Question

A data related to air pollution in 10 U.S. cities. The dependent variable Y is the annual mean concentration of sulfur dioxide, in micrograms per cubic meter. The explanatory variable X records the number of manufacturing enterprises employing 20 or more workers. Below is Routput of the relationship between X and Y.

Coefficients: Estimate Std. Error t value Pro> tl) (Intercept) 9.4764 9.6266 0.98 0.354 2.0315 0.0070 4.51 0.CO2 ** X Signif. codes: 9 ****' 0.001 ***' 0.01 **' 0.05, 0.1' '1 Residual standard error: 17.9 on 8 degrees of freedom Multiple R-squared: 0.717, Adjusted R-squared: 0.682 F-statistic: 20.3 on 1 and 8 DF, p-value: 0.00198
a) Write the regression equation with parameters from the R output.b) Suppose that the number of manufacturing enterprises employing 20 or more workers in Irvine is 250, could you predict that the annual mean concentration of sulfur dioxide in Irvine?c) What is the residual if in Irvine the annual mean concentration of sulfur dioxide is 15 micrograms per cubic meter.d) What is the value of the correlation coefficient?e) Calculate a 95% confidence interval for the slope of the model.f) Based on the confidence interval, is there a linear relationship between X and Y?

Answer 1

Y = 0.0315x + 9.4764

Residual = 2.35

Correlation Coefficient = 0.847

Explanation:

From the R output given :

Intercept = 9.4764

Slope = 0.0315

x = number of manufacturing enterprise employing 20 or more workers

y = annual mean concentration of Sulphur dioxide

The regression equation :

y = bx + c

b = slope ; c = intercept

y = 0.0315x + 9.4764

Prediction using the regression equation :

The predicted y value, when x = 250

y = 0.0315(250) + 9.4764

y = 17.3514

The residual, if actual annual concentration = 15

Y residual = 17.35 - 15 = 2.35

The correlation Coefficient value, R

R = √R²

R = √0.717

R = 0.847