Question

Question 1: Is the magnitude of an earthquake related to the depth below the surface at which the quake occurs? In order to answer this question an analysis is conducted. Let we wish to explain depth (in kilometers) of the quake below the surface at the epicenterx by magnitude of an earthquake (on the Richter scale). Data are as follows: 1 2 3 4 5 6 7 8 9 10 3.9 4.3 3.3 4.6 3.9 3.2 3.4 4.5 5.1 2.6 Subject Magnitude of earthquake Depth of quake 8.8 10.0 11.2 9.5 12.8 3.9 5.5 7.1 8.5 7.9 11 12 13 14 15 16 17 18 19 20 3.3 3.0 4.5 2.9 4.5 2.8 5.1 4.7 3.5 2.2 Subject Magnitude of earthquake Depth of quake 5.3 5.0 6.5 7.6 10.0 5 9 9 2 9 a) Determine the independent and dependent variables. b) What is the simple linear regression model? c) Calculate the least squares estimates of the slope and intercept, and write fitted simple linear regression model. d) What is an estimate of the mean earthquake depth when the magnitude is 3.3? e) What is residual when the magnitude is 3.3? f) Test the significance of regression. i. State the hypotheses. ii. Calculate ANOVA table. iii. State your rejection criterion. iv. Interpret your result. Use a=0.05.

Answer 1

Magnitude of earthquake Depth of quake 3.3 8.8 4.6 10 b) Linear regression, Depth of quake = a + b*Magnitude of quake Magnitude of earthquake Depth of quake a) Independent variable Dependent variable 3.3 11.2 SUMMARY OUTPUT c) Slope 0.98 Intercept 4.00 Depth of quake = 4.00 +0.98 * Magnitude of quake 4.6 9.5 3.9 3.2 d) Mean Depth of quake when magnitude 12.8 3.9 5.5 7.1 7.2 3.4 is 3.3 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 4.5 0.33 0.11 0.06 2.57 20.00 5.1 8.5 7.9 e) Residual (y - ) 1.6 Actual y when magnitude = 3.3 is 8.8 2.6 3.3 5.3 3 5 f) ii) ANOVA df SS MS F 4.5 2.9 6.5 7.6 Significance F 0.16 f) i) Ho: B1 = 0 VS Ha : B10 ii) Reject Ho if F> Fa Fa (1,18,0.05) 4.41 iv) Fail to reject Ho 1.00 2.14 4.5 10 Regression Residual Total 18.00 14.14 14.14 118.82 132.95 6.60 2.8 5 19.00 5.1 9 4.7 9 Coefficients 4.00 3.5 2 Intercept Magnitude of earthquake Standard Error t Stat P-value 2.58 1.55 0.14 0.67 1.46 0.16 Lower 95% Upper 95% -1.41 9.42 -0.43 2.39 2.2 9 0.98