Psychology and Human Behavior Parents with children under age 16 often spend a lot of time during the day driving their kids to various places, for example, to/from after-school activities, music practice, sports practices and games, the library, and a friend's home. Suppose a family has k child(ren) under 16 (k = 1, 2, 3, 4, 5), and let the random variable Xk be the time (in hours) spent taxiing during the day. Xk has a uniform distribution with a = 0 and b = k. For example, for a family with two children, X2 has a uniform distribution with a = 0 and b = 2.
a. For a family with three children, what is the probability that parents will spend less than one hour driving kids on a randomly selected day?
b. For a family of four children, what is the mean number of hours spent driving kids? What is the probability that the driving time will be greater than two standard deviations from the mean?
c. For a family with five children under 16, find a time t such that the probability of driving kids more than t hours is 0.25.
d. Suppose five families are selected at random, the first with one child under 16, the second with two children under 16, etc. What is the probability that all five families drive less than 30 minutes on a randomly selected day? What is the probability that all five families drive more than 90 minutes on a randomly selected day?