Q1.

The dataset “Power Plant” records variables which the company’s engineers believe are important factors in the operation of the plant. The company is interested in maximizing net hourly electrical energy output (recorded as PE in the dataset). For each hour of energy output recorded, other variable “Temperature” (AT) in the range 1.81°C and 37.11°C is recorded.

Steps:

1. Run a linear regression model for PE over AT. Record the value for the slope beta bar 1 and take it as the actual population parameter beta 1.

2. For 1000 iterations:

a. Take 50 random samples from the dataset. Run the regression model and using the expression for CI for a beta 1, that we found in the lecture, find a 95% CI for beta 1.

b. Find what percentage of the CIs generated in step 2 would contain the beta 1 that you got in step 1.

Q2.

If X1 and X2 are independent random samples from the Uniform distribution U(0,1), by generating random samples find P (|X1 – X1| < 1/4).

Q3.

If Xis are independent random samples from the Beta distribution beta(1, 1 + theta), by generating random samples for 3 different values for theta find

 and show that the result is independent of theta.

b. Using the distribution in Q2, show that the result is even independent of the distribution of Xis.