Lecture 16 Dustin Lueker

 n≥30  n<30 STA 291 Summer 2008 Lecture 162

 Start with the confidence interval formula assuming that the population standard deviation is known  Mathematically we need to solve the above equation for n 3STA 291 Summer 2008 Lecture 16

 The sample proportion is an unbiased and efficient estimator of the population proportion ◦ The proportion is a special case of the mean 4STA 291 Summer 2008 Lecture 16

 As with a confidence interval for the sample mean a desired sample size for a given margin of error (E) 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? 5STA 291 Summer 2008 Lecture 16

 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 6STA 291 Summer 2008 Lecture 16

 Take independent samples from both groups  Sample sizes are denoted by n 1 and n 2 ◦ To use the large sample approach both samples should be greater than 30  Subscript notation is same for sample means 7STA 291 Summer 2008 Lecture 16

 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  Is it plausible that the mean was the same in both years? 8STA 291 Summer 2008 Lecture 16

 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 p 1 -p 2 9STA 291 Summer 2008 Lecture 16

 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 10STA 291 Summer 2008 Lecture 16

 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 (UK fans), 3 partners became infected with HIV  55 couples who did not always use a condom (U of L fans), 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? 11STA 291 Summer 2008 Lecture 16