Question

The time taken to assemble a car in a certain plant is a random variable having a normal distribution of $20$ hours and a standard deviation of $2$ hours. What is the probability that a car can be assembled at this plant in a period of time between $20$ and $22$ hours?

• A. $0.3513$
• B. $0.3216$
• C. $0.3413$
• D. $0.3613$

Answer 1

The correct option is C $0.3413$

TIme taken to assemble follows normal distribution.

Given mean i.e. $\mu$ $=20$ hours

Given standard deviation i.e. $\sigma$ $=2$ hours

The normal random variable of a standard normal distribution is called a standard score or a z-score. Every normal random variable X can be transformed into a z score via the following equation:

$z=(X-\mu )/\sigma$

where $X$ is a normal random variable, $\mu$ is the mean of $X$, and $\sigma$ is the standard deviation of $X$.

For $X=22$

$Z=(22-20)/2=1$

For $X=20$

$Z=(20-20)/2=0$

$P(a<X<b)=P(X<b)–P(X<a)$ as understood from diagram

$P(20<X<22)=P(X<22)-P(X<20)$

$=P(Z<1)-P(Z<0)$

$=0.3413$