An article gave a scatter plot along with the least squares line of x = rainfall

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.

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

(c) Calculate a point estimate of the true average runoff volume when rainfall volume is 45. (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.)

x 6 12 14 18 23 30 40 51 55 67 72 79 96 112 127 y 4 10 13 14 15 25 27 47 38 46 53 76 82 99 104

Explanation / Answer

Here we can use excel. First we copy the data set in excel in two columns, then we go to Data, we select Data Analysis. Under Data Analysis we can find a list of tests. We select Regression. Next we select the data for Y and X. Finally we click OK; our regression output will be ready.

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.986750567

R Square

0.9737

Adjusted R Square

0.97165181

Standard Error

5.57

Observations

15

ANOVA

df

SS

MS

F

Significance F

Regression

1

14924.25652

14924.26

480.8587

0.0000

Residual

13

403.476812

31.03668

Total

14

15327.73333

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-2.33572

2.538612799

-0.92008

0.374291

-7.820057508

3.14862152

X Variable 1

0.85790

0.039122618

21.92849

1.19E-11

0.773380685

0.94241924

Question b)

Slope = 0.85790

Intercept = -2.33572

Question c)

y^ = -2.33572 + 0.85790x

x = 45

y^ = -2.33572 + (0.85790*45) = 36.27

Question d)

s (Standard deviation) = 5.57

Question e)

The proportion of the observed variation in runoff volume can be attributed to the simple linear regression relationship between runoff and rainfall is 0.9737

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.986750567

R Square

0.9737

Adjusted R Square

0.97165181

Standard Error

5.57

Observations

15

ANOVA

df

SS

MS

F

Significance F

Regression

1

14924.25652

14924.26

480.8587

0.0000

Residual

13

403.476812

31.03668

Total

14

15327.73333

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-2.33572

2.538612799

-0.92008

0.374291

-7.820057508

3.14862152

X Variable 1

0.85790

0.039122618

21.92849

1.19E-11

0.773380685

0.94241924

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