Question

Suppose that grade point averages of undergraduate students at one university have a bell-shaped distribution with a mean of 2.57 and a standard deviation of 0.44. Using the empirical rule, what percentage of the students have grade point averages that are at least 1.69? Please do not round your answer.

Answer 1

Given:

$\begin{gathered} mean(\mu)=2.57 \\ Standard\text{ }deviation(\sigma)=0.44 \end{gathered}$

To Determine: The percentage of the students that have grade point averages that are at least 1.68

Solution

Please note that according to empirical rule,

$\begin{gathered} \mu-\sigma=68\% \\ \mu-2\sigma=95\% \\ \mu-3\sigma=99.7\% \end{gathered}$

$\begin{gathered} 2.57-0.44=2.13 \\ 2.57-2(0.44)=2.57-0.88=1.69 \end{gathered}$

For at least 1.69, we can see from the above, calculation that it is the mean subtracted from twice the standard deviation that gives 1.69. Therefore, the percentage for at least 1.69 would be the average of the percentage for twice and thrice of the standard deviation from the mean. This is as calculated below

$\begin{gathered} Average=\frac{(99.7+95)\%}{2} \\ Average=\frac{194.7}{2} \\ Average=97.35\% \end{gathered}$