lecture 15 dustin lueker. the width of a confidence interval ◦ increases as the confidence level...

STA 291Summer 2010

Lecture 15Dustin Lueker

The width of a confidence interval◦ Increases as the confidence level increases◦ Increases as the error probability decreases◦ Increases as the standard error increases◦ Increases as the sample size n decreases

Facts about Confidence Intervals

2STA 291 Summer 2010 Lecture

15

n≥30

n<30

Confidence Intervals for μ

STA 291 Summer 2010 Lecture 15 3

n

sZx 2/

n

stx 2/

Start with the confidence interval formula assuming that the population standard deviation is known

Mathematically we need to solve the above equation for n

Choice of Sample Size

4

MExn

sZx 2/

2

2/2

ME

Zsn

STA 291 Summer 2010 Lecture 15

Confidence Interval for a Proportion The sample proportion is an unbiased and

efficient estimator of the population proportion◦ The proportion is a special case of the mean

5

n

ppZp

)ˆ1(ˆˆ 2/


6

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


p̂

4

2~

n

xp

4

)~1(~~2

n

ppZp

Sample Size As with a confidence interval for the sample

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

◦ This formula requires guessing before taking the sample, or taking the safe but conservative approach of letting = .5 Why is this the worst case scenario? Or the

conservative approach?

7

2

2/)ˆ1(ˆ

ME

Zppn

p̂

p̂


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


15

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


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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

10

2

22

1

21

2/21 )(n

s

n

sZxx


Example In the 1992 General Social Survey, 350

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

In the 2004 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 1992 and 2004. Is it plausible that the mean was the same in both

years?


15

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

12

2

22

1

112/21

)ˆ1(ˆ)ˆ1(ˆˆˆ

n

pp

n

ppZpp


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?


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lecture 15 dustin lueker. the width of a confidence interval ◦ increases as the confidence level...

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