Answer to Question #161471 in Statistics and Probability for Aemie

Question

The weights of the students in a certain year level are normally distributed with a mean of 60kg and a standard deviation of 3.5kg. Find the probability that a student randomly selected from this group weighs. i) 55kg or less. ii) 68kg or more. iii) between 48kg and 53kg

Answer 1

The weight of the students in a certain year level are normally distributed with mean $60kg$ and standard deviation $3.5kg$ .

Let $X$ be random variable which defines weight of the students.

Then , $\mu =60$ and $\sigma =3.5$

Let us take $Z=\frac{X-\mu}{\sigma }$ . Then $Z=\frac{X-60}{3.5}$

(i) Probability that a student randomly selected is less than or equal to $55$ kg is $P(X\leq 55)$ .

$\therefore P(X\leq 55)=P(Z \leq \frac{55-60}{3.5})$

$=P(Z \leq -1.43)$

$=0.5-P(0<Z<1.43)$

$=0.5-0.4236$ [ from normal distribution table]

$=0.076$ (approximately)

(ii) Probability that a student randomly selected is greater than or equal to $68$ kg is $P(X\geq 68)$ .

$\therefore P(X\geq 68)=P(Z \geq \frac{68-60}{3.5})$

$=P(Z\geq 2.29)$

$=0.5-P(0<Z<2.29)$

$=0.5-0.4889$

$=0.011$ (approximately)

(iii)Probability that a student randomly selected is greater than $48$ kg and less than 53 kg is $P(48<X<53)$

$\therefore P(48<X<53)=P(\frac{48-60}{3.5}<Z<\frac{53-60}{3.5})$

$=P(-3.43<Z<-2)$

$=P(0<Z<3.43)-P(0<Z<2)$

$=0.4996-0.4772$

$=0.022$ (approximately)