Chapter PI, Problem 25RE

(a)

To determine

To use the $68-95-99.7$ rule to draw the normal model.

Explanation of Solution

It is given in the question that the time measured from the moment the driver first sees the danger has a normal model with mean of $1.5$ seconds and standard deviation of $0.18$ seconds. Thus, using the $68-95-99.7$ rule to draw the normal model as follows we have,

(b)

To determine

To write a few sentences describing driver reaction times.

Explanation of Solution

It is given in the question that the time measured from the moment the driver first sees the danger has a normal model with mean of $1.5$ seconds and standard deviation of $0.18$ seconds. Thus, as we can see that according to the normal model, reaction times are symmetric with centre at $1.5$ seconds. And about $95\%$ of all reaction times are between $1.14\text{ and 1}\text{.86}$ seconds.

(c)

To determine

To find out what percent of drivers have a reaction time less than $1.25$ seconds.

Answer to Problem 25RE

  $8.2\%$ .

Explanation of Solution

It is given in the question that the time measured from the moment the driver first sees the danger has a normal model with mean of $1.5$ seconds and standard deviation of $0.18$ seconds. Thus, we have that,

  $\begin{array}{l}z=\frac{1.25-1.5}{0.18}\\ =-1.39\\ P(z<-1.39)=normalcdf(-E99,-1.39,0,1)\\ =8.2\%\end{array}$

Thus, $8.2\%$ of drivers have a reaction time less than $1.25$ seconds.

(d)

To determine

To find out what percent of drivers have a reaction time between $1.6\text{ and 1}\text{.8}$ seconds.

Answer to Problem 25RE

  $24.1\%$ .

Explanation of Solution

It is given in the question that the time measured from the moment the driver first sees the danger has a normal model with mean of $1.5$ seconds and standard deviation of $0.18$ seconds. Thus, we have that,

  $\begin{array}{l}z=\frac{1.6-1.5}{0.18}\\ =0.556\\ z=\frac{1.8-1.5}{0.18}\\ =1.667\\ P(0.556<z<1.667)=normalcdf(0.556,1.667,0,1)\\ =24.1\%\end{array}$

Thus, $24.1\%$ of drivers have a reaction time between $1.6\text{ and 1}\text{.8}$ seconds.

(e)

To determine

To find out what is the interquartile range of reaction times.

Answer to Problem 25RE

  $0.24\text{ seconds}$ .

Explanation of Solution

It is given in the question that the time measured from the moment the driver first sees the danger has a normal model with mean of $1.5$ seconds and standard deviation of $0.18$ seconds. Thus by looking at the diagram in part (a) we can say that,

  $\begin{array}{l}\text{1st quartile}=1.38\\ \text{3rd quartile}=1.62\\ IQR=1.62-1.38\\ =0.24\text{ seconds}\end{array}$

(f)

To determine

To describe the reaction times of the slowest $\frac{1}{3}$ of all drivers.

Explanation of Solution

It is given in the question that the time measured from the moment the driver first sees the danger has a normal model with mean of $1.5$ seconds and standard deviation of $0.18$ seconds. Thus, the reaction times of the slowest $\frac{1}{3}$ of all drivers will be the reaction time of $\text{1}\text{.58}$ seconds or more.