Question

The speeds of cyclists on city bikeways follow a normal distribution with mean 26.4 km/h and standard deviation 4.2 km/h. Find the probability that a cyclist travels a) at more than 30 km/h, b) at less than 20 km/h, c) between 20 and 30 km/h. d) What minimum speed will place a cyclist in the top fastest 5%? e) What range of speeds will place a cyclist in the central 90%?

Answer 1

Solution :

Given that ,

mean = = 26.4

standard deviation = = 4.2

(a)

P(x > 30) = 1 - P(x < 30)

= 1 - P[(x - ) / < (30 - 26.4) / 4.2]

= 1 - P(z < 0.86)

= 0.1949

(b)

P(x < 20) = P[(x - ) / < (20 - 26.4) / 4.2]

= P(z < -1.52)

= 0.0643

(c)

P(20 < x < 30) = P[(20 - 26.4)/ 4.2) < (x - ) /  < (30 - 26.4) / 4.2) ]

= P(-1.52 < z < 0.86)

= P(z < 0.86) - P(z < -1.52)

= 0.8051 - 0.0643

= 0.7408

(d)

Central z has two z values : -1.645 and +1.645

Using z-score formula,

x = z * +

x = -1.645 * 4.2 + 26.4 = 19.491

x = 1.645 * 4.2 + 26.4 = 33.309

Range = 19.491 to 33.309