1. STA 291 Spring 2008
Lecture 16
Dustin Lueker

2. Confidence Intervals for μ
σ known
σ unknown, n≥30
σ unknown, n<30
STA 291 Spring 2008 Lecture 16

3. Choice of Sample Size
Start with the confidence interval formula that includes the population standard deviation
Mathematically we need to solve the above equation for n
STA 291 Spring 2008 Lecture 16

4. 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
Only applies to large samples (n≥30)
STA 291 Spring 2008 Lecture 16

5. Sample Size
As with a confidence interval for the sample mean a desired sample size for a given margin of error (B) 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?
STA 291 Spring 2008 Lecture 16

6. 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
STA 291 Spring 2008 Lecture 16

7. 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
STA 291 Spring 2008 Lecture 16

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?
STA 291 Spring 2008 Lecture 16

9. 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
STA 291 Spring 2008 Lecture 16

10. 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
STA 291 Spring 2008 Lecture 16

11. 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?
STA 291 Spring 2008 Lecture 16