Question

35 6. (15 points) Data from the United States Geological Survey can be used to predict the magnitude of earthquakes based on the depth of the fault below the surface. The magnitude is given according to the Richter scale and the depth is in kilometers. magnitude 7.7 6.7 6.9 6.8 4.0 3.8 7.1 5.9 depth 18 17 26 5 10 23 10 The standard deviation for the magnitude is 1.45, and the standard deviation for the depth is 9.83. The correlation between these two variables is 0.843. (a) Given this information, find an appropriate regression line for these observations. (b) Find and interpret the coefficient of determination for the model. (c) Find the residual for the fourth observation (magnitude 6.8, depth 26). (d) Suppose an earthquake occurs where the fault is 20 kilometers below the surface. Estimate the magnitude of this earthquake.

Answer 1

Solution: a) We have to find an app &opciate begression line of obse&vations: Given that, Sy=1.45 Soc = 9.83 & 8 = 0.843 Line of degression is, y = atba whese a is inte ocept b is the slope. 2 is independent vadrouble y is dependent vaziable from the given values we can calculate the Slope b by wing formula, है Sy an ta Sx .. b = 0.843 * 1.45 9.83 b = 0.12 43 . Slope Slope = b =0-1243 Intercept a=ý boc from the given observation, we calculate the Ý f 5. observation table Seina depth o magnitude(y). X = 1 x = 1 x 144 7.7 18 6.7 1x=18 8 35 2 3 17. 26 ý = 1 ² = 1 x 48.9= 6.1125 n in 8 6-8 4.0 3.8 5 5 6 y = 6.1125 7 23. 8 10 5.9 Ey=48.9 EN=144.

= ý - boo -6.1125-0.1243 X 18 - 6.1125 -2.2374 Q.3 3.8751 Regression line yzatba is, 9 = 3.8751 +0.1243*(0) b) we have to find Coefficient of detemination d² here correlation ben depth of magnitude is, 2=0.843 "Coefficient of determination & ² (0.843)² = Koefficient of determination e²= 0.71 heae & Zo.71 It means that on the basis of the sample app coximatel 71% of the vadiation in vaciable magnitude cawed by the variation of the depth variable. The remaining 29% variation in magnitude is unexplained by variation in depth. c) we have find the besidual for the fourth obsee veetion (magnitude 6-8 depth 26). Formula : Residual = y- ŷ here, degression line is, ŷ - 3.8751 +0.1243400 here the value of ac. lie. value of a vadable obsee vaton is 26 4th

i ý = 3.8751 +0,1243x26 = 3.8751 +3.2318 19 - 7.1069 here y = 6.8 Residual = y- 6.8-7.1069 Residual --0.3069 . Residual fore the fourth obseevation is, Residual = -0.3069 d) We have to estimate the magnitude of earthquake when depth is 20 kilometees we know the line of degression, ie. ŷ = 3.8751 +0.124 3 x x. hede x=20 + in ŷ = 3.8751.40.1243720 - 3.8751 +2.486 Û - 6.3611 i Estimated value of magnitude at depth is 20 kilometeo is 6.3611