Statistics in WR: Lecture 8

• Key Themes– Replication in Monte Carlo experiments– Testing paired differences and analysis of

variance– Correlation

• Reading: Helsel and Hirsch Chapter 9 Simple Regression

Statistics of the summation:20 random series of 1000 uniform (0,1) random numbers

Mean of Sum: 503.92Variance of Sum: 46.51Theoretical Variance of Sum: 83.33

Number of series

Statistics of the summation:50 random series of 1000 uniform (0,1) random numbers

Mean of Sum: 502.29Variance of Sum: 74.16Theoretical Variance of Sum: 83.33

Number of series

Statistics of the summation:100 random series of 1000 uniform (0,1) random numbers

Mean of Sum: 501.47Variance of Sum: 83.18Theoretical Variance of Sum: 83.33

Number of series

Statistics of Mean of Replicated Series

Statistics of Mean of Replicated Series

Matched-Pair TestsTo determine the effectiveness of an acid solution in developing wells in carbonate rock, yields of twenty wells were measured both before and after treatment of the wells with acid. Factoring out the differences in yield between wells, have the yields changed as a result of using the acid?What is the magnitude of this change?

Annual sediment loads are measured at two sites over a period of twenty-four years. Both drainage basins are of essentially the same size, and have the same basin characteristics. However, logging has occurred in one basin during the period, but not in the other. Can the year to year variation in load (due to differences in precipitation) be compensated for, to determine whether the site containing logging produced generally higher loads than the other?

Helsel and Hirsch p.137

H&H p.148

Comparing Several Independent Groups

Concentrations of volatile organic compounds are measured in shallow ground waters across a several county area. The wells sampled can be classified as being contained in one of sevenland-use types: undeveloped, agricultural, wetlands, low-density residential, high-density residential, commercial, and industrial/transportation. Do the concentrations of volatiles differ between these types of surface land-use, and if so, how?

Analysis of Variance

H&H p.165

Multifactor Analysis of Variance

Patterns of data that all have correlation between x and y of 0.7

Monotonic nonlinear correlation

Linear correlation

Non-monotonic correlation