Problems 1-5 in notes.001.

Problem 5 and Problem 6 in notes.004.

Problem 9, notes.007, and Problem 10, notes.008.

Problem 11, notes.010.

Problem 12, notes.011.

Problem 13, notes.012.

Problem 13a, notes.012

Write proofs of the following (where A, B are 2 x 2 matrices):

1. (A * B)' = B' * A'.

2. (inv(A))' = inv(A').

3. inv(A * B) = inv(B) * inv(A).

4. det(A * B) = det(B * A).

Problem 15, parts 1. and 2., notes.017

Problem 16, parts 1. and 2., notes.017

Problem 17, parts 1. and 2., notes.018

Problem 18, notes.019

Problem 19, notes.020

Problem 20, notes.021

Problem 21, notes.022

Problem 22, notes.022

Problem 23 and Problem 24, notes.023

Exact confidence interval problem:

Assume you have a sample of size n, and that m people in this sample have a specified characteristic. The sample proportion of people who have this characteristic is denoted by pobs. It has the distribution of a binomial proportion, i.e., the number m has a binomial distribution binom(n, p), where p is the true (and unknown) proportion. The object of this problem is to write an algorithm in SAS or Splus or R which computes an exact confidence interval (plower, pupper) for the true proportion p, given the sample data (n, m).

This is explained in a 1934 paper in Biometrika by Clopper and Pearson.

Clopper and Pearson show that an exact 95% confidence interval has the following property:

If plower is the lower bound for the exact confidence interval, then if the true proportion were equal to plower, the probability that you would observe a sample proportion of size pobs or lower is .975.

If pupper is the upper bound for the exact confidence interval, then if the true proportion were equal to pupper, the probability that you would observe a sample proportion of size pobs or higher is .975.

Given a value for plower, you can find the desired probability by using the cumulative binomial distribution function: in SAS, you would use the function probbnml.

You can find plower by starting with a guess at what it should be (for example, a good starting value might be pobs/2), and then doing a binary search to improve the guess.

So the assignment is to write a program (or a macro) which has as input: n, m, and alpha, and has as output the lower and upper (1 - alpha) exact confidence interval limits, plower and pupper.

Test your program with the following values for n and m:

n = 100, m = 40

n = 1000, m = 30

n = 10000, m = 5

n = 100000, m = 5

n = 100000, m = 1

For of these values, compare your results with the confidence interval that can be obtained with the usual normal approximation.

Also: Problem 26, notes.024

Homework #12 - due December 14, 2004.

Problem 28, parts 1. and 2., notes.026

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Web address: http://www.biostat.umn.edu/~john-c/assign5421.s2004.html

Most recent update: December 8, 2004