Answer to Question #261832 in Statistics and Probability for shann

Question

1.  On a recent math test, the mean score was 75 and the standard deviation was 5. Mike scored an 85.

a)     What percent of the students scored below Mike?

b)    What percent of the students scored above Mike?

c)     If the class had 150 students in it, how many of them would you expect to score higher than Mike?

d)    If the class had 220 students in it, how many of them would you expect to score lower than Mike? 

Answer 1

From this data,

$\bar{x}=75,\space \sigma=5$

a.

Let Mike's score be $x=85$

In order to find the percent of the students scored below Mike, we first determine the probability that $x\lt85$ that is $p(x\lt 85)$ and is given by,

$p(x\lt85)=p((x-\mu)/\sigma\lt (85-\mu)/\sigma)=p(Z\lt (85-75)/5)=p(Z\lt2)$

Using the standard normal tables we have,

"p"(Z\\lt 2)=0.9773". To find the percentage we multiply this probability by 100%.

Therefore, the number of students who scored below mike's score in percent is 0.9773*100=97.73%

b.

The probability that students performed higher than mark is given as,

$p(x\gt85)=1-p(x\lt 85)=1-p(Z\lt2)=1-0.9773= 0.0227$ and the percentage of students who scored above Mark is 0.0227*100=2.27%

c,

Given that the number of students in the class is 150 then, the number of students expected to score higher than Mike is,

$150*p(x\gt85)=150*0.0227=3.405\approx4\space students$

Therefore in a class of 150 students we would expect that only 4 students scored above Mike's score.

d.

In a class of 220 students the number of students who scored below Mike's score is given as, $220*p(x\lt85)=220*0.9773=215.006\approx216$

Hence we would expect that 216 students scored lower than Mike.