STA 291 Spring 2008 Lecture 15 Dustin Lueker

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

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

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

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

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

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

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

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

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