STA 291Winter 09/10

Lecture 8Dustin Lueker

Comparison of Two Groups Two independent samples

◦ Different subjects in the different samples◦ Two subpopulations

Ex: Male/Female◦ The two samples constitute independent samples from two

subpopulations Two dependent samples

◦ Natural matching between an observation in one sample and an observation in the other sample Ex: Two measurements of the same subject

Left/right hand Performance before/after training

◦ Important: Data sets with dependent samples require different statistical methods than data sets with independent samples

2STA 291 Winter 09/10 Lecture 8

Confidence Interval for the Difference of Two Means Take independent samples from both groups Sample sizes are denoted by n1 and n2

◦ To use the large sample approach both samples should be greater than 30

Subscript notation is same for sample means

3

2

22

1

21

2/21 )(n

s

n

sZxx

STA 291 Winter 09/10 Lecture 8

Example In the 1982 General Social Survey, 350

subjects reported the time spend every day watching television. The sample yielded a mean of 4.1 and a standard deviation of 3.3.

In the 1994 survey, 1965 subjects yielded a sample mean of 2.8 hours with a standard deviation of 2.◦ Construct a 95% confidence interval for the

difference between the means in 1982 and 1994. Is it plausible that the mean was the same in both

years?

4STA 291 Winter 09/10 Lecture 8

Comparing Two Proportions For large samples

◦ For this we will consider a large sample to be those with at least five observations for each choice (success, failure) All we will deal with in this class

Large sample confidence interval for p1-p2

5

2

22

1

112/21

)ˆ1(ˆ)ˆ1(ˆˆˆ

n

pp

n

ppZpp

STA 291 Winter 09/10 Lecture 8

When would this be useful? Is the proportion who favor national health

insurance different for Democrats and Republicans?◦ Democrats and Republicans would be your two samples◦ Yes and No would be your responses, how you’d find

your proportions Is the proportion of people who experience pain

different for the two treatment groups?◦ Those taking the drug and placebo would be your two

samples Could also have them take different drugs

◦ No pain or pain would be your responses, how you’d find your proportions

6STA 291 Winter 09/10 Lecture 8

Example Two year Italian study on the effect of condoms

on the spread of HIV◦ Heterosexual couples where one partner was infected

with HIV virus 171 couples who always used condoms, 3 partners

became infected with HIV 55 couples who did not always use a condom, 8 partners

became infected with HIV◦ Estimate the infection rates for the two groups◦ Construct a 95% confidence interval to compare them

What can you conclude about the effect of condom use on being infected with HIV from the confidence interval? Was your Sex Ed teacher lying to you?

7STA 291 Winter 09/10 Lecture 8

8

Confidence Interval for p To calculate the confidence interval, we

use the Central Limit Theorem (np and nq ≥ 5)◦ What if this isn’t satisfied?

Instead of the typical estimator, we will use

Then the formula for confidence interval becomes

STA 291 Winter 09/10 Lecture 8

p̂

4

2~

n

xp

4

)~1(~~2

n

ppZp

Suppose a student in an advertising class is studying the impact of ads placed during the Super Bowl, and wants to know what the proportion of students on campus watched it. She takes a random sample of 25 students and finds that all 25 watched the Super Bowl. ◦ Find a 95% confidence interval using first method

learned for p◦ Find a 95% confidence interval using the new

method if np, nq condition fails

Example

STA 291 Winter 09/10 Lecture 8 9