1. STA 291 Spring 2008
Lecture 15
Dustin Lueker

2. Confidence Intervals for μ
σ known
σ unknown, n≥30
σ unknown, n<30
STA 291 Spring 2008 Lecture 15

3. Facts about Confidence Intervals
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
STA 291 Spring 2008 Lecture 15

4. 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 15

5. Confidence Interval for Unknown σ
To account for the extra variability of using the sample standard deviation instead of the population standard deviation the student’s t-distribution is used instead of the normal distribution
α=.02, n=13
tα/2=
STA 291 Spring 2008 Lecture 15

6. 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 15

7. 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? (conservative approach)
STA 291 Spring 2008 Lecture 15

8. Example ABC/Washington Post poll (December 2006) Sample size of 1005
Question
Do you approve or disapprove of the way George W. Bush is handling his job as president?
362 people approved
Construct a 95% confidence interval for p
What is the margin of error?
STA 291 Spring 2008 Lecture 15

9. Previous Example
If we wanted B=2%, using the sample proportion from the Washington Post poll, recall that the sample proportion was .36
n=2212.7, so we need a sample of 2213
What do we get if we use the conservative approach?
STA 291 Spring 2008 Lecture 15

10. Relationship Between Margin of Error and Sample
A confidence interval has been calculated with a sample size of 80
The calculated interval is (17, 21), meaning B=2
Based on this result, what sample size is needed to achieve a precision of B=1? Why?
What would happen to the margin of error if we doubled the sample size?
These properties also work for a confidence interval of the population mean
STA 291 Spring 2008 Lecture 15