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

2. Chi-Square Distribution

3. t-, z and ChiSquare
Source:

4. Normal and t-distributions
t-dist for ν = 1
t-dist for ν = 3
t-dist for ν = 2
t-dist for ν = 5
t-dist for ν = 10
t-dist for ν = 30

5. Standard Normal and Student - t
Standard Normal z
X1, … , Xn are independently distributed (μ,σ), and
then
is normally distributed with mean 0 and std dev 1
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. 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

7. One-sided test

8. Two-sided test

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 lower ranks
Test sum of higher ranks
p-value is for Rank sum of 78.5