Bivariate Linear Regression: Least Squares Method
Recall from high school geometry the definition of a straight line
Y=a+bX   a=the Y interceptof the X axis
  b=the slope of the line
Switched for plotting
  Y X X Y  
Dealer Sales (000) Permits Permits Sales (000)    
1 77 86 86 77    
2 79 93 93 79    
3 80 95 95 80    
4 83 104 104 83    
5 101 139 139 101    
6 117 180 180 117    
7 129 165 165 129    
8 120 147 147 120    
9 97 119 119 97    
10 106 132 132 106    
11 99 126 126 99    
12 121 156 156 121    
13 103 129 129 103    
14 86 96 96 86    
15 99 108 108 99    
       
Correlation Permits Sales (000)
Permits 1
Sales (000) 0.9356056 1
Step 1: Estimate Beta - solve for Beta hat     Y Y square X X square XY
Dealer Sales (000)   Permits    
1 77 5929 86 7396 6622
2 79 6241 93 8649 7347
3 80 6400 95 9025 7600
4 83 6889 104 10816 8632
5 101 10201 139 19321 14039
6 117 13689 180 32400 21060
7 129 16641 165 27225 21285
8 120 14400 147 21609 17640
9 97 9409 119 14161 11543
beta hat = 15(193,345)-((1,497)*(1,875) 10 106 11236 132 17424 13992
15(245,759)-(1875)^2 11 99 9801 126 15876 12474
12 121 14641 156 24336 18876
beta hat = 93300 = 0.546 13 103 10609 129 16641 13287
170760 14 86 7396 96 9216 8256
15 99 9801 108 11664 10692
Step 2: Estimate the intercept - solve for a hat Sums 1497 153283 1875 245759 193345
   
Y-Bar 99.8 X-bar 125
a hat = 99.8-0.54638(125) = 31.5
Step 3: Estimate Y based on the regression, i.e. find Y hat
Y hat = 31.5+(0.546)X
We can now generate the line of best fit or the trend line.
Plot Y hat (or the best linear estimate of Y) for each X.
Actually, pick a low and a high X plot those and draw a straight line
  Y X   deviation Explained error Error
Dealer Sales (000) Permits Y Hat yhat-ybar Variance y-yhat Variance
1 77 86 78.46 -21.344 455.56634 -1.456 2.119936
2 79 93 82.28 -17.522 307.02048 -3.278 10.745284
3 80 95 83.37 -16.43 269.9449 -3.37 11.3569
4 83 104 88.28 -11.516 132.61826 -5.284 27.920656
5 101 139 107.39 7.594 57.668836 -6.394 40.883236
6 117 180 129.78 29.98 898.8004 -12.78 163.3284
7 129 165 121.59 21.79 474.8041 7.41 54.9081
8 120 147 111.76 11.962 143.08944 8.238 67.864644
9 97 119 96.47 -3.326 11.062276 0.526 0.276676
10 106 132 103.57 3.772 14.227984 2.428 5.895184
11 99 126 100.30 0.496 0.246016 -1.296 1.679616
12 121 156 116.68 16.876 284.79938 4.324 18.696976
13 103 129 101.93 2.134 4.553956 1.066 1.136356
14 86 96 83.92 -15.884 252.30146 2.084 4.343056
15 99 108 90.47 -9.332 87.086224 8.532 72.795024
Y-bar 99.8 Sum of Squares 3393.79   483.95
Step 4: F test for Statistically Significant Differences
Total Deviation = Devaition explained by the regression + Deviation not explained (residual error)
Y - Ybar = (Y hat - Y bar)+(Y-Y hat)
Total Variation = Variation explained by the regression + Variation not explained (residual error)
Sources of Variance Sum of Squares
df
Mean Square
 F-value p
Explained 3393.79
(K-1) = 1
 3393.79 91.1649 < .01
Error 483.95
(n-k) = 13
 37.2269
Total 3877.74
Step 5: Calculate the percentage of variance accounted for or r-square
r-square = 1- (SS-unexplained/SS-total)
r-square = 0.8752