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 8 12 14 18 23 30 40 50 55 67 72 85 96 112 127
y 4 10 13 15 15 25 27 45 38 46 53 75 82 99 104

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

slope =     
intercept =


(c) Calculate a point estimate of the true average runoff volume when rainfall volume is 47. (Round your answer to four decimal places.)
  m3

(d) Calculate a point estimate of the standard deviation ?. (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.)

Solutions

Expert Solution

The statistical software output for this problem is:

Simple linear regression results:
Dependent Variable: y
Independent Variable: x
y = -2.6491939 + 0.85381695 x
Sample size: 15
R (correlation coefficient) = 0.99017899
R-sq = 0.98045444
Estimate of error standard deviation: 4.779818

Parameter estimates:

Parameter Estimate Std. Err. Alternative DF T-Stat P-value
Intercept -2.6491939 2.185153 ? 0 13 -1.2123609 0.2469
Slope 0.85381695 0.033435167 ? 0 13 25.536494 <0.0001


Analysis of variance table for regression model:

Source DF SS MS F-stat P-value
Model 1 14898.593 14898.593 652.11252 <0.0001
Error 13 297.00659 22.84666
Total 14 15195.6


Predicted values:

X value Pred. Y s.e.(Pred. y) 95% C.I. for mean 95% P.I. for new
47 37.480203 1.2557268 (34.76737, 40.193035) (26.803629, 48.156776)

Hence,

B) Slope = 0.85382

Intercept = -2.64919

c) Point estimate = 37.4802 m3

d) Point estimate for standard deviation = 4.78

e) Proportion explained = R-square = 0.9805


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