Homework 4  Due March 30

1. Suppose we have the following independent observations

-5.42  3.23  4.74 -0.07  4.48  0.59 -8.82  5.56 -0.34

from a Normal distribution N ( \mu, \sigma )
where both \mu and \sigma are unknown parameters.

a. Find a 90% joint confidence region for parameters ( \mu, \sigma )
(Use either Wilks or Wald pivotal, your choice). Sketch your region
on a picture. 

b. suppose we know \sigma = 3, and only the \mu is the unknow parameter
Find a 90% confidence interval for \mu.



2. Suppose the following data are from a Gamma distribution

0.37 2.72 2.96 1.31 0.67 1.58 3.19 1.32 0.62 0.50 2.48 1.87 1.46 0.81 2.26

with two parameters (\alpha, \lambda).
(the density of a gamma, see the appendix of the online book we use.)

a. Use the Wilks pivotal.  Find a 90% confidence region for 
parameters (\alpha, \lambda). Sketch the region on a picture.

b. Now suppose \lambda is known to be 2.4, only the \alpha is the 
parameter. Find a 90% confidence interval for \alpha.

Notice the gamma function is available in R as  gamma(5)  etc.

(On bottom of page 347 of our book, there is a 
likelihood function of gamma written in R, 
except I do not like the name: l --- this is "el" not one.
may be we should call it loglik instead of l.)