Statistics 4C03/6C03 - Test #1

2007-02-12 17:30-19:30


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.

  1. 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

  2. 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:


  1. 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.
  2. 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.
  3. 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.]

Statistics 4C03/6C03