Question

NHTSA crash safety tests. Refer to Exercise 4.21 (p. 224)and the NHTSA crash test data for new cars. One of the variablessaved in the accompanying file is the severity ofa driver’s head injury when the car is in a head-on collision with a fixed barrier while traveling at 35 miles per hour. The more points assigned to the head-injury rating,the more severe the injury. The head-injury ratings can be shownto be approximately normally distributed with a meanof 605 points and a standard deviation of 185 points.One of the crash-tested cars is randomly selected from the data, and the driver’s head-injury rating is observed.

a. Find the probability that the rating will fall between 500 and700 points.

b. Find the probability that the rating will fall between 400and 500 points.

c. Find the probability that the rating will be less than 850points.

d. Find the probability that the rating will exceed 1,000points.

Short answer

a. The probability that the rating will fall between 500and 700 points is 0.4110

b. The probability that the rating will fall between 400and 500 points is 0.1513

c. The probability that the rating will be less than 850 points is.0.9073

d. The probability that the rating will exceed 1,000 points is 0.0164

Step-by-step solution

Given information

Let X denotes the head injury rating.

X follows a normal distribution with the parameter $\mu$and ${\sigma }^2$

$z=\frac{X-\mu }{\sigma }$follows normal distribution with mean 0 and a variance 1

Also, given$mean=605SD=185$

Calculating the probability that the rating will fall between 500 and 700 points

a.

We have to calculate the probability that the rating will fall between 500 and 70 points.

That is,$P\left(500<X<700\right)$

role="math" localid="1660714332516" $$\begin{gathered} P\left(500<X<700\right)=P\left(700\right)-P\left(X-\le 500\right) \\ =P\left(Z<\frac{X-\mu }{\sigma }\right) \\ =P\left(Z<\frac{700-605}{185}\right)-P\left(Z\le \frac{700-605}{185}\right) \\ P\left(500<X<700\right)\approx P\left(Z<0.513\right)-P\left(Z\le -0.567\right) \\ P\left(500<X<700\right)\approx 0.6962-0.2852 \\ =0.411 \end{gathered}$$

Hence, the probability that rating will fall between 500 and 70 points. That is,$P\left(500<X<700\right)$ is 0.411.

Calculating the probability that the rating will fall between 400 and 500 points

b.

For calculating the probability that rating will fall between 400 and 500,that is$P\left(400<X<500\right)$ .

$$\begin{gathered} P\left(400<X<500\right)=P\left(500\right)-P\left(X-\le 400\right) \\ =P\left(Z<\frac{X-\mu }{\sigma }\right) \\ =P\left(Z<\frac{500-605}{185}\right)-P\left(Z\le \frac{400-605}{185}\right) \\ P\left(400<X<500\right)\approx P\left(Z<0.567\right)-P\left(Z\le -1.108\right) \\ \approx 0.2852-0,1339 \\ =0.1513 \end{gathered}$$

Hence, the probability that rating will fall between 400 and 500,that is$P\left(400<X<500\right)$ .is 0.1513.

Calculating the probability that the rating will beless than 850 points.

c.

We have to calculate the probability that the rating will be less than 850, that is,,

$P\left(X<850\right)$

Hence,

$$\begin{gathered} P\left(X<850\right)=P\left(Z<\frac{X-\mu }{\sigma }\right) \\ P\left(z<\frac{850-605}{185}\right) \\ \approx P\left(z<1.3243\right) \\ P\left(X<850\right)\approx 0.9073 \end{gathered}$$

Hence, the probability that the rating will be less than 850, that is$P\left(X<850\right)$ is 0.9073.

Calculating the probability that therating will exceed 1000 points.

d.

We will find the probability that rating will exceed 1000 points that is$P\left(X>1000\right)$

$$\begin{gathered} P\left(X>1000\right)=1-P\left(X>1000\right) \\ =1-P\left(Z<\frac{X-\mu }{\sigma }\right) \\ =1-P\left(Z<\frac{1000-605}{185}\right) \\ P\left(X>1000\right)\approx 1-P\left(Z>2.1351\right) \\ \approx 1-0.9836 \\ \approx 0.0164 \end{gathered}$$

Hence, the probability that rating will exceed 1000 points that is$P\left(X>1000\right)$ is 0.0164 .