Statistics in Water Resources, Lecture 6

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
http//en.wikipedia.org/wiki/Chi-square_distributi
on
3
t-, z and ChiSquare
Source http//en.wikipedia.org/wiki/Student's_t-d
istribution
4
Normal and t-distributions
Normal
t-dist for ? 1
t-dist for ? 3
t-dist for ? 2
t-dist for ? 30
t-dist for ? 5
t-dist for ? 10
5
Standard Normal and Student - t

• Standard Normal z
• X1, , Xn are independently distributed (µ,s),
  and
• then
• is normally distributed with mean 0 and std dev 1

• Students t-distribution
• Applies to the case where the true standard
  deviation s 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 (lt0.05 or
0.025) then reject Ho If p-value is larger than
a 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 lt 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
gt X) or P(Wrs lt 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 0.024 for Rank sum of 78.5
14
(No Transcript)
15
(No Transcript)