Question

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.)

Answer 1

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