In: Statistics and Probability
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.)
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 =