STA 291Fall 2009

Lecture 17Dustin Lueker

◦ α=.02, n=16 tα/2=

Confidence Interval for µ

2

nstx 2/

STA 291 Fall 2009 Lecture 17

Sample Size As with a confidence interval for the sample

proportion, a desired sample size for a given margin of error (ME) and confidence level can be computed for a confidence interval about the sample mean

◦ Found solving for ME in following confidence interval formula

3STA 291 Fall 2009 Lecture 17

MExnstx 2/

22/2

MEtsn

4

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 Fall 2009 Lecture 17

p̂

42~

nxp

4)~1(~~

2

n

ppZp

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

5STA 291 Fall 2009 Lecture 17

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

6

2

22

1

21

2/21 )(ns

nsZxx

STA 291 Fall 2009 Lecture 17

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?7STA 291 Fall 2009 Lecture 17

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

8

2

22

1

112/21

)ˆ1(ˆ)ˆ1(ˆˆˆnpp

nppZpp

STA 291 Fall 2009 Lecture 17

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

9STA 291 Fall 2009 Lecture 17

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?

10STA 291 Fall 2009 Lecture 17