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

Hint: First, we will use the equation that every normal random variable \[X\] can be transformed into a \[z\] score, \[z = \dfrac{{\left( {X - \mu } \right)}}{\sigma }\], where \[X\] is the normal random variable, \[\mu \] represents mean and \[\sigma \] represents the standard deviation.

Complete step-by-step answer:
Apply this property, and then use the given conditions to find the required value.
Given that 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.
We know that the time taken to assemble follows a normal distribution.
Let us assume that \[\mu \] represents mean and \[\sigma \] represents the standard deviation.
Thus, we have \[\mu = 20{\text{ hours}}\] and \[\sigma = 2{\text{ hours}}\] from the given conditions.
We know that a standard score or a \[z\]–score is a normal random variable of a standard normal distribution.
We also know that every normal random variable \[X\] can be transformed into a \[z\] score using the equation, \[z = \dfrac{{\left( {X - \mu } \right)}}{\sigma }\], where \[X\] is the normal random variable, \[\mu \] represents mean and \[\sigma \] represents the standard deviation.
First, we will take the normal random variable to be 22 hours.
Substituting the values of \[\mu \] and \[\sigma \] in the above formula of \[z\] score.
\[
  z = \dfrac{{\left( {22 - 20} \right)}}{2} \\
   = \dfrac{2}{2} \\
   = 1 \\
\]
We will now take the normal random variable to be 20 hours.
Substituting the values of \[\mu \] and \[\sigma \] in the above formula of \[z\] score.
\[
  z = \dfrac{{\left( {20 - 20} \right)}}{2} \\
   = \dfrac{0}{2} \\
   = 0 \\
\]
We also know the property of probability, that is, \[P\left( {a < X < b} \right) = P\left( {X < b} \right) - P\left( {X < a} \right)\].
Replacing 20 for \[a\] and 22 for \[b\] in the above property of the probability, we get
\[
  P\left( {20 < X < 22} \right) = P\left( {X < 22} \right) - P\left( {X < 20} \right) \\
   = P\left( {z < 1} \right) - P\left( {z < 0} \right) \\
   = 0.3413 \\
\]
Therefore, the probability that a car can be assembled at this plant in a period of time between 20 and 22 hours is \[0.3413\].
Hence, the option C is correct.

Note: In solving these types of questions, you should be familiar with the formula of transferring normal random variables into \[z\] score. Then use the given conditions and values given in the question, and find the sum of given numbers, and substitute the values in the formula. Also, we are supposed to write the values properly to avoid any miscalculation.