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	5	12	14	18	23	30	40	46	55	67	72	80	96	112	127
y	4	10	13	15	15	25	27	46	38	46	53	72	82	99	101

(a) Does a scatter plot of the data support the use of the simple linear regression model?

Yes, the scatterplot shows a reasonable linear relationship. Yes, the scatterplot shows a random scattering with no pattern.     No, the scatterplot shows a reasonable linear relationship. No, the scatterplot shows a random scattering with no pattern.

(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 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.)

Answer 1

	X	Y	X * Y	X2	Y2	Ŷ	( Y - Ŷ )2
	5	4	20.0000	25	16	3	1
	12	10	120.0000	144	100	9	2
	14	13	182.0000	196	169	10	7
	18	15	270.0000	324	225	13.7333	1.6047
	23	15	345.0000	529	225	17.9078	8.4555
	30	25	750.0000	900	625	23.7523	1.5569
	40	27	1080.0000	1600	729	32.1014	26.0245
	46	46	2116.0000	2116	2116	37.1109	79.0156
	55	38	2090.0000	3025	1444	44.6252	43.8930
	67	46	3082	4489	2116	54.6442	74.7219
	72	53	3816	5184	2809	58.8188	33.8580
	80	72	5760	6400	5184	65.4981	42.2747
	96	82	7872	9216	6724	78.8568	9.8799
	112	99	11088	12544	9801	92.2154	46.0302
	127	101	12827	16129	10201	104.7392	13.9816
Total	797	646	51418	62821	42484	646.0000	390.9746

Part a)

Yes, the scatter plot shows a reasonable linear relationship.

Part b)

Equation of regression line is Ŷ = a + bX

b = ( 15 * 51418 - 797 * 646 ) / ( 15 * 62821 - ( 797 )²)
b = 0.834917
a =( Σ Y - ( b * Σ X) ) / n
a =( 646 - ( 0.8349 * 797 ) ) / 15
a = -1.29525
Equation of regression line becomes Ŷ = -1.29525  + 0.834917 X

Part c)

When X = 47
Ŷ = -1.295 + 0.835 X
Ŷ = -1.295 + ( 0.835 * 47 )
Ŷ = 37.9458

Part d)

Standard Error of Estimate S = √ ( Σ (Y - Ŷ )² / n - 2) = √(390.9746 / 13) = 5.48

Part e)

r = 0.987

Coefficient of Determination
R² = r² = 0.9733

Explained variation = 0.9733* 100 = 97.33%