STAT 330 Lecture 10

Reading for Today's Lecture: 9.1, 9.2, 9.4

Goals of Today's Lecture:

• Describe small sample CIs and HTs for difference of two means.
• Describe CIs and HTs for differences of two proportions.

Today's notes

Theorem: If

1. X and Y populations normal
2. X and Y samples independent
3. 

then:

1. 

   has exactly a standard normal distribution.

2. 

   has a t distribution on m+n-2 degrees of freedom.

Example: Hypertension data as before but assume m=10 (treatment) and n=15 (control). Then

which works out to 331.52 so that .

Now get

and, from t tables with 9+14=23 df we get a (one-sided) P value of about 10% which is mild evidence against the null hypothesis of no treatment effect.

A confidence interval for the effect of the treatment is

which works out to

or

Two Proportions

Example 1: two independent surveys, one of 400 in Quebec and 625 in the rest of Canada find 65% and 75% support for Capital Punishment respectively.

Example 2: two groups of patients, on of 30 receives placebo and other of 30 receives new pain killer as treatment for headache. In the control group 12 report a decrease in pain while 18 in the treatment report a decrease in pain.

Model: X has a Binomial distribution and Y has a Binomial distribution. X and Y are independent.

In the examples we observe X=240 and Y=468 (for experiment 1) and X=12 and Y=18 (for experiment 2).

The model applies to the following situation:

• Two independent samples or experiments.
• First experiment has m Bernoulli trials, that is, m independent trials each with the same probability, of ``Success''.
• Second experiment has n Bernoulli trials, each with probability, of ``Success''.

Natural estimate of is

where and .

Properties of :

• 

• 

• The standard error of is the square root of the last quantity.
• If m and n are large then has approximately a normal distribution. (It is a linear combination of two things which each are approximately normal; linear combinations of independent things are more normal than the things being combined.)

This leads to the following confidence intervals:

To compute the estimated standard error we just replace each with the obvious estimate:

In our survey example we get

which works out to

for a 95% confidence interval for the difference between the true levels of support in the two regions.

Hypothesis Tests

The only null hypothesis which is at all common is the corresponding usual test statistic is

Crucial Point: The null hypothesis specifies . We use the notation p for this common value of the two probabilities. We can then use the assumption that is true to get a better estimate of p, namely:

which becomes

In our example we had and Y=468 and get

and

which leads to a miniscule P value and the clear conclusion that the support for CP is definitely lower in Quebec than in the rest of Canada.