Q:

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?

Accepted Solution

A:
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 [tex]\mu=540[/tex].Also, 95% of his classmates scored between 400 and 680. Then, by empirical rule , 95% of data falls in between [tex]mu\pm 2\sigma[/tex] i.e. [tex]540- 2\sigma=400[/tex]    (1)[tex]540+2\sigma=680[/tex]       (2)Subtracting (1) from (2), we get[tex]4\sigma=680-400=280\\\\\Rightarrow\ \sigma=\dfrac{280}{4}=70[/tex]Let x be the random variable to represent the scores of every student.Statistic z-score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]For x= 490, we have[tex]z=\dfrac{490-540}{70}\approx-0.71[/tex]The p-value = [tex]P(z<-0.71)=0.2388521\approx23.89\%[/tex]Hence, 23.89% of his classmates who scored lower than he did .