Question

Suppose the average length of a TTC delay is 30 minutes with a standard deviation of 5.5 minutes. A local business estimates that the cost to their business associated with these delays 150H^2 −8.50, where H is the duration of a TTC delay in hours. How much will a TTC delay cost, at least 70% of the time? Show all your work and include any annotations you think would be helpful in explaining your process. You may find it helpful to know that E[H^4] = 0.07.

Answer 1

TTC delay cost is given as,

X = 150H^2 −8.50

Given,

E[H] = 30 minutes = 0.5 hour

SD[H] = 5.5 minutes = 0.09167 hour

Var[H] = 0.09167² = 0.0084

E[H^2] = Var[H] + [E[H]]² = 0.0084 + 0.5² = 0.2584

E[X] = E[150H^2 −8.50] = 150 * E[H^2] - 8.50 = 150 * 0.2584 - 8.50 = 30.26

E[X²] = E[(150H^2 −8.50)²] = E[150² H^4 + 8.50² - 2 * 150 H^2 * 8.50]

= 22500 E[H^4] + 72.25 - 2550 E[H^2]

= 22500 * 0.07 + 72.25 - 2550 * 0.2584 = 988.33

Var[X] = E[X²] - E[X]²  = 988.33 - 30.26² = 72.6624

SD[X] = = 8.524224

Using normal distribution, z value for p = 0.70 is  0.5244

TTC delay cost = 30.26 + 0.5244 * 8.524224 =  34.73