Question

Final grades for a class are approximately Normally distributed with a mean of $76$and a standard deviation of $8$. A professor says that the top $10\%$of the class will receive an A, the next $20\%$a B, the next $40\%$a C, the next $20\%$a D, and the bottom $10\%$an F. What is the approximate maximum average a student could attain and still receive an F for the course?

(a) $70$

(b)$69.27$

(c) $65.75$

(d) $62.84$

(e)$57$

Short answer

The maximum average a student could attain and still receive an F for the course is $65.75$i.e., option (c).

Step-by-step solution

Given Information

We are given that the mean is $\mu =76$and standard deviation is $\sigma =8$and we are also given that anything below $10\%$is considered as an F.

We have to find out the maximum average a student can attain and still receive an F for the course.

Explanation

Now using Z score formula as the normal distribution table of mean and standard deviation is not given,

$P(Z\le z_0)=0.10$and then

we will use $z_0=\frac{x_0-\mu }{\sigma }$let this be equation $1$....

Simplify

Now see the standard table and find the value of $Z$that will correspond to $10\%$.

which gives $z_0=-1.28$

Now put the value in equation $1$...

we get $-1.28=\frac{x_0-76}{8}$

which results $x_0=65.75$

Hence, the maximum average a student could attain is$65.75$.