Pollard homework 8
Statistics 101106, Fall 1998
Pollard homework sheet 8
Due: Thursday 5 November
(8.1)
Rearrange the data from Homework problem 6.2 into a 2by12
(or 12by2) table, containing the counts of summonses
not delivered and counts of summonses (presumed) delivered for each month.
(The total numbers of summonses mailed for each month should then be one
of the marginal totals for the table.)

Carry out the chisquare test for noassociation between month of
delivery and fate of the summons (delivered or not).

Generate tables of ``expected'' values and the table of ``chi'' values.

What does the table of chi values suggest about the lack of fit of the
noassociation model? (The model
asserts a constant probability, across months, for a summons to be delivered.)
(8.2)
For this problem you will simulate observations on a categorical
variable, and calculate the X^2 statistic many times. The aim is to
produce a histogram for the X^2 distribution, and maybe even compare it with the
appropriate chisquare.

Put values 1,2, ..., k in column C1, for a k of your choosing. (I
used k=4.) Put probabilities p1, p2,...,pk, of your choosing in
column c2, making sure that they sum to one. Generate many samples of
size n (I used n=100 for testing purposes, but you might like a larger
n), from the discrete distribution defined by c1 and c2, then
determine the counts in each category. Calculate the X^2 statistic
for each of the many repetitions, and display them with a density
histogram.
[The macro shown below seems to work reasonably well. You
will need to adapt it to your purposes.]

Draw the density function for the chisquare distribution on k1
degrees of freedom. [One way: Fill a column with numbers like 0,
0.1,0.2,0.3,...; use the calc menu to calculate the density at those
points; use plot, with points connected, to draw a smooth curve.]
gmacro
hw8
# c1 contains values for discrete distribution, c2 contains probabilities
# generate samples of size 100 from the discrete distribution
# do it 7 times, storing in c4c10
Random 100 c4c10;
Discrete C1 C2.
# get a tally of the counts, overwrite columns c4c10
do k100 =1:7
let k99 = k100 + 3
stats ck99;
count ck99;
by ck99.
# calculate X^2, save values in c3
let c3(k100) = sum((ck99  100*c2)**2/(100*c2))
enddo
endmacro