Chapter 6.3, Problem 72E

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.

Consumer Satisfaction In a survey, consumers were asked to rate a new toothpaste on a scale of 1-10. The resulting data are modeled by a normal distribution with $\mu =4.5$ and $\sigma =1.0$. The percentage of consumers who rated the toothpaste with a score 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 $n=10$ to estimate the percentage of customers who rated the toothpaste 5 or above. (Use the range 4.5 to 10.5.)

b. What percentage of customers rated the toothpaste 0 or 1? (Use the range – 0.5 to 1.5.)