Statistical Inference and t-Distribution: Formulas for Standard Errors and Confidence Intervals

1. STAT 101 Dr. Kari Lock Morgan Inference Using Formulas • Chapter 6 • t-distribution • Formulas for standard errors • Normal and t based inference • Matched pairs

2. Confidence Interval Formula • IF SAMPLE SIZES ARE LARGE… From N(0,1) From original data From bootstrap distribution

3. Formula for p-values • IF SAMPLE SIZES ARE LARGE… From original data From H0 From randomization distribution Compare z to N(0,1) for p-value

4. Standard Error • Wouldn’t it be nice if we could compute the standard error without doing thousands of simulations? • We can!!!

5. Standard Error Formulas

6. SE Formula Observations • n is always in the denominator (larger sample size gives smaller standard error) • Standard error related to square root of 1/n • Standard error formulas use population parameters… (uh oh!) • For intervals, plug in the sample statistic(s) as your best guess at the parameter(s) • For testing, plug in the null value for the parameter(s), because you want the distribution assuming H0 true

7. Null Values • Single proportion: H0: p = p0=> use p0 for p • Difference in proportions: H0: p1 = p2 • use the overall sample proportion from both groups (called the pooled proportion) as an estimate for both p1 and p2 • Means: Standard deviations have nothing to do with the null, so just use sample statistic s • Correlation: H0:ρ= 0 => use ρ = 0

8. t-distribution • For quantitative data, we use a t-distributioninstead of the normal distribution • This arises because we have to estimate the standard deviations • The t distribution is very similar to the standard normal, but with slightly fatter tails (to reflect the uncertainty in the sample standard deviations)

9. Degrees of Freedom • The t-distribution is characterized by itsdegrees of freedom (df) • Degrees of freedom are based on sample size • Single mean: df = n – 1 • Difference in means: df = min(n1, n2) – 1 • Correlation: df = n – 2 • The higher the degrees of freedom, the closer the t-distribution is to the standard normal

10. t-distribution

11. Aside: William Sealy Gosset

12. Matched Pairs • A matched pairs experiment compares units to themselves or another similar unit • Data is paired(two measurements on one unit, twin studies, etc.). • Look at the difference for each pair, and analyze as a single quantitative variable • Matched pairs experiments are particularly useful when responses vary a lot from unit to unit; can decrease standard deviation of the response (and so decrease the standard error)

13. Golden Balls: Split or Steal? http://www.youtube.com/watch?v=p3Uos2fzIJ0 Both people split: split the money One split, one steal: stealer gets all the money Both steal: no one gets any money Would you split or steal? • Split • Steal Van den Assem, M., Van Dolder, D., and Thaler, R., “Split or Steal? Cooperative Behavior When the Stakes Are Large,” available at SSRN: http://ssrn.com/abstract=1592456, 2/19/11.

14. To Do • Do Project 1 (due Friday, 3pm) • Read Chapter 6 • Do HW 5 (due Wednesday, 3/19)