The mean weight of $500$ students in a certain college is $151$ pounds and the standard deviation is $15$ pounds. Assuming the weights are normally distributed, find how many students weigh
- between $120$ and $155$ pounds;
- more than $185$ pounds.
Given: Mean $= \mu = 151$; $\;$ Standard deviation $= \sigma = 15$
Let $X$ be a normal variate.
Standard normal variate $Z = \dfrac{X - \mu}{\sigma}$
- When $X = 120$, $\;$ $Z = \dfrac{120 - 151}{15} = -2.07$
When $X = 155$, $\;$ $Z = \dfrac{155 - 151}{15} = 0.27$
$\therefore \;$ Probability of students weighing between $120$ and $155$ pounds is
$\begin{aligned} P \left(120 < X < 155\right) & = P \left(- 2.07 < Z < 0.27\right) \\\\ & = P \left(- 2.07 < Z \leq 0\right) + P \left(0 \leq Z < 0.27\right) \\\\ & = P \left(0 \leq Z < 2.07\right) + P \left(0 \leq Z < 0.27\right) \;\;\; \left[\text{by symmetry}\right] \\\\ & = 0.4808 + 0.1064 \;\;\; \left[\text{from Normal Distribution Table}\right] \\\\ & = 0.5872 \end{aligned}$
$\therefore \;$ Number of students who weigh between $120$ and $155$ pounds is $= 500 \times 0.5872 = 293.6$
i.e. $\;$ approximately $294$ students
- When $X = 185$, $\;$ $Z = \dfrac{185 - 151}{15} = 2.27$
$\therefore \;$ Probability of students weighing more than 185 pounds is
$\begin{aligned} P \left(X > 185\right) & = P \left(Z > 2.27\right) \\\\ & = \left(\text{area between } Z = 0 \text{ to } Z = \infty\right) \\\\ & \hspace{1.5cm} - \left(\text{area between } Z = 0 \text{ to } Z = 2.27\right) \\\\ & = 0.5 - 0.4884 \\\\ & = 0.0116 \end{aligned}$
$\therefore \;$ Number of students weighing more than $185$ pounds is $= 500 \times 0.0116 = 5.8$
i.e. $\;$ approximately $6$ students