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 13 14 15 16 17 18 19 20 11 3.3 12 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 6 2 9 4 2 Show your hand writing answers step by step with the formulas and calculations for the following: 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. State your rejection criterion. iv. Interpret your result. Use a=0.05. 111.

Answer 1

Solution:

a)

Independent variable (X) = Magnitude of earth quake

Dependent variable (Y) = Depth of quake

b)

Regression line:

Y^ = b+m*X (Single independent variable)

c)

Excel > Data > Data Analysis > Regression

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.026974584
R Square	0.000727628
Adjusted R Square	-0.054787504
Standard Error	2.986219457
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.116880366	0.116880366	0.013106844	0.910120618
Residual	18	160.5151196	8.917506646
Total	19	160.632
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	7.622721633	3.067158648	2.485271389	0.023000926	1.178860429	14.06658284	1.178860429	14.06658284
Magnitude (X)	-0.091028322	0.795110481	-0.114485124	0.910120618	-1.761493456	1.579436811	-1.761493456	1.579436811

Intercept (b) = 7.6227

Slope (m) = -0.0910

Regression equation:

Y^ = 7.6227-0.0910*X

d)

If X = 3.3

Y^ = 7.6227-0.0910*X

Y^ = 7.6227-0.0910*3.3 = 7.3224

e)

Y = 5.3 when X = 3.3

Y^ = 7.3224

Residual = Y-Y^ = 5.3-7.3224 = -2.0224

f)

i)

Hypothesis:

H0: β1 = 0

Ha: β1 not = 0

ii)

ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.116880366	0.116880366	0.013106844	0.910120618
Residual	18	160.5151196	8.917506646
Total	19	160.632

iii)

Fcritical = 4.4139

Rejection region:

If Fstat > Fcritical, reject H0

iv)

Fstat < Fcritical, Do not reject H0

P value = 0.9101 > 0.05, Do not reject H0

There is not enough evidence to conclude that regression model is statistically significant at 5% significance level

Note:

Calculations:

	X	Y	X^2	Y^2	XY
	3.9	8.8	15.21	77.44	34.32
	4.3	10	18.49	100	43
	3.3	11.2	10.89	125.44	36.96
	4.6	9.5	21.16	90.25	43.7
	3.9	12.8	15.21	163.84	49.92
	3.2	3.9	10.24	15.21	12.48
	3.4	5.5	11.56	30.25	18.7
	4.5	7.1	20.25	50.41	31.95
	5.1	8.5	26.01	72.25	43.35
	2.6	7.9	6.76	62.41	20.54
	3.3	5.3	10.89	28.09	17.49
	3	5	9	25	15
	4.5	6.5	20.25	42.25	29.25
	2.9	7.6	8.41	57.76	22.04
	4.5	10	20.25	100	45
	2.8	5	7.84	25	14
	5.1	4	26.01	16	20.4
	4.7	3	22.09	9	14.1
	3.5	3	12.25	9	10.5
	2.2	11	4.84	121	24.2
SUM	75.3	145.6	297.61	1220.6	546.9
n	20
Mean	3.765	7.28
k	2
SSxx	14.1055	Sum(x^2) - ((Sum(x))^2 /n)	SSR	0.11688	slope * Ssxy	MSR	0.11688	SSR/k-1
Ssyy	160.632	Sum(y^2) - ((Sum(y))^2 /n)	SSE	160.5151	SST-SSR	MSE	8.917507	SSE/n-k
Ssxy	-1.284	Sum(xy) - (Sum(x)*Sum(y)/n)	SST	160.632	Ssyy	F	0.013107	MSR/MSE
slope	-0.09103	Ssxy/SSxx
intercept	7.622722	Mean Y - Mean X * Slope
Se	2.986219	SQRT(SSE/(n-2))
Sb1	0.79511	Se/SQRT(SSxx)
r	-0.02697	Ssxy/SQRT(SSxx*Ssyy)
r^2	0.000728