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 2.9644 meters, appears in the table as 644. 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 644 646 657 668 675 690 698 700 715 718 726 743 759

(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)

(b)

Following is the output of regression analysis:

		SUMMARY OUTPUT
		Regression Statistics
	Multiple R	0.993013115
	R Square	0.986075047
	Adjusted R Square	0.984809142
	Standard Error	4.530945876
	Observations	13
	ANOVA
		df	SS	MS	F	Significance F
	Regression	1	15991.40659	15991.40659	778.948808	1.46159E-11
	Residual	11	225.8241758	20.52947053
	Total	12	16217.23077
		Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 99.0%	Upper 99.0%
	Intercept	-63.72527473	27.23334946	-2.339971983	0.03917423	-123.6654727	-3.785076746	-148.3067889	20.85623941
	Year	9.373626374	0.335856053	27.90965438	1.4616E-11	8.634412185	10.11284056	8.330522455	10.41673029

The  equation of the least-squares line is

y' = 9.374x - 63.725

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

0.9861 or 98.61%

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

(8.33, 10.42)