STA 291 Spring 2008 Lecture 16 Dustin Lueker

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

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

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

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

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

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

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

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

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

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