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# 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 more than $185$ 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 more than 185 pounds (i.e) $P(X > 185)$
When $X=185$
$Z=\large\frac{185-151}{15}$
$\;\;\;=\large\frac{34}{15}$
$\;\;\;=2.27$
$P(X > 185)=P(Z >2.27)$
$\qquad\qquad\;\;\;=0.5-P(0 < Z < 2.27)$
$\qquad\qquad\;\;\;=0.5-0.4884$
$\qquad\qquad\;\;\;=0.0116$
Step 3:
The number of male students in the college=500
$\therefore$ the number of students expected to weigh more than 185 pounds =$500\times 0.0116$
$\Rightarrow 6$(approx)