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 2.9646 meters, appears in the table as 646. 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 646 648 660 671 676 691 700 701 717 721 729 746 760 (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	0.994106714
R Square	0.98824816
Adjusted R Square	0.987179811
Standard Error	4.123711314
Observations	13
ANOVA
	df	SS	MS	F	Significance F
Regression	1	15730.02198	15730.02198	925.0236165	5.74301E-12
Residual	11	187.0549451	17.004995
Total	12	15917.07692
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 99.0%	Upper 99.0%
Intercept	-55.64835165	24.7856572	-2.245183623	0.046278983	-110.2012153	-1.09548799	-132.6278072	21.33110394
Year	9.296703297	0.305669819	30.4142009	5.74301E-12	8.623928561	9.969478033	8.347351981	10.24605461

The  equation of the least-squares line is

y' = 9.297x - 55.648

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

0.988 or 98.8%

(c)

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

(8.35, 10.25)