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 2.9645 meters, appears in the table as 645. 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	645	647	659	670	675	690	699	700	716	719	727	744	759

(a) Plot the data. Consider whether or not the trend in lean over time appears to be linear. (Do this on paper. Your instructor may ask you to turn in this graph.)

(b) 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.)
%

(c) 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

Independent variable, X: Year

Dependent variable, Y: Lean

(a)

Following is the scatter plot of the data:

Scatter plot shows that there is a strong linear positive relationship between the variables.

(b)

Following is the output of regression analysis:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.993486809
R Square	0.987016039
Adjusted R Square	0.985835679
Standard Error	4.306443375
Observations	13
ANOVA
	df	SS	MS	F	Significance F
Regression	1	15507.69231	15507.69231	836.199095	9.9432E-12
Residual	11	204	18.54545455
Total	12	15711.69231
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 99.0%	Upper 99.0%
Intercept	-51.53846154	25.88397226	-1.991134167	0.07188712	-108.5087	5.431777256	-131.9290712	28.85214811
Year	9.230769231	0.319214821	28.91710731	9.9432E-12	8.52818215	9.933356314	8.239349761	10.2221887

The  equation of the least-squares line is:

y' = 9.231*x -51.538

The percent of the variation in lean is explained by this line, the r-square, is

0.987 or 98.7%

(c)

A 99% confidence interval for the average rate of change (tenths of a millimeter per year) of the lean is

(8.24, 10.22)