Question

Engineers concerned about a tower's stability have done extensive studies of its increasing tilt. Measurements of the lean of the tower over time provide much useful information. The following table gives measurements for the years 1975 to 1987. The variable "lean" represents the difference between where a point on the tower would be if the tower were straight and where it actually is. The data are coded as tenths of a millimeter in excess of 2.9 meters, so that the 1975 lean, which was 2.9643 meters, appears in the table as 643. Only the last two digits of the year were entered into the computer.

Year	75	76	77	78	79	80	81	82	83	84	85	86	87
Lean	643	646	657	668	675	689	697	699	715	718	726	744	759

(a) What is the equation of the least-squares line? (Round your answers to three decimal places.)
y =  +  x

What percent of the variation in lean is explained by this line? (Round your answer to one decimal place.)
%

(b) Give a 99% confidence interval for the average rate of change (tenths of a millimeter per year) of the lean. (Round your answers to two decimal places.)
(  ,  )

Answer 1

(a) following data is given and we find the equation y=a+bx=629.62+9.35x

the R² is 0.987, so 98.7 percent of the variation in lean is explained by this line

(b)

(1-alpha)*100% confidence interval for slope=b ±z(alpha/2,n-2)*SE(b)

99% confidence interval =9.35±t(0.01/2,11)*SE(b)=9.35±3.11*0.32=9.35±1.00=(8.35,10.35)

Year	x	Lean
75	1	643
76	2	646
77	3	657
78	4	668
79	5	675
80	6	689
81	7	697
82	8	699
83	9	715
84	10	718
85	11	726
86	12	744
87	13	759

followign regression anlaysis information has been generated using ms-excel

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.99364
R Square	0.98732
Adjusted R Square	0.986167
Standard Error	4.310849
Observations	13
ANOVA
	df	SS	MS	F	Significance F
Regression	1	15916.51	15916.51	856.4897	8.73E-12
Residual	11	204.4176	18.58342
Total	12	16120.92
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	629.6154	2.536281	248.2435	5.69E-22	624.0331	635.1977
X Variable 1	9.351648	0.319541	29.26585	8.73E-12	8.648343	10.054