Question

3. (Ch 2, Ch 3) A study to assess the capability of subsurface flow wetland systems to remove biochemical oxygen demand and various other chemical constituents resulted in the accompanying data on = BOD mass loading (kg/ha/d) and y=BOD mass removal (kg/ha/d) ("Subsurface Flow Wetlands: A Performance Evaluation," Water Envir. Res., 1995: 244-247): 2: 3 8 10 11 13 16 27 30 35 37 38 44 103 142 y: 4 7 8 8 10 11 16 26 21 9 31 30 75 90 (a) Ch 1, Ch 2A: In R, construct histograms and box plots of both mass loading and mass removal (for a total of four graphs). Describe the distribution of each variable (remember there are four aspects to discuss). Each graph should indicate the sample size, variable name, units, and include your R code in a separate file. (b) (See Ch 2C notes): Make the normal quantile for both variables. Include the line "y = 82 + ". Based on these plots, discuss whether these variables are sampled from normal distributions. (C) Ch 3.1-3.3: In R. construct a fully labeled scatter plot of the data. Describe the pattern in scatterplot and comment on any interesting features In particular, address this question: "Do the values of y appear to be linearly related to the I values?" (d) Determine the equation of the least squares line. Include your R code. (e) Find the proportion of the observed variation in BOD mass removal that can be attributed to the approximate linear relationship between it and BOD mass loading dict the value of BOD mass removal when the BOD mass loading is 20. Show your work (8) Is it appropriate to use the least squares line to predict BOD mass removal when the BOD mass loading is 200? Explain. You can read in the data: dat = read.csv"Wetland_Data.csv", header-T) plot (dat) names (dat) y - dat $y X - dat $x To answer these questions, look at the class notes for things you can do with "10". For example... mod - ln(y" x) summary (mod) anova(mod) mod$coefficients plot(x,y) abline (a-mod$coefficients[1], b-mod$coefficients [2])

Answer 1

Histogram of x 6 4 Frequency 2 Ilh 00 0 0 50 100 150

Histogram of y 8 6 Frequency 4 2 0 0 40 80

box plot of x

0 40 80 120

box plot of y

2060

y variable is seen to have a normal distribution however x is assumed to have normal distn

Normal Q-Q Plot 80 120 Sample Quantiles 40 0000000000 0 -1 0 1 Theoretical Quantiles

for variable y

Normal Q-Q Plot 60 Sample Quantiles 20 TTTT 00000 bo o 00000 -1 0 1 Theoretical Quantiles

Ο 40 80 120 ΤΙ ΙΙΙΙΙΙΟ 2000 αφού 5000

x and y are said to have sampled from normal distribution

scatterplot between x and y is

60 20 о 40 80 120

x and y are likely to have linear relationship

m=lm(y~x)
> summary(m)

Call:
lm(formula = y ~ x)

Residuals:
Min 1Q Median 3Q
-15.761 -1.694 0.762 1.352
Max
7.188

Coefficients:
Estimate Std. Error
(Intercept) 0.62614 2.13541
x 0.65229 0.04041
t value Pr(>|t|)
(Intercept) 0.293 0.774
x 16.142 1.67e-09

(Intercept)
x ***
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’
0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.715 on 12 degrees of freedom
Multiple R-squared: 0.956,   Adjusted R-squared: 0.9523
F-statistic: 260.6 on 1 and 12 DF, p-value: 1.675e-09
the line is

y=0.626+0.652*x

required proportion is nothing but r squared that is coefficient of determination =0.956

when x=20

predict(m,20)

predicted value=13.66

when x is 200 it is not a good idea to use least squares method since the value lies outside the interval of x thus it would give a wrong prediction