Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.

a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
b. What proportion of the vehicles would be going less than 50 mph?
c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?
d. In what way do you think the actual distribution of speeds differs from a normal distribution?

a. Using our z-score calculator, we see that P(x<65) = 22.66%

b. Using our z-score calculator, we see that P(x<50) = 0.4269%

c. Inverse of normal for 90% percentile = 1.281551566
Plug into z-score formula: (x - 71)/8 = 1.281551566
x = 81.25241252

d. The shape/ trail differ because the normal distribution is symmetric with relatively more values at the center. Where the actual has a flatter trail and could be expected to occur.