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. (data41.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

Let xi represent the ith year given and yi represent the lean in that year.

Setting 1981 as 0 and rewriting xi accordingly,

the least square equation is: y = 700.84 + 10.16x, x reckoned from 1981 ANSWER 1

Percent variation in lean that is explained by the regression line = 96 ANSWER 2 [= 100 x r2 where r = sample correlation coefficient]

99% Confidence Interval for the average rate of change of the lean is: (778.31, 789.99)

Back-up Theory and Detailed Excel Calculations

Let X and Y be two variables such that Y depends on X by the model Y = + X + , where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2. Then,

Y is termed ‘dependent variable’ and X is termed ‘independent variable’.

Let (xi, yi) be a pair of sample observation on (X, Y), i= 1, 2, …., n where n = sample size.

Then, Mean X = Xbar = (1/n)sum of xi over I = 1, 2, …., n; ……………….(1)

Sxx = sum of (xi – Xbar)2 over i = 1, 2, …., n ………………………………..(2)

Similarly, Mean Y = Ybar =(1/n)sum of yi over i= 1, 2, …., n;…………….(3)

Syy = sum of (yi – Ybar)2 over i = 1, 2, …., n ………………………………………………(4)

Sxy = sum of {(xi – Xbar)(yi – Ybar)} over i = 1, 2, …., n………(5)

Correlation Coefficient of X and Y = rXY= Sxy/sq.rt(Sxx.Syy). …………………………(6)

Estimated Regression of Y on X is given by: Y = a + bX, where

b = Sxy/Sxx and a = Ybar – bYX.Xbar..…………………….(7)

Estimate of 2 is given by s2 = (Syy – b2Sxx)/(n - 2).

Standard Error of b is sb, where sb2 = s2/Sxx

100(1 - )% Confidence Interval (CI) for ycap at x = x0 is (a + bx0) ± tn – 2,/2xs[(1/n) + {(x0 – Xbar)2/Sxx}] Calculations

Details of Excel Calculations

n

13

xbar

0

ybar

700.846154

Sxx

182

Syy

18871.6923

Sxy

1849

b

10.1593407

a

700.846154

s

2.81346484

sb

0.20854789

sa

0.78031475

r

0.9976904

r^2

0.99538614

CIYcapLB

778.314733

CIYcapUB

789.990762

DATA

i

xi

yi

1

-6

642

2

-5

650

3

-4

660

4

-3

668

5

-2

682

6

-1

692

7

0

698

8

1

714

9

2

718

10

3

726

11

4

744

12

5

755

13

6

762

DONE

n

13

xbar

0

ybar

700.846154

Sxx

182

Syy

18871.6923

Sxy

1849

b

10.1593407

a

700.846154

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