0.75/0.95 points | Previous Answers IDCollabStat2 12 HW 005 general The height (

Question

0.75/0.95 points | Previous Answers IDCollabStat2 12 HW 005 general The height (sidewalk to roof) of notable tall buildings in America is compared to the number o Height (in feet) Stories 1050 428 362 529 790 401 380 1454 1127 700 29 25 40 60 38 110 100 46 Part (a) O Part (b) a Part (c) Part (d) Find the correlation coefficient r.(Round your answer to four decimal places.) 09383 Is it significant? Yes No Part (e) Find the estimated height for 33 stories. (Use your equation from part (c).Round your answer to one decimal place) 4939

Explanation / Answer

Back-up Theory

Let X = number of stories and Y = height of the building (in feet). The regression model is:

Y = + X + , where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2. Then,

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

where n = sample size (10).

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 – b.Xbar..………………………………………..(7)

Calculations (based on Excel)

n

10

xbar

53

ybar

722.1

Sxx

8272.5

Syy

1285830.9

Sxy

96771.5

b

11.6979752

a

107.956301

r

0.93829015

r^2

0.8803884

So, the estimated regression line is: ycap = 107.956 + 11.698x.

Part (1)

Correlation coefficient, r = 0.9383 ANSWER

Part (2)

To test for significance of r, test statistic is: t = r.sqrt{(n - 2)/(1 – r2)} = 7.6740. This test statistic has t-distribution with 8 degrees (n -2) of freedom.

So, p-value = P(| t8 | > 7.6740) = 5.88E-05 (i.e., 0.0000588)

Since p-value is very low, r is significant ANSWER

Part (3)

Estimated height of a building with 33 stories is obtained by substituting x = 33 in the estimated regression equation, to get ycap = 493.99 = 494.0 ft ANSWER

Part (4 & 5)

Estimated height of a building with 3 stories cannot be found using this regression equation, because the range of x-values used for this estimation is 22 to 110 and 3 is far below the minimum. ANSWER. Second option

Part (6)

Since b = 11.698, every addition of one story would increase the height of the building by 11.698 ft. ANSWER

Part (7)

Slope is represented by b = 11.698 ANSWER 1

Interpretation:

Every addition of one story would increase the height of the building by 11.698 ft. ANSWER 2

n

10

xbar

53

ybar

722.1

Sxx

8272.5

Syy

1285830.9

Sxy

96771.5

b

11.6979752

a

107.956301

r

0.93829015

r^2

0.8803884

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