Question

The composite scores in a statistics course have a normal distribution with a mean of 68 and a standard deviation of 8. If a teacher decides to give the top 15% of the students an A grade, what is the lowest grade needed for an A?

Answer 1

Denote the random variable for composite scores by \(X\). You're looking for a cutoff score \(k\) such that
\[\mathbb P(X>k)=0.15\]To do that, you can transform \(X\) to \(Z\), which follows a standard normal distribution.
\[Z=\frac{X-\mu}{\sigma}\iff X=\mu+\sigma Z\]where \(\mu\) is the mean and \(\sigma\) the standard deviation for \(X\). So,
\[\mathbb P(X>k)=\mathbb P\left(\frac{X-68}{8}>\frac{k-68}{8}\right)=\mathbb P(Z>\hat k)=0.15\]where \(\hat k\) is the transformed cutoff score with respect to \(Z\).

You can find the value of \(\hat k\) by looking for it in a \(z\) table, or use the one given here in the right-tail row of the table:
http://www.wolframalpha.com/input/?i=p(z%3Ek)%3D0.15
\[\frac{k-68}{8}=1.036\implies k=\cdots\]