Question

1) 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 47 mph and that the standard deviation was 17 mph. A histogram revealed that the vehicle speed at impact distribution was approximately normal.

•(a) Roughly what proportion of vehicle speeds were between 30 and 64 mph? (Round the answer to the nearest whole number.)

•(b) Roughly what proportion of vehicle speeds exceeded 67 mph? (Round the answer to the nearest whole number.)

•(c) For these 5,000 fatal automobile accidents, what is the speed for which about 85% of drivers were driving below this speed? In other words, find the 90th percentile.

2) The percentage of juice lost after thawing for 19 different strawberry varieties appeared in an article about their different degrees of resistance to red scale insects. They are given : 48, 53, 50, 35, 48, 60, 43, 56, 48, 55, 47, 42, 50, 52, 48, 43, 41

•(a) Find the Five Number Summary

•(b) Find the Interquartile Range

Answer 1

let X denotes the average speed of cars of 5000 car accidents in mph

then from histogram X~N(47,17²) approximately.

a)

so the probability that a vehicle's spped was between 30 and 64 mph is

P[30<X<64]=P[(30-47)/17<(X-47)/17<(64-47)/17]=P[-1<Z<1] where Z~N(0,1)

=P[Z<1]-P[Z<-1]= 0.8413447-0.1586553=0.6826894  

so approximately 5000*0.6826894=3413.447=3413 cars had speed between 30 and 64 mph

b) the probability of cars exceeding speed 67 mph is

P[X>67]=P[(X-47)/17>(67-47)/17]=P[Z>1.176471]=1-P[Z<1.176471]=1-0.8802966=0.1197034

so roughly 5000*0.1197034=598.517=599 cars had speed exceeding 67 mph

c) let the required speed be k mph

then P[X<k]=0.85

P[(X-47)/17<(k-47)/17]=0.85

or, P[Z<(k-47)/17]=0.85=P[Z<1.036433]

so (k-47)/17=1.036433

or, k=47+17*1.036433=64.61936

so the required speed is 64.61936 mph