Chapter 10, Problem AP3.20CPT

To determine

To find out from the option that what is the approximate maximum grade a student could attain and still receive an F for the course.

Answer to Problem AP3.20CPT

Option (c) is the correct option.

Explanation of Solution

It is given in the question that:

Let $X\sim \text{Normal}(\mu =76,\sigma =8)$ represent the distribution of the final grades of the class.

Since, the question states that scores below $10\%$ will be considered as F grade. So, we want to find out the value that is the ${\text{10}}^{\text{th}}$ percentile of this normal distribution $X$ .

Thus, we want to find the value $x_0$ such that $P(X\le x_0)=0.10$ .

This implies that, $P(Z\le z_0)=0.10$ .

Now we will use the z -score as:

  $z_0=\frac{x_0-\mu }{\sigma }$

Now, using the standard normal table, we will look for $10\%$ within the table then find the value of Z that corresponds to $10\%$ .

So, the closest value is then, $z_0=-1.28$ .

Thus by equating in the equation above, we have,

  $\begin{array}{l}{z}_0=\frac{x_0-\mu }{\sigma }\\ \Rightarrow -1.28=\frac{x_0-76}{8}\\ \text{(Solving by multiplying by 8 and adding 76)}\\ \Rightarrow x_0=-1.28\times 8+76\\ \Rightarrow x_0=65.75\end{array}$

Therefore, the approximate maximum grade a student could attain and still receive an F for the courseis $65.75$ . Thus, option (c) is the correct option.