Subject: Belated suggestions for today's recitation sections

From: "Gary Lorden" <[email-masked]>
Date: Thu, 22 Feb 2007 09:08:27 -0800

Sorry to be so late suggesting some things, but in case
it might help...

You probably will be asked about t-tests and should prepare
to run through an example of the two-population version and/or
the paired x/y version, which by taking differences becomes the
one-population version.

To help with problem 4 where they're asked to derive a test statistic,
etc., I'd suggest doing an easy problem like testing a hypothesis
about the variance of a normal distribution with mean zero-- where
the variance is --say-- less than or equal to 5 under the null.  The
book tells how to do a chi-square test for it, but you can derive the
statistic using the likelihood ratio of "variance = a fixed value >5"
to "variance =5".   They might ask "how do you know in a problem
(like number 3) which hypothesis should be the null hypothesis?"
The best answer is "consider the value of 'theta' that defines the
boundary between the hypotheses, and decide the answer to
the question 'assuming that value is true, which decision do you
want to make 95% of the time?'"   That's the null hypothesis.

The one problem on confidence intervals, that I just started talking
about in lecture yesterday, will surely lead to questions.   Might want
to do an easy example of a standard interval (from the book?).  And
to help them approach number 5, I'd suggest working the following

You have 5 iid observations from a density that's symmetrical around
an unknown 'theta'-- so that theta is the median of the distribution.
Show that the interval  [2nd smallest of the 5 X's,  2nd largest of the
5 X's] is a p% confidence interval for theta-- and determine p.

The answer is p=52/64 (X100) %, derived by calculating

Prob(2nd largest is >= theta)= Prob(at least 2 of the 5 X's are > median)=
Prob(at least 2 successes in 5 flips of a fair coin)= 6/64,  and a
similar calculation for Prob(2nd smallest is <= theta).  This is very
similar
to the technique needed for number 5.

G.L.