Question

A curious student in a large economics course is interested in calculating the percentage of his classmates who scored lower than he did on the GMAT; he scored 490. He knows that GMAT scores are normally distributed and that the average score is approximately 540. He also knows that 95% of his classmates scored between 400 and 680. Based on this information, calculate the percentage of his classmates who scored lower than he did?

Answer 1

Answer:  23.89%

Step-by-step explanation:

The empirical rule says that the 95% of the data falls in between two standard deviations of the mean.

Given : GMAT scores are normally distributed and the the average score is approximately $\mu=540$.

Also, 95% of his classmates scored between 400 and 680.

Then, by empirical rule , 95% of data falls in between $mu\pm 2\sigma$

i.e. $540- 2\sigma=400$    (1)

$540+2\sigma=680$       (2)

Subtracting (1) from (2), we get

$4\sigma=680-400=280\\\\\Rightarrow\ \sigma=\dfrac{280}{4}=70$

Let x be the random variable to represent the scores of every student.

Statistic z-score : $z=\dfrac{x-\mu}{\sigma}$

For x= 490, we have

$z=\dfrac{490-540}{70}\approx-0.71$

The p-value = $P(z$

Hence, 23.89% of his classmates who scored lower than he did .