Statistics: Unlocking the Power of Data Lock 5 Inference Using Formulas STAT 101 Dr. Kari Lock Morgan Chapter 6 t-distribution Formulas for standard errors Normal and t based inference Matched pairs

Statistics: Unlocking the Power of Data Lock 5 Confidence Interval Formula From original data From bootstrap distribution From N(0,1) IF SAMPLE SIZES ARE LARGE…

Statistics: Unlocking the Power of Data Lock 5 Formula for p-values From randomization distribution From H 0 From original data Compare z to N(0,1) for p-value IF SAMPLE SIZES ARE LARGE…

Statistics: Unlocking the Power of Data Lock 5 Standard Error Wouldn’t it be nice if we could compute the standard error without doing thousands of simulations? We can!!!

Statistics: Unlocking the Power of Data Lock 5 ParameterDistributionStandard Error Proportion Normal Difference in Proportions Normal Meant, df = n – 1 Difference in Meanst, df = min(n 1, n 2 ) – 1 Correlationt, df = n – 2 Standard Error Formulas

Statistics: Unlocking the Power of Data Lock 5 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 H 0 true

Statistics: Unlocking the Power of Data Lock 5 Null Values Single proportion: H 0 : p = p 0 => use p 0 for p Difference in proportions: H 0 : p 1 = p 2  use the overall sample proportion from both groups (called the pooled proportion) as an estimate for both p 1 and p 2 Means: Standard deviations have nothing to do with the null, so just use sample statistic s Correlation: H 0 : ρ = 0 => use ρ = 0

Statistics: Unlocking the Power of Data Lock 5 For quantitative data, we use a t-distribution instead 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) t-distribution

Statistics: Unlocking the Power of Data Lock 5 The t-distribution is characterized by its degrees of freedom (df) Degrees of freedom are based on sample size Single mean: df = n – 1 Difference in means: df = min(n 1, n 2 ) – 1 Correlation: df = n – 2 The higher the degrees of freedom, the closer the t-distribution is to the standard normal Degrees of Freedom

Statistics: Unlocking the Power of Data Lock 5 t-distribution

Statistics: Unlocking the Power of Data Lock 5 Aside: William Sealy Gosset

Statistics: Unlocking the Power of Data Lock 5 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) Matched Pairs

Statistics: Unlocking the Power of Data Lock 5 Golden Balls: Split or Steal? 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? a) Split b) Steal Van den Assem, M., Van Dolder, D., and Thaler, R., “Split or Steal? Cooperative Behavior When the Stakes Are Large,” available at SSRN: 2/19/11.

Statistics: Unlocking the Power of Data Lock 5 To Do Do Project 1 (due Friday, 3pm)Project 1 Read Chapter 6 Do HW 5 (due Wednesday, 3/19)