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Formula's to Determine the Least Squares Regression and Correlation Coefficient Assignment Solution


Instructions

(X) Weight (lbs)(Y) Fuel Economy (mi/gal
346621.5
250136.5
473119.0
389313.0
492519.5
322222.0
2899

46.0

A) Calculate the least squares line
        r=
C) Using the standard deviation for the residuals are there any data values that are more than 2 standard deviations above or below the least squares line?
Assignment Solution
(X) Weight (lbs)(Y) Fuel Economy (mi/gal
346621.5
250136.5
473119.0
389313.0
492519.5
322222.0
2899  46.0
A)
B) Calculate the least squares line
(X) Weight (lbs)(Y) Fuel Economy (mi/gal)XYX2Y2
346621.57451912013156462.25
250136.591286.562550011332.25
4731198988922382361361
3893135060915155449169
492519.596037.524255625380.25
3222227088410381284484
28994613335484042012116
Σ=25637Σ=177.5Σ=606579Σ=98847077Σ=5304.75

Thus, Regression: y = bx+a = -0.0088*b + 57.52

C) Find the correlation coefficient r.
Least Squares Regression Analysis

i) Using the standard deviation for the residuals are there any data values that are more than 2 standard deviations above or below the least squares line?
(X) Weight (lbs)(Y) Fuel Economy (mi/galyhaty-yhatres-mean(res-mean)2
346621.527.09011-5.59011-5.5817931.15638
250136.535.562810.937190.945510.893989
47311915.983413.016593.024919.150081
38931323.34105-10.3411-10.3327106.7653
492519.514.280095.219915.2282327.33439
32222229.23243-7.23243-7.2241152.18777
28994632.0683713.9316313.93995194.3222



Mean=-0.00832
Σ= 421.8101
SD for residuals =Least Squares Regression Analysis =8.3846
Using this value of the standard deviation of residuals, the points below and above two standard deviations from the least square line are calculated below:
(X) Weight (lbs)(Y) Fuel Economy (mi/gal)yhatyhat-2*SDYhat+2*SD
346621.527.0901110.3208943.85933
250136.535.5628118.7935952.33203
47311915.98341-0.7858132.75263
38931323.341056.5718340.11027
492519.514.28009-2.4891331.04931
32222229.2324312.4632146.00165
28994632.0683715.2991548.83759
It can be seen that all the data values lie between below and above two standard deviations from the least square line.