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 2.9643 meters, appears in the table as 643. 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	643	646	658	669	674	689	698	699	715	718	727	744	758

(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

a)

trend in lean over time appears to be linear

b)

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
75	643	36.00	2728.05	313.38
76	646	25.00	2423.67	246.15
77	658	16.00	1386.13	148.92
78	669	9.00	688.05	78.69
79	674	4.00	450.75	42.46
80	689	1.00	38.82	6.23
81	698	0.00	7.67	0.00
82	699	1.00	14.21	3.77
83	715	4.00	390.82	39.54
84	718	9.00	518.44	68.31
85	727	16.00	1009.28	127.08
86	744	25.00	2378.44	243.85
87	758	36.00	3939.98	376.62

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	1053	9038	182	15974.3	1695.00
mean	81.00	695.23	SSxx	SSyy	SSxy

sample size ,   n =   13          
here, x̅ = Σx / n=   81.00   ,     ȳ = Σy/n =   695.23  
                  
SSxx =    Σ(x-x̅)² =    182.0000          
SSxy=   Σ(x-x̅)(y-ȳ) =   1695.0          
                  
estimated slope , ß1 = SSxy/SSxx =   1695.0   /   182.000   =   9.3132
                  
intercept,   ß0 = y̅-ß1* x̄ =   -59.1374          
                  
so, regression line is   Ŷ =   -59.137   +   9.313   *x

R² =    (Sxy)²/(Sx.Sy) =    0.9882

98.8 percent of the variation in lean is explained by this line

c)

confidence interval for slope                  
α=   0.01              
t critical value=   t α/2 =    3.106   [excel function: =t.inv.2t(α/2,df) ]      
estimated std error of slope = Se/√Sxx =    4.13913   /√   182.00   =   0.307
                  
margin of error ,E= t*std error =    3.106   *   0.307   =   0.953
estimated slope , ß^ =    9.3132              
                  
                  
lower confidence limit = estimated slope - margin of error =   9.3132   -   0.953   =   8.36
upper confidence limit=estimated slope + margin of error =   9.3132   +   0.953   =   10.27