Statistics 4C03/6C03 - Test #1
2007-02-12 17:30-19:30
- This test is to be written in the BSB Student Technology Centre. The duration of the test is 2 hours.
- Any aids and resources are permitted. You may consult any web pages but you may not use e-mail or communicate with anyone other than the instructor.
- When no particular method is specified and there is a choice, you are free to use any method you like.
Part A.
Analyse the following two examples. Include an ANOVA table and a 99% confidence interval for the residual variance. State your assumptions and your conclusions.
- The percentage of hardwood concentration in raw pulp,
the freeness and the cooking time of the pulp, were investigated for
their effects on the strength of paper.
Cooking time 1.5 hr |
|
Freeness |
|
|
350 |
500 |
650 |
% Hardwood |
10 |
96.6, 96.0 |
97.7, 96.0 |
99.4, 99.8 |
|
15 |
98.5, 97.2 |
96.0, 96.9 |
98.4, 97.6 |
|
20 |
97.5, 96.6 |
95.6, 96.2 |
97.4, 98.1 |
Cooking time 2.0 hr |
|
Freeness |
|
|
350 |
500 |
650 |
% Hardwood |
10 |
98.4, 98.6 |
99.6, 100.4 |
100.6, 100.9 |
|
15 |
97.5, 98.1 |
98.7, 96.0 |
99.6, 99.0 |
|
20 |
97.6, 98.4 |
97.0, 97.8 |
98.5, 99.8 |
- The following data come from a fractional factorial
design used to study the effect of temperature, pressure,
concentration and stirring rate on filtration rate in a chemical
process.
run |
temp |
press |
conc |
stir |
filtration |
1 |
- |
- |
- |
- |
45 |
2 |
+ |
- |
- |
+ |
100 |
3 |
- |
+ |
- |
+ |
45 |
4 |
+ |
+ |
- |
- |
65 |
5 |
- |
- |
+ |
+ |
75 |
6 |
+ |
- |
+ |
- |
60 |
7 |
- |
+ |
+ |
- |
80 |
8 |
+ |
+ |
+ |
+ |
96 |
Part B.
Consider a negative binomial distribution with the following
parameterization:
- Show that this distribution is in the exponential family if k
is known, but k is not a dispersion parameter as defined by
McCullagh & Nelder. Find the exponential family components: cumulant
function,
canonical parameter, canonical link, mean, and variance function.
- Here is a vector of 9 observations from this distribution:
yT = (4, 7, 6, 0, 1, 2, 3, 5, 11). Find 95% Wald and
Score confidence intervals for the mean, assuming k = 3.
Estimate k by moments.
- Show that in the limit as k goes to infinity with mu fixed,
this distribution becomes the Poisson. [Hint: You can take the limit
of the probability density function, or you can use the cumulant
function you found in (1) to get the cumulant-generating function in
terms of mu and show that it approaches the Poisson cumulant-generating
function as k goes to infinity.]