Find the percentage of vehicles passing through this construction zone that are(a) Exceeding speed limit (b) traveling at a speed of 55 and 60 miles per hour.

Question

a construction zone on a highway has a posted speed limit of 40 per hour. The speed of vehicles passing through this construction zone are normally distributed with a mean of 45 miles per hour and a standard deviation of 4 miles per hour. Find the percentage of vehicles passing through this construction zone that are

a) Exceeding the posted speed limit.

b) Traveling at speeds between of 55 and 60 miles per hour.

Answer 1

speed (X) is Normal with mean of 45 and st.dev of 4.

a) P(X>40)

First convert to z-score, then find the corresponding area using z-table.

z=(40-45)/4=-1.25

area accumulating to z-score of -1.25 is 0.1056

hence, those exceeding the speed limit = 1 - 0.1056 = 0.8944 or 89%

or

P(X>40) = P(z>(40-45)/4)

= P(z>-1.25)

= 1 - P(z<-1.25)

= 1 - 0.1056

= 0.8944

b) P(55<X<60)

Similarly, convert each speeds into z-scores, then find the area between the the accumulated areas (from the left) via subtraction.

z1 = (55-45)/4 = 2.5

area accumulating to the left of z-score of 2.5 is 0.9938

z2 = (60-45)/4 = 3.75

area accumulating to the left of z-score of 3.75 is essentially 1

hence, area in-between is 1 - 0.9938 = 0.0062 or 0.6%

or

P(55 < X < 60) = P((55-45)/4 < z < (65-45)/4)

= P(2.5 < z < 3.75)

= P(z<3.75) - P(z<2.5)

= 1 - 0.9938

= 0.0062