Question

Is lack of sleep causing traffic fatalities? A study conducted under the auspices of the National Highway Traffic Safety Administration found that the average number of fatal crashes caused by drowsy drivers each year was 1550 (Business Week, January 26, 2015). Assume the annual number of fatal crashes per year is normally distributed with a standard deviation of 300 and a mean of 1550.

1.What is the probability of the number of fatal crashes will be between 1000 and 2000 for a year?

2. For a year to be in the top 2.5% of the distribution with respect to the number of fatal crashes, how many fatal crashes would have to occur?

Answer 1

Solution :

Given that,

mean = = 1550

standard deviation = = 300

1 ) P (1000 < x < 331 )

P ( 1000 - 1550 / 300) < ( x -  / ) < ( 2000 - 1550 / 300)

P ( - 550 / 300 < z < 450 / 300 )

P (-1.83 < z < 1.5 )

P ( z < 1.5 ) - P ( z < -1.83 )

Using z table

= 0.9332 - 0.0336

= 0.8996

Probability = 0.8996

2 ) P( Z > z) = 2.5%

P(Z > z) = 0.025

1 - P( Z < z) = 0.025

P(Z < z) = 1 - 0.025

P(Z < z) = 0.975

z = 1.96

Using z-score formula,

x = z * +

x = 1.96 * 300+ 1550

x = 2138