Transcript Normal and t-distributions

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
Chi-Square Distribution
http://en.wikipedia.org/wiki/Chi-square_distribution
t-, z and ChiSquare
Source: http://en.wikipedia.org/wiki/Student's_t-distribution
Normal and t-distributions
Normal
t-dist for ν = 1
t-dist for ν = 5
t-dist for ν = 2
t-dist for ν = 10
t-dist for ν = 3
t-dist for ν = 30
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
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
6
One-sided test
Two-sided test
Helsel and Hirsch p.120
Box and Whisker Plots of the N data
Precipitation Water Quality at two
sites
Ranked Precipitation Quality Data
Mean concentration is nearly the same
but ranks suggest residential
concentration is smaller. Is this so?
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
Test sum
of higher
ranks
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.
p-value is 0.024 for
Rank sum of 78.5
Test sum
of lower
ranks