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Inference Using Formulas

This chapter focuses on the use of the t-distribution and standard error formulas in statistical inference. It covers concepts such as matched pairs confidence intervals, the impact of sample sizes on standard error, and how to compute standard errors without extensive simulations. The discussion includes how to formulate hypotheses for proportions and means, as well as the correlation in data analysis. Additionally, it emphasizes the importance of degrees of freedom and the application of the t-distribution for estimating standard deviations, making statistical inference more accurate.

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Inference Using Formulas

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  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)

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