STAT 330 Lecture 9

Reading for Today's Lecture: 6.1, 9.1, 9.2

Goals of Today's Lecture:

• Introduce some jargon of point estimation
• Compute standard error for difference of two means.
• Describe large sample CIs and HTs for difference of two means.
• Introduce small sample case.
• Define pooled estimate of SD.

Today's notes

Two Independent Samples

Model: iid mean and SD (or , );

iid mean and SD (or , );

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

Parameter of interest:

Natural point estimate:

Jargon:

• Point Estimate: 1 number = best guess
• Interval estimate: e.g. CI

Properties of estimate

Jargon: is unbiased.

so that

is the Standard Error of .

Theorem:

1. If m and n are both large then

   

   has approximately a standard normal distribution.

2. The same is true if each is replaced by the corresponding sample SD, , i=1,2.

   

Randomized Trial Example:

Recall two samples: measure blood pressure (in mm of Hg):

and a confidence interval for is

which is roughly .

Test versus the only interesting alternative .

so

Conclusion: Strong evidence that BP is reduced by this treatment.

What if m, n not large?

Need extra assumptions:

• Normal populations (both).
• : equal population variances.

In this case let denote the common variance. How do we estimate ?

Solution: Pooled estimate:

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.