Data is collected on the time between arrivals of consecutive taxis at

a downtown hotel. We collect a data set of size 45 with sample mean x = 5.0 and sample

standard deviation s = 4.0.

(a) Assume the data follows a normal random variable.

(i) Find an 80% confidence interval for the mean of X.

(ii) Find an 80% ?

2

-confidence interval for the variance?

(b) Now make no assumptions about the distribution of of the data. By bootstrapping,

we generate 500 values for the differences ?

? = x

? ? x. The smallest and largest 150 are

written in non-decreasing order on the next page.

Use this data to find an 80% bootstrap confidence interval for .

(c) We suspect that the time between taxis is modeled by an exponential distribution, not

a normal distribution. In this case, are the approaches in the earlier parts justified?

(d) When might method (b) be preferable to method

A matrix P = | | Pull_1 is called stochastic if : (i) Pi; 2 0 for all i and j = 1, 2, ... and (ii) _ Puj = 1 for all i = 1, 2, ... =1 A matrix P is called doubly stochastic if it satisfies (i) and (ii) above, in addition have [ Pis = 1, for all j = 1, 2, ... i=1 Show that if a finite-state irreducible Markov chain has a doubly stochastic transition probability matrix, then all the stationary probabilities are equal.Consider the contingent states insurance Model. John faces the following probability distribution of material loss L = (0, 1000; a, 1 - a), where a is the probability of the normal state, and 1 a is the probability of the bad state. John has an initial wealth of W0 = $10,000, and his utility function u(x) = In x, where ln denotes natural log, and x is wealth. Assume his optimal coverage is partial coverage. If a is reduced 0 John faces an unfavorable first-degree stochastic dominance (FSD) shift in the distribution of Consider an episodic MDP with one state and two actions (left and right). The left action has stochastic reward 1 with probability p and 3 with probability 1-p. The right action has stochastic reward 0 with probability q and 10 with probability 1-q. What relationship between p and q makes the actions equally optimal? C) 13+2p=40q C) 7+3p=10q C) 7+2p=40q C) 13+3p=10q C) 13+3p=40q C) 13+2p=1oq O 7+2p=10q 3. Assuming degrees of freedom = 22, and you are interested in obtaining a confidence interval for the middle 90% of values, what would your / value be? 4. According to national employment statistics, employees at Company Y, a local manufacturing company, earned a starting salary of u= $50,000 with o= $10,440. Assume you take a sample of 36 new employees. What is the probability the average salary of this sample will be between $48,000 and $52,000