in a study investigating the effect of car speed on accident severity, 5000 reports of fatal automobile…

in a study investigating the effect of car speed on accident severity, 5000 reports of fatal automobile accidents were examined, and the vehicle speed at impact was recorded for each one. for these 5000 accidents, the average speed was 42 mph with standard deviation 15 mph. a histogram revealed that the vehicle speed at impact distribution was approximately normal.\n\n a. about how many of the 5000 fatal automobile accidents were between 27 and 57 mph?

in a study investigating the effect of car speed on accident severity, 5000 reports of fatal automobile accidents were examined, and the vehicle speed at impact was recorded for each one. for these 5000 accidents, the average speed was 42 mph with standard deviation 15 mph. a histogram revealed that the vehicle speed at impact distribution was approximately normal.\n\n a. about how many of the 5000 fatal automobile accidents were between 27 and 57 mph?

Answer

Explanation:

Step1: Calculate z - scores

For $x_1 = 27$, the z - score $z_1=\frac{x_1-\mu}{\sigma}=\frac{27 - 42}{15}=\frac{- 15}{15}=-1$. For $x_2 = 57$, the z - score $z_2=\frac{x_2-\mu}{\sigma}=\frac{57 - 42}{15}=\frac{15}{15}=1$.

Step2: Find the proportion between z - scores

Using the standard normal distribution table, the proportion of values between $z=-1$ and $z = 1$ is $P(-1<Z<1)=\Phi(1)-\Phi(-1)$. Since $\Phi(-z)=1 - \Phi(z)$, then $\Phi(-1)=1-\Phi(1)$. So $P(-1<Z<1)=\Phi(1)-(1 - \Phi(1)) = 2\Phi(1)-1$. Looking up $\Phi(1)$ in the standard - normal table, $\Phi(1)=0.8413$, so $P(-1<Z<1)=2\times0.8413 - 1=0.6826$.

Step3: Calculate the number of accidents

Multiply the proportion by the total number of accidents. The number of accidents between 27 and 57 mph is $n = 0.6826\times5000 = 3413$.

Answer:

3413