# The mean weight of $500$ male students in a certain college in $151$ pounds and the standard deviation is $15$ pounds. Assuming the weights are normally distributed, find how many students weigh between $120$ and $155$ pounds

Toolbox:
• Standard normal distribution:
• In a standard normal distribution $\mu=0,\sigma ^2=1$
• The random variable $X$ can be converted to the standard normal variable $Z$ by the transformation
• $Z=\large\frac{X-\mu}{\sigma}$
Step 1:
Let $X$ be the random variable denoting the weight of a male student in the college.
$X\sim N(150,15^2)$
Step 2:
To find the probability that a student weighs between 120 and 155 pounds.
To find $P(120 < X < 155)$
Step 3:
Let $Z$ be the standard normal variable
$Z=\large\frac{X-\mu}{\sigma}$
$\;\;=\large\frac{X-151}{15}$
When $X=120$
$Z=\large\frac{120-151}{15}$
$\;\;=\large\frac{-31}{15}$
$\;\;=-2.07$
When $X=155$
$Z=\large\frac{155-151}{15}$
$\;\;=\large\frac{-4}{15}$
$\;\;=0.27$
Step 4:
$P(120 < X < 155)=P(-2.07 < Z < 0.27)$
$\qquad\qquad\qquad\quad\;\;=P(-2.07 < Z < 0)+P(0 < Z < 0.27)$
$\qquad\qquad\qquad\quad\;\;=P(0 < Z < 2.07)+P(0 < Z < 0.27)$ by symmetry
$\qquad\qquad\qquad\quad\;\;=0.4808+0.1064$
$\qquad\qquad\qquad\quad\;\;=0.5872$
Step 5:
The number of male students in the college=500
$\therefore$ the number of students expected to weigh between 120 and 155 pounds =$500\times 0.5872$
$\Rightarrow 244$(approx)
edited Sep 19, 2013