Question

Vehicle speed on a particular bridge in China can be modeled as normally distributed.

If 5% of all vehicles travel less than 39.15 m/h and 10% travel more than 73.27 m/h, what are the mean and standard deviation of vehicle speed?

What is the probability that a randomly selected vehicle's speed is between 50 and 65 m/h?

What is the probability that a randomly selected vehicle's speed exceeds the speed limit of 70 m/h?

Answer 1

a) P(X < 39.15) = 0.05

or, P((X - )/ < (39.15 - )/) = 0.05

or, P(Z < (39.15 - )/) = 0.05

or, (39.15 - )/ = -1.645

or, = 39.15 + 1.645

P(X > 73.27) = 0.10

or, P((X - )/ > (73.27 - )/) = 0.10

or, P(Z > (73.27 - )/) = 0.10

or, P(Z < (73.27 - )/) = 0.90

or, (73.27 - )/ = 1.28

or, = 73.27 - 1.28

39.15 + 1.645 = 73.27 - 1.28

or, 2.925 = 34.12

or, = 11.66

= 39.15 + 1.645 * 11.66 = 58.33

b) P(50 < X < 65)

= P((50 - )/ < (X - )/ < (65 - )/ )

= P((50 - 58.33)/11.66 < Z < (65 - 58.33)/11.66)

= P(-0.71 < Z < 0.57)

= P(Z < 0.57) - P(Z < -0.71)

= 0.7157 - 0.2389

= 0.4768

c) P(X > 70)

= P((X - )/ > (70 - )/)

= P(Z > (70 - 58.33)/11.66)

= P(Z > 1)

= 1 - P(Z < 1)

= 1 - 0.8413

= 0.1587