Question

A large group of students took a test in Physics and the final grades have a mean of $70$ and a standard deviation of $10$. If we can approximate the distribution of these grades by a normal distribution, what percent of the students should fail the test (grades$<60$)?

• A. $15.21$
• B. $23.21$
• C. $15.87$%
• D. $16.23$

Answer 1

The correct option is C $15.87$%

Final grades follows normal distribution.

Given mean i.e. $\mu$ $=70$

Given standard deviation i.e. $\sigma$ $=10$

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 $X=60$

$Z=(60-70)/10=-1$

$P(X<60)=P(Z<-1)$

$=0.1587$

Hence percent of students failed in test is $15.87\%$