STAT 330 Lecture 11

Reading for Today's Lecture: 9.3

Goals of Today's Lecture:

• Learn to recognize matched pairs designs.
• Describe CIs and HTs for differences of two means in a matched pairs design.

Today's notes

Two sample problems covered so far (all of which have 2 independent samples:

• Large n,m general population.
• Small n,m normal population
• Large n,m proportions (0,1 populations)
• Large n,m Poisson data with both population means large: see assignment.

Matched Pairs

Examples: for each of a sample of n pitchers record

Are throwing arms longer?

Can we use

or

NO: we do not have independent samples so

Correct Solution Define so that . Let . Then we can translate our hypotheses:

and

After subtracting to get the 's this is a 1 sample problem with n D's.

Use

where n is the number of pitchers or pairs not the number of arms or

and use n-1 degrees of freedom for small n (assuming as always that the population of differences is approximately normal).

Example: 9 patients. Measure the concentration of T4 cells before and after administering some new drug.

Patient # Before After After - Before

1 114 110 -4

2 94 115 21

3 117 134 17

4 110 121 11

5 96 124 28

6 88 104 16

7 105 121 16

8 109 117 8

9 85 100 15

Average 102 114.9 12.9

SD 11.6 11.4 8.6

Paired t-test:

with 8 degrees of freedom. A one sided P value is about 0.1% so that there is a clear increase in T4 counts after treatment.

Wrong way:

where we have used

Notice that the P value, calculated now with 16 df will be much larger. The point is that the Before and After measurements on the same patient are related:

Notice that there is a trend upward to the right. Large values Before are associated with large values After.

Some Theory of Estimation

Concepts of Estimation

Setup: We have data and a model: a family of possible probability densities for .

Goal: Inference about a parameter which describes the population or true density.

Point Estimate: A function of the data whose values are `` in or near'' a set of possible values. Typically, is a real number and so is .

Notice: X's are random variables so is a random variable.

Problems of Point Estimation:

1. What is a ``good' estimate?
2. Given 2 estimates how can we tell which one is better?
3. How can we find even 1 sensible point estimate?