A study carried out to investigate the distribution of total

A study carried out to investigate the distribution of total braking time (reaction time plus accelerator-to-break movement time, in ms) during real driving conditions at 60 km/hr gave the following summary information on the distribution of times (\"A Field Study on Braking Responses Dring Driving,\" Ergonomics, 1995: 1903-1910): mean = 535 , s = 80 . Assume the braking times have approximately a bell shaped distribution.

A) Approximately what percentage of braking times are in the interval (535, 615)?

B) What value do the top 2.5% of braking times exceed?

Solution

let X be the random variable denoting the braking times.

Assume the braking times have approximately a bell shaped distribution(normal distribution) with mean = 535 , s = 80

so X~N(535,80²)

A) we need to find Approximately what percentage of braking times are in the interval (535, 615)

so P[535<X<615]=P[(535-535)/80<(X-535)/80<(615-535)/80]=P[0<Z<1] where Z~N(0,1)

                          =P[Z<1]-P[Z<0]=0.841345-0.5=0.341345 [answer]  

B) let M be the value do the top 2.5% of braking times exceed

so P[X>M]=0.025

or, 1-P[X<M]=0.025

or, P[X<M]=0.975

or P[(X-535)/80<(M-535)/80]=0.975

or P[Z<(M-535)/80]=0.975=P[Z<1.96]   using MINITAB   where Z~N(0,1)

so (M-535)/80=1.96 or M=691.8

so the required value is 691.8 km/hr [answer]