Chapter 13.3, Problem 71E

The Normal Curve The normal distribution curve, which models the distributions of data in a wide range of applications, is given by the function

$p\left(x\right)=\frac{1}{\sqrt{2\pi \sigma }}{e}^{-{\left(x-\mu \right)}^2/\left(2{\sigma }^2\right)}$ where $\pi =\mathrm{3.14159265\dots .}$ and $\sigma$ and $\mu$ are constants called the standard deviation and the mean, respectively. The graph of the normal distribution (when $\sigma =1$ and $\mu =2$

) is shown in the figure. Exercises 71 and 72 illustrate its use.

Test Scores Enormous State University's Calculus I test scores are modeled by a normal distribution with $\mu =72.6$ and $\sigma =5.2$. The percentage of students who obtained scores between a and b on the test is given by

${\displaystyle {\int }_a^{b}p\left(x\right)}dx$.

a. Use a Riemann sum with $n=40$ to estimate the percentage of students who obtained between 60 and 100 on the test.

b. What percentage of students scored less than 30?