Question

A large group of students took a test in physics and have a mean of 70 and a standard deviation of 10.If we approximate the distribution of these grades by a normal distribution, what percentage of the students

a) scored higher than 80?

b) should pass the test(grades>=60)?

c) should fail the test(grades<60)?

Answer 1

Let $X$ be the random variable denoting the mark in the test scored by the students.

Given, $\mu = 70, \sigma = 10.$ Let $z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 70}{10}$

a. $\begin{aligned} P(\text{students score more than 80)} &= P(X > 80)\\ &= P\left(\dfrac{X-\mu}{\sigma} > \dfrac{80 - 70}{10}\right)\\ &= P(z > 1)\\ &= 1 - P(z < 1)\\ & = 1 - 0.8413\quad(\text{Using Normal distribution table})\\ &= 0.1587 \end{aligned}$

Therefore, 15.87% of students scored above 80 marks.

b.

$\begin{aligned} P(\text{students pass the test)} &= P(X \ge 60)\\ &= P\left(\dfrac{X-\mu}{\sigma} \ge \dfrac{60 - 70}{10}\right)\\ &= P(z \ge -1)\\ &= 1 - P(z < -1)\\ & = 1 - 0.1587\quad(\text{Using Normal distribution table})\\ &= 0.8413 \end{aligned}$

Therefore, 84.13% of students passed in the test.

c.

$\begin{aligned} P(\text{students fail the test)} &= P(X < 60)\\ &= P\left(\dfrac{X-\mu}{\sigma} < \dfrac{60 - 70}{10}\right)\\ &= P(z < -1)\\ & = 0.1587\quad(\text{Using Normal distribution table})\\ \end{aligned}$

Therefore, 15.87% of students failed in the test.