STAT 330 Lecture 8

Reading for Today's Lecture: 7.2 (see especially page 288 top half); some of today's lecture is covered only in exercises in the text. Look up Poisson in the index.

Goals of Today's Lecture:

• Describe Poisson model for rates.
• Derive CI from more complicated pivots
• Describe tests, CIs for rates.

Today's notes

Rates and the Poisson Distribution

Example: huge vat of bleach in mine. Count how many Cl atoms converted to Ar in 3 months, get say 25. Suppose standard physics model for Solar neutrino emission predicts rate of 160 per year. Is 25 too few?

Data: X = count of events in 3 months.

Model: X has a Poisson distribution.

Null hypothesis: (divide by 4 because 3 months is quarter of year).

Poisson approximations are used for the distribution of the total number of rare events in some time period provided the presence or absence of events in one time period is independent of the presence or absence in another time period:

• Radiation counts (eg. neutrinos)
• Leukaemia cases in a region
• Accidents at a corner

Recall: means

Fact: if is large then a Poisson( ) rv is approximately Normal. Recall that implies and . Conclusion:

and

Note on aggregation: if are iid Poisson( ) then is Poisson( ). This is why the CLT shows large means nearly normal. It is also why I don't need more than the aggregate count in the example; there is no more information about in more detailed records. But more detailed information concerning when the counts happened over the 3 months would help check the Poisson model.

Deriving CIs from Pivots

Method A:

Since

we solve to get

so that is a level CI for .

Method B:

Use

and solve the inequalities:

implies

The equation for -z replacing z leads to the same thing after you square it out and the two roots are

Now you have to check to see that the inequality is satisfied between the two roots (and not outside the range of the two roots) but this is true.

Example: for our count of X=25 we get using z=2:

Method A:

Method B

So the intervals are about the same width but shifted over from each other. Method B is slightly more accurate: the probability approximation is a bit better with B.

Hypothesis tests

To test versus we compute

and look up a 2 tailed P value in normal tables getting

Conclusion: Fairly strong evidence that the mean Solar Neutrino Count is not 160 per year.

Two Sample Problems

Experimental Designs:

A: Two independent samples:

• Sample 400 in Québec, 625 in Rest of Canada. (Numbers fixed in advance, not luck of draw.)
• Get 65%, 75% support for CP respectively.
• Is the difference real?
• How big is the difference?

B: Randomized Trial:

• 60 patients (hypertension, say)
• Pick 30 at random for treatment, rest control.
• measure blood pressure (in mm of Hg):

  

• Treatment reduces blood pressure?

C: Matched Pairs

• 60 patients (hypertension, say)
• Form 30 pairs of ``similar'' patients
• Pick 1 of each pair for treatment.

C: Sample of Pairs

• Sample of 100 baseball pitchers
• Measure length of throwing arm and length of non-throwing arm.
• Notice measurements come naturally in pairs.
• Measure stretching effect of throwing.

Designs A and B are analyzed as two independent samples:

Model: iid mean and SD (or , );

iid mean and SD (or , );

ASSUMPTION: X's and Y's independent of each other.

Designs C and D are analyzed by subtracting and they always have n=m. We do not have two independent samples in this case.

Now consider model for A, B -- two independent samples.

Parameter of interest:

Natural estimate: