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.

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. х 5 12 14 20 23 30 40 51 55 67 72 85 96 112 127 y 4 10 13 15 15 25 27 45 38 46 53 75 82 99 103 in USE SALT (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. O No, the scatterplot shows a reasonable linear relationship. O 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 four decimal places.) slope intercept (c) Calculate a point estimate of the true average runoff volume when rainfall volume is 54. (Round your answer to four decimal places.) m3 (d) Calculate a point estimate of the standard deviation o. (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.) Need Help? Read It Talk to a Tutor

x	5	12	14	20	23	30	40	51	55	67	72	85	96	112	127
y	4	10	13	15	15	25	27	45	38	46	53	75	82	99	103

(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 four decimal places.)

slope
intercept

(c) Calculate a point estimate of the true average runoff volume when rainfall volume is 54. (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

a) We have to draw a scatter plot. We perform the following steps in minitb-

1.Enter the given data.

2. Select graph then scatterplot then 3rd option.

3. Enter y in y variable and x in x variable.

4. Click OK.

After running the above stpes we get the following output-

Scatterplot of y vs x 120 100 80 60 40 20 0 0 20 40 60 80 100 120 140 х

Yes, the scatterplot shows a reasonable linear relationship.

b) Here, we have to find the value of slope and intercept.

Steps for regression-

1. Enter the given values in different columns.

2. Go to “Stat” then “Regression” then “Regression” then “Fit regression model”.

3. Enter Y in “response” and X in “continuous predictor”.

4. Click OK.

After running the above steps we get the following ouptut-

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant -2.38 2.17 -1.10 0.292
x 0.8476 0.0331 25.58 0.000 1.00

Thus we get,

Slope = 0.8476

Intercept = -2.38

c) Here, we have to calculate a point estimate of the true average runoff volume when rainfall volume is 54.

Thus, average runoff volume = -2.38+0.8476*54 = 42.9704 m³

d) Here, we have to alculate a point estimate of the standard deviation σ.

Model Summary

S R-sq R-sq(adj) R-sq(pred)
4.75344 98.05% 97.90% 97.48%

From running the above steps we get the standard error of the model = 4.75344.

$\sigma =\sqrt{n}*s$

= $\sqrt{15}*4.75344$

= 18.41.