The Leaning Tower of Pisa is an architectural wonder. Engineers concerned about

Question

The Leaning Tower of Pisa is an architectural wonder. Engineers concerned about the 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 2.9642 meters, appears in the table as 642. Only the last two digits of the year were entered into the computer. (data152.dat)

(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 two 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.)
(  ,  )

Explanation / Answer

PART A.

Line of Regression Y on X i.e Y = bo + b1 X

calculation procedure for regression

mean of X = X / n = 81

mean of Y = Y / n = 698.3846

(Xi - Mean)^2 = 182

(Yi - Mean)^2 = 19207.08

(Xi-Mean)*(Yi-Mean) = 1867

b1 = (Xi-Mean)*(Yi-Mean) / (Xi - Mean)^2

= 1867 / 182

= 10.25824

bo = Y / n - b1 * X / n

bo = 698.3846 - 10.25824*81 = -132.53298

value of regression equation is, Y = bo + b1 X

Y'=-132.53298+10.25824* X

rounding 2 decilmals regression equation, Y'=-132.53+10.256* X

PART B.

calculation procedure for correlation
sum of (x) = x = 1053
sum of (y) = y = 9079
sum of (x^2)= x^2 = 85475
sum of (y^2)= y^2 = 6359841
sum of (x*y)= x*y = 737266
to caluclate value of r( x,y) = covariance ( x,y ) / sd (x) * sd (y)
covariance ( x,y ) = [ x*y - N *(x/N) * (y/N) ]/n-1
= 737266 - [ 13 * (1053/13) * (9079/13) ]/13- 1
= 143.6154
and now to calculate r( x,y) = 143.6154/ (SQRT(1/13*737266-(1/13*1053)^2) ) * ( SQRT(1/13*737266-(1/13*9079)^2)
=143.6154 / (3.7417*38.4378)
=0.9986
value of correlation is =0.9986
value of variation in lean is explained by this line is  = r^2 = 0.997 ~ 99.7%

PART C.

CI calculated is (-162.0978 -102.9681 )

X Y (Xi - Mean)^2 (Yi - Mean)^2 (Xi-Mean)*(Yi-Mean) 75 635 36 4017.6075 380.3076 76 645 25 2849.9155 266.923 77 660 16 1473.3775 153.5384 78 668 9 923.22392 91.1538 79 678 4 415.53192 40.7692 80 688 1 107.83992 10.3846 81 701 0 6.84032 0 82 712 1 185.37912 13.6154 83 716 4 310.30232 35.2308 84 726 9 762.61032 82.8462 85 741 16 1816.0723 170.4616 86 750 25 2664.1495 258.077 87 759 36 3674.2267 363.6924

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