Question

Assume the speed of vehicles along an open stretch of a certain highway in Texas that is not heavily traveled has an approximately Normal distribution with a mean of 71 mph and a standard deviation of 3.125 mph.

1. The current posted speed limit is 65 mph. What is the proportion of vehicles going above the current posted speed limit?
2. What proportion of the vehicles would be going less than 50 mph?
3. What proportion of the vehicles would be going between 60 and 75 mph?  
4. State authorities are cognizant of the road not being heavily trafficked, and that it can handle a higher speed limit. However, they now will implement a high fee for speeding over the new speed limit in order to ensure some level of overall safety. Speeds will be checked by radar. Assume the same Normal distribution of vehicle speeds continues into the future as in the past. What should be the new speed limit such that only about 10% of vehicles will be speeding over the new posted speed limit? Show all of your reasoning/work in answering this.

Answer 1

a) P(X > 65)

= P((X - )/ > (65 - )/)

= P(Z > (65 - 71)/3.125)

= P(Z > -1.92)

= 1 - P(Z < -1.92)

= 1 - 0.0274

= 0.9726

B) P(X < 50)

= P((X - )/ < (50 - )/)

= P(Z < (50 - 71)/3.125)

= P(Z < -6.72)

= 0.000

C) P(60 < X < 75)

= P((60 - )/ < (X - )/ < (75 - )/)

= P((60 - 71)/3.125 < Z < (75 - 71)/3.125)

= P(-3.52 < Z < 1.28)

= P(Z < 1.28) - P(Z < -3.52)

= 0.8997 - 0

= 0.8997

d) P(X < x) = 0.1

Or, P((X - )/ < (x - )/) = 0.1

Or, P(Z < (x - 71)/3.125) = 0.1

Or, (x - 71)/3.125 = -1.28

Or, x = -1.28 * 3.125 + 71

Or, x = 67