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 2.9648 meters, appears in the table as 648. 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 648 649 661 673 678 693 702 704 718 723 731 747 763

(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

Part (a) SCATTER PLOT

760														•
750
													•
740
730												•
720											•
										•
710
700								•	•
L 690							•
E 680
						•
A 670					•
N 660				•
650
		•	•
640
		75	76	77	78	79	80	81	82	83	84	85	86	87
						Y	E	A	R

Scatter plot exhibits a clear upward linear trend. Answer 1

Back-up Theory for Parts (b) and (c)

The linear regression model: Y = β₀ + β₁X + ε, …………………….....................................................…………………..(1)

where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y_cap = β_0cap + β_1capX, ……..............................................……………….(2)

where β_1cap = Sxy/Sxx = r.√(Syy/Sxx) = r.(SDy/SDx) and

β_0cap = Ybar – β_1cap.Xbar..………………………..............................................................................................…….…..(3)

Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i …………………......................................................……………………….……….….(4)

Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i ………......................................................………………………………….……….….(5)

Sxx = Σ(i = 1 to n)(x_i – Xbar)² ……………….........................................................………………………………..…………....(6)

Syy = Σ(i = 1 to n)(y_i – Ybar)² …………………..........................................................…………………………..………………(7)

Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)} ……………........................................................………………………………….(8)

Correlation coefficient, r = Sxy/sqrt(Sxx. Syy) ……….....................................................…………………………….. ..(9)

Interpretation of regression coefficients, r and r²:

In the estimated regression of Y on X given by: Y = β_0cap + β_1capX,

β_0cap represents the y-intercept mathematically and physically represents the expected value of

the response (dependent) variable when the predictor (independent) variable is zero ……....................……(10)

β_1cap represents the slope of the regression line mathematically and physically represents the

expected change (increase/decrease) in value of the response (dependent) variable when the predictor (independent) variable changes (increases/decreases) by one unit…………….…........................................... .. (11)

The correlation coefficient between two variables, r_xy is a measure of the linear relationship between the

two variables. It takes values in the range [- 1, 1]. Both – 1 and 1 signify a perfect linear relationship and

as a corollary, as the value of r gets closer to ± 1, the linearity gets closer to perfection................................ (12)

r² represents the proportion of the variation in the response variable that is explained by the variation in the predictor variable and is called coefficient of determination............................................................................. (13)

Now, to work out the solution,

Let x = year and y = lean of the tower in the given units.

Part (b)

Vide (2), Equation of the least-squares line: y = 699.231 + 9,346x [year 1981 is reckoned as x = 0 and accordingly 1980 is – 1, 1982 is 1 and so on] Answer 2

r² = 0.9879

So, vide (13), 98.8 percent of the variation in lean is explained by this line. Answer 3

Part (c)

Vide (11), the required CI is that for β₁ and the interval is: [8.378, 10.314] Answer 4

Details of Calculations

n	13
Xbar	0.0000
ybar	699.2308
Sxx	182
Syy	16092.30769
Sxy	1701
β1cap	9.346153846
β0cap	699.2307692
s^2	17.68181818
s β1cap ^2	0.097152847
s	4.204975408
s β1cap	0.311693515
r	0.993938368
r^2	0.98791348
α	0.01
n-2	11
tn-2,α/2	3.105806514
CI β1 LB	8.378094098
CI β1UB	10.31421359

DONE