Math/Stat 571B Spring 2009

• Syllabus, including course policies
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Projects

• Project 1

Homework Assignments

Homework 1 - Due January 27, 2009

Homework 2 - Due February 5, 2009

Problem 1

Homework 3 - Due March 3, 2009

• Please write your homework in fully electronic format.
• Use LaTeX (preferred), Word 2003-, OpenOffice 2.2+ or TeXmacs to typeset your main document. If submitting in LaTeX or TeXmacs, please generate a PDF of your main document.
• Attach all programs and supporting materials.
• Build a zip archive of your solution.
• Submit via e-mail with "Subject: Math571B_H3".

Homework 4 - Due April 7, 2009

• Submit electronically as per instructions of Homework 3.

• Submit electronically as per instructions of Homework 3-4.

• Submit electronically as per instructions of Homework 3-5.

Midterm 1 Questions

• Describe the difference (if any) between an experiment and an observational study.
• What is the primary goal of statistics in the context of experimental design?
• Give an example of things that are under the control of the researcher which lead to reduction of the variance of the experimental error.
• Describe the difference between a completely randomized design and randomized complete block design.
• In some experiment, the following responses were received from two treatment groups:

  Treatment Group	Response level
  A	1
  B	2
  A	3
  B	7

  Should the hypothesis H₀, that the means of both groups are the same, be accepted or rejected with confidence level .7?
• Define Fisher information.
• What is the Fisher information I(μ) where μ is the mean of the family of normal distributions f(x|μ) with mean μ and fixed variance σ²?
• Let I(μ) be the Fisher information for the family of normal distributions f(x|μ) with a fixed variance σ². Why does I(μ) NOT depend on μ?
• Graphically justify the formula for the number of replications:

  r ≥ 2(z_α/2+z_β)²(σ/δ)²

• (Multiple choice) The χ²-distribution is an example of a
  1. Γ-distribution;
  2. β-distribution;
  3. Laplace distribution;
  4. None of the above.
• Give an argument supporting the use of the sample mean over the median as an estimator of the population mean, when the distribution is a normal distribution.
• The difference of the means between treatments A and B is tested for equality, using the Student t-test. There are r₁ observational units in treatment group A and r₂ observational units in treatment group B. What is the number of degrees of freedom used in the Student t-distribution?
• What distribution do we use to estimate the power of the Student t-test?
• True or false: When the population distribution is normal, the sample mean and sample variance are independent random variables.
• True or false: For all population distributions, the sample mean and sample variance are independent random variables.
• True or false: Two normal random variables are independent iff they are uncorrelated.
• (Source: Wikipedia)

  In 1747, while serving as surgeon on HM Bark Salisbury, James Lind, the ship's surgeon, carried out a controlled experiment to develop a cure for scurvy.

  Lind selected 12 men from the ship, all suffering from scurvy, and divided them into six pairs, giving each group different additions to their basic diet for a period of two weeks. The treatments were all remedies that had been proposed at one time or another. They were:

  • A quart of cider every day
  • Twenty five gutts of elixir vitriol three times a day upon an empty stomach,
  • One half-pint of seawater every day
  • A mixture of garlic, mustard, and horseradish in a lump the size of a nutmeg
  • Two spoonfuls of vinegar three times a day
  • Two oranges and one lemon every day.

  The men who had been given citrus fruits recovered dramatically within a week. One of them returned to duty after 6 days and the other became nurse to the rest. The others experienced some improvement, but nothing was comparable to the citrus fruits, which were proved to be substantially superior to the other treatments.

  In this study his subjects' cases "were as similar as I could have them", that is he provided strict entry requirements to reduce extraneous variation. The men were paired, which provided replication.

  Please find one problem with this experimental design from the point of view of modern statistical theory.