Question

In a departmental examination on statistics, the mean grade was 64 and the standard deviation was 10. If the grades are approximately normally distributed and 40 students got grades between 60 and 70, how many students took the examination?

Answer 1

Solution

first, we obtain the P(60<x<70)

x~N(64,10²)

$$P(60<x<70)=P(x<70)-P(x<60)$$

$$= \phi({70-64 \over 10})- \phi({60-64 \over 10})= \phi(0.6) - \phi (-0.4)$$ $$= 0.7257 - 0.3446=0.3811$$

This implies that 0.3811 or 38.11% of the students scored between 60 and 70.

But we know that 40 students scored between 60 and 70, therefore, the total number of students who took the examination is:

$$={1 * 40 \over 0.3811} = 104.96 \approx 105$$

$\therefore 105$ students took the examination