EX 4.16 reaction time

The time that it takes a driver to react to the brake lights on a decelerating vehicle is critical in helping to avoid rear-end collisions.

• The article “Fast-Rise Brake Lamp as a Collision-Prevention Device” (Ergonomics, 1993: 391-395) suggests that reaction time for an in-traffic response to a brake signal from standard brake lights can be modeled with
• a normal distribution having mean value 1.25 sec and standard deviation of .46 sec.

Question

What is the probability that reaction time is between 1.00 sec and 1.75 sec?

If we let $X$ denote reaction time, then standardizing gives

$$1.00\le X\le 1.75$$

if and only if

$$\frac{1.00-1.25}{.46}\le \frac{X-1.25}{.46}\le \frac{1.75-1.25}{.46}$$

Thus

$$\begin{array}{rl}& P\left(1.00\le X\le 1.75\right)\\ & =P\left(\frac{1.00-1.25}{0.46}\le Z\le \frac{1.75-1.25}{0.46}\right)\\ & =P\left(-0.54\le Z\le 1.09\right)\\ & =\Phi \left(1.09\right)-\Phi \left(-0.54\right)\\ & =0.8621-0.2946\\ & =0.5675\end{array}$$

This is illustrated in Figure 4.22.

Figure 4.22 Normal curves for Example 4.16

Similarly, if we view 2 sec as a critically long reaction time, the probability that actual reaction time will exceed this value is

$$\begin{array}{rl}& P\left(X>2\right)\\ & =P\left(Z>\frac{2-1.25}{0.46}\right)\\ & =P\left(Z>1.63\right)\\ & =1-\Phi \left(1.63\right)\\ & =0.0516\end{array}$$