Question

Given a random variable X having a normal distribution with µ = 40 , σ = 10 ,Find the probability that X assumes a value between 55, 70

Answer 1

6.55%

Explanation:

Given a random variable X having a normal distribution with:

• Mean, µ = 40

,

• Standard Deviation, σ = 10

We want to find the probability that X is between 55 and 70.

In order to do this, first, we find the z-scores for X=55 and X=70.

The z-score formula is:

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

Thus:

`\begin{gathered} At\text{ X=55, }z=(55-40)/(10)=(15)/(10)=1.5 \\ At\text{ X=70, }z=(70-40)/(10)=(30)/(10)=3 \end{gathered}`

From the z-table:

`\begin{gathered} P\mleft(x<1.5\mright)=0.93319 \\ P\mleft(x>3\mright)=0.0013499 \\ P\mleft(x<1.5orx>3\mright)=0.93319+0.0013499=0.93454 \end{gathered}`

Therefore, using a z-score table, the probability that X assumes a value between 55, 70 is:

`\begin{gathered} P\mleft(1.53) \\ =1-0.93454 \\ =0.065457 \end{gathered}`

The required probability is 6.55%.