Question

In: Statistics and Probability

Engineers concerned about a tower's stability have done extensive studies of its increasing tilt. Measurements of...

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.)
(  ,  )

Solutions

Expert Solution

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)


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