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 7 12 14 17 23 30 40 51 55 67 72 80 96 112 127
y 4 10 13 15 15 25 27 46 38 46 53 68 82 99 100

(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 51. (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.)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 7 12 14 17 23 30 40 51 55 67 72 80 96 112 127
y 4 10 13 15 15 25 27 46 38 46 53 68 82 99 100

(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 51. (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.)

Solutions

Expert Solution

a)

Yes, the scatterplot shows a reasonable linear relationship.

b) Formula sheet

Sxx Syy Sxy
x y (X-Xbar)2 (Y-Ybar)2 (X-Xbar)(Y-Ybar)
7 4 =(B4-$B$20)^2 =(C4-$C$20)^2 =(B4-$B$20)*(C4-$C$20)
12 10 =(B5-$B$20)^2 =(C5-$C$20)^2 =(B5-$B$20)*(C5-$C$20)
14 13 =(B6-$B$20)^2 =(C6-$C$20)^2 =(B6-$B$20)*(C6-$C$20)
17 15 =(B7-$B$20)^2 =(C7-$C$20)^2 =(B7-$B$20)*(C7-$C$20)
23 15 =(B8-$B$20)^2 =(C8-$C$20)^2 =(B8-$B$20)*(C8-$C$20)
30 25 =(B9-$B$20)^2 =(C9-$C$20)^2 =(B9-$B$20)*(C9-$C$20)
40 27 =(B10-$B$20)^2 =(C10-$C$20)^2 =(B10-$B$20)*(C10-$C$20)
51 46 =(B11-$B$20)^2 =(C11-$C$20)^2 =(B11-$B$20)*(C11-$C$20)
55 38 =(B12-$B$20)^2 =(C12-$C$20)^2 =(B12-$B$20)*(C12-$C$20)
67 46 =(B13-$B$20)^2 =(C13-$C$20)^2 =(B13-$B$20)*(C13-$C$20)
72 53 =(B14-$B$20)^2 =(C14-$C$20)^2 =(B14-$B$20)*(C14-$C$20)
80 68 =(B15-$B$20)^2 =(C15-$C$20)^2 =(B15-$B$20)*(C15-$C$20)
96 82 =(B16-$B$20)^2 =(C16-$C$20)^2 =(B16-$B$20)*(C16-$C$20)
112 99 =(B17-$B$20)^2 =(C17-$C$20)^2 =(B17-$B$20)*(C17-$C$20)
127 100 =(B18-$B$20)^2 =(C18-$C$20)^2 =(B18-$B$20)*(C18-$C$20)
Total =SUM(B4:B18) =SUM(C4:C18) =SUM(D4:D18) =SUM(E4:E18) =SUM(F4:F18)
Mean =B19/15 =C19/15
beta1 =F19/D19
beta0 =C20-B21*B20

Values

Sxx Syy Sxy
x y (X-Xbar)2 (Y-Ybar)2 (X-Xbar)(Y-Ybar)
7 4 2165.351 1500.271 1802.391
12 10 1725.018 1071.471 1359.524
14 13 1562.884 884.0711 1175.458
17 15 1334.684 769.1378 1013.191
23 15 932.2844 769.1378 846.7911
30 25 553.8178 314.4711 417.3244
40 27 183.1511 247.5378 212.9244
51 46 6.417778 10.67111 -8.27556
55 38 2.151111 22.40444 -6.94222
67 46 181.3511 10.67111 43.99111
72 53 341.0178 105.4044 189.5911
80 68 700.4844 638.4044 668.7244
96 82 1803.418 1541.871 1667.524
112 99 3418.351 3165.938 3289.724
127 100 5397.351 3279.471 4207.191
Total 803 641 20307.73 14330.93 16879.13
Mean 53.53333 42.73333
beta1 0.831168
beta0 -1.76185

Intercept = beta0= -1.76185

slope=beta1=0.831168

c) True average = -1.76185 + 0.831168 (51) = 40.62771

d) standard deviation =


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