Question

In: Statistics and Probability

An article gave a scatter plot along with the least squares line of x = rainfall...

An article gave a scatter plot along with the least squares line of x = rainfall volume (m3) and y = runoff volume (m3) for a particular location. The accompanying values were read from the plot.

x 4 12 14 18 23 30 40 47 55 67 72 80 96 112 127
y 4 10 13 14 15 25 27 44 38 46 53 69 82 99 100

(b) Calculate point estimates of the slope and intercept of the population regression line. (Round your answers to five decimal places.)

slope   0.82834  
intercept     -1.41247


(c) Calculate a point estimate of the true average runoff volume when rainfall volume is 51. (Round your answer to four decimal places.)
40.8329m3


(d) Calculate a point estimate of the standard deviation σ. (Round your answer to two decimal places.)
____________m3

(e) What proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall? (Round your answer to four decimal places.)

____________________

Please solve D and E, the rest of the answers are correct.

Solutions

Expert Solution

Sol;

with lm fucntion in R fit a linear model of Y on X

x <- c(   4   ,12   ,14   ,18,   23,   30,   40,   47,   55,   67,   72,   80,   96,   112,   127)
y <- c(4,   10,   13   ,14,   15,   25,   27,   44,   38,   46,   53,   69,   82,   99,   100)
lmodel <- lm(y~x)
coefficients(lmodel)
summary(lmodel)
anova(lmodel)

Output:

> coefficients(lmodel)
(Intercept) x
-1.4124839 0.8283403
> anova(lmodel)
Analysis of Variance Table

Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x 1 14106 14105.6 566.04 4.206e-12 ***
Residuals 13 324 24.9
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> sqrt( 24.9)
[1] 4.98999
> summary(lmodel)

Call:
lm(formula = y ~ x)

Residuals:
Min 1Q Median 3Q Max
-8.086 -4.254 1.472 3.354 7.638

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.41248 2.25466 -0.626 0.542
x 0.82834 0.03482 23.792 4.21e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.992 on 13 degrees of freedom
Multiple R-squared: 0.9775,   Adjusted R-squared: 0.9758
F-statistic: 566 on 1 and 13 DF, p-value: 4.206e-12

ANSWER(D)

y^= -1.4124839 +0.8283403 *x

point estimate of the standard deviation=sqrt(MSE)=sqrt(24.9 )= 4.98999

=4.99(rounded to 2 decimals)

ANSWER(E)

proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall is= R sq

=0.9775

ANSWERS(D)-4.99

ANSWERS(E)-0.9775


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