StatWRLecture6.ppt
- 1. Statistics in Water Resources, Lecture 6 • Key theme – T-distribution for distributions where standard deviation is unknown – Hypothesis testing – Comparing two sets of data to see if they are different • Reading: Helsel and Hirsch, Chapter 6 Matched Pair Tests
- 3. t-, z and ChiSquare Source: http://en.wikipedia.org/wiki/Student's_t-distribution
- 4. Normal and t-distributions Normal t-dist for ν = 1 t-dist for ν = 30 t-dist for ν = 5 t-dist for ν = 3 t-dist for ν = 2 t-dist for ν = 10
- 5. • Standard Normal z – X1, … , Xn are independently distributed (μ,σ), and – then is normally distributed with mean 0 and std dev 1 Standard Normal and Student - t • Student’s t-distribution – Applies to the case where the true standard deviation σ is unknown and is replaced by its sample estimate Sn
- 6. 6 p-value is the probability of obtaining the value of the test-statistic if the null hypothesis (Ho) is true If p-value is very small (<0.05 or 0.025) then reject Ho If p-value is larger than α then do not reject Ho
- 9. Helsel and Hirsch p.120
- 10. Box and Whisker Plots of the N data
- 11. Precipitation Water Quality at two sites
- 12. Ranked Precipitation Quality Data Mean concentration is nearly the same but ranks suggest residential concentration is smaller. Is this so?
- 13. Wilcoxon Rank Sum Test Helsel and Hirsch p. 462 This is < 0.05 for a one-sided test, thus reject Ho and say residential concentration is lower than industrial p-value in middle is for P(Wrs > X) or P(Wrs < X*) for m = n = 10 Note that the sum of n = 1, 2, …. 20 = 210 and X + X* = 210 in all cases in this table. Test sum of higher ranks Test sum of lower ranks p-value is 0.024 for Rank sum of 78.5