1. Lecture 15 Dustin Lueker

2.  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 2STA 291 Summer 2010 Lecture 15

3.  n≥30  n<30 STA 291 Summer 2010 Lecture 153

4.  Start with the confidence interval formula assuming that the population standard deviation is known  Mathematically we need to solve the above equation for n 4STA 291 Summer 2010 Lecture 15

5.  The sample proportion is an unbiased and efficient estimator of the population proportion ◦ The proportion is a special case of the mean 5STA 291 Summer 2010 Lecture 15

6. 6  To calculate the confidence interval, we use the Central Limit Theorem (np and nq ≥ 5) ◦ What if this isn’t satisfied?  Instead of the typical estimator, we will use  Then the formula for confidence interval becomes STA 291 Summer 2010 Lecture 15

7.  As with a confidence interval for the sample mean a desired sample size for a given margin of error (ME) 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? 7STA 291 Summer 2010 Lecture 15

8.  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 8STA 291 Summer 2010 Lecture 15

9.  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 9STA 291 Summer 2010 Lecture 15

10.  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 10STA 291 Summer 2010 Lecture 15

11.  In the 1992 General Social Survey, 350 subjects reported the time spent every day watching television. The sample yielded a mean of 4.1 and a standard deviation of 3.3.  In the 2004 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 1992 and 2004.  Is it plausible that the mean was the same in both years? 11STA 291 Summer 2010 Lecture 15

12.  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 12STA 291 Summer 2010 Lecture 15

13.  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? 13STA 291 Summer 2010 Lecture 15