Question

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

  i) scored higher than 80?                                                                                                                                          (3mks)

  ii) should pass the test?                                                                                                                                           (3mks)

Answer 1

Let X be random variable of grades a random student scores.

X is normally distributed with a mean $\mu = 70$ and a standard deviation $\sigma = 10$. Therefore, $X = \sigma Y +\mu$ where random variable Y has a standard Gaussian distribution with the cumulative distribution function $P(Y<y) =\Phi(y) = \frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{y}e^{-t^2/2}dt$

$P(x\leq X) = P(x\leq \sigma Y +\mu) = P(\frac{x-\mu}{\sigma} \leq Y) = 1 - \Phi(\frac{x-\mu}{\sigma})$

Let $x=80$ then $(x-\mu)/\sigma = (80 - 70)/10 = 1$

$P(80\leq X) = 1 - \Phi(1)= 1-0.84134 = 0.15866=15.866\%$

Therefore, approximately 15.866% of students score 80+.

A student passes the test if he scores 60+.

Let $x=60$ then $(x-\mu)/\sigma = (60 - 70)/10 = -1$

$P(60\leq X) = 1 - \Phi(-1)= 1-0.15866=0.84134$

Therefore, approximately 84.134% of students pass the test on Physics.

Answer.

Approximately 15.866% of students score 80+

Approximately 84.134% of students pass the test