normal distribution; the distribution of such a random variable x, is said to be the Lognormal distribution. It is a continuous distribution.

11. For a lognormal random variable, what is the value of mean?
a) $$μ=e^{\theta+\frac{w^2}{2}}$$
b) $$μ=e^{\frac{w^2+\theta}{2}}$$
c) $$μ=e^{w+\frac{\theta^2}{2}}$$
d) $$μ=e^{\theta+w^2}$$

Explanation: if for x, which has x=ew, and w has a normal distribution with mean θ and variance w2; x is a lognormal random variable with mean expressed as
μ=e(θ+ w2/2).

12. The exponential distribution is given by _____
a) f(x)=xeλ
b) f(x)= e-λx
c) f(x)= λe-λx
d) f(x)=ew

Explanation: The continuous probability distribution is said to be exponential if the distribution follows this equation,
f(x)= λe-λx.

13. The cumulative distribution function for the lognormal distribution is given by _____
a) $$\phi[\frac{ln⁡(a)-\theta}{\omega}]$$
b) $$\phi[\frac{ln⁡(a)-\theta}{a}]$$
c) $$\phi[\frac{ln⁡(a)-\omega}{\omega}]$$
d) $$\phi[\frac{ln⁡(a)-\theta}{\theta}]$$

Explanation: The lognormal cumulative distribution function defines the probability that the variable x is less than or equal to a is given by,
$$\phi[\frac{ln⁡(a)-\theta}{\omega}]$$

14. If a card is chosen from a deck of cards, what is the probability that it is either 7 or 9?
a) 4/52
b) 7/52
c) 9/52
d) 8/52

Explanation: There are 8 cards in the deck which are either 7 or 9. There are total 52 cards in the deck, so the probability that the card is either a 7 or a 9 is 8/52; based upon the outcomes of interest divided by the total possible outcomes.

15. The rule of multiplication of probability is possible only _____________
a) When events are independent
b) When events are mutually exclusive
c) When events are Bayesian
d) When events are Empirical