Manual Calculations: First, summarize these data graphically in form of a scatter-plot. Do these data points group reasonable well along a sloped line? All Lines are defined by the general form Y = a + bX, where a is its intercept (the value of Y if X is zero) and b represents the slope (the rate of change in Y with X). Follow these step-by-step instructions to determine the line that best fits your data by obtaining estimates for a and b.

cases [i]	X	Y	Xi-	SSx = (Xi-)2	Yi-	SSy = (Yi-)2	Cross Productxy = (Xi-) * (Yi-)
1	0	-1.02
2	0	1.42
3	0	-0.26
4	0	-0.05
5	0	1.86
6	0	0.68
7	0	0.05
8	0	0.27
9	0	2.47
10	0	-1.39
11	1	-0.84
12	1	0.12
13	1	-0.57
14	1	-1.09
15	1	0.21
16	1	-0.13
17	1	0.94
18	1	0.87
19	1	-0.35
20	1	-0.40
Σ =

• Calculate the sum of all X: ΣX_i =
• Calculate the mean for X: = ΣX_i / n =
• Calculate for each case: (X_i-)
• Calculate for each case: (X_i-)²
• Sum the squares for SS_x =
• Calculate the sum of all Y: ΣY_i =
• Calculate the mean for Y:  = ΣY_i / n =
• Calculate for each case: (Y_i-)
• Calculate for each case: (Y_i-)²
• Sum the squares for SS_y =
• Count the number of cases: n =
• Calculate the Crossproduct_xy for each case: (X_i-) * (Y_i-)
• Sum the Crossproducts_xy =
• Calculate the slope: b = Σ(X_i-)(Y_i-) / Σ(X_i-)² =
• Calculate the intercept: a =  - b =
• The prediction equation for your regression line is: Y =
• 
  

Indicate the location of the regression line on your scatter plot. To test whether the data points are significantly associated with this line, we partition out the total sums of squares [Σ(Y_i-)²] into two components, those explained by the line [b Σ(X_i-)(Y_i-)] and those associated with deviations from it [Σ(Y_i-)²- b Σ(X_i-)(Y_i-)]. Mean squares are obtained by dividing the regression and error sum of squares by their respective degrees of freedom. Calculate your test statistic (F) by dividing the mean squares for the straight line model (MS_M) by the mean squares for the error (MS_E). Look up the F-ratio in the table of critical values for the F-distribution to obtain significance (p-value = 0.05) for 1 and n-2 degrees of freedom.

Source	Degrees of Freedom (df)	Sum of Squares (SS)	Mean Square (MS)	F-ratio
Regression	1	b Σ(Xi-)(Yi-)	SSM/1	MSM/MSE
Error	n-2	Σ(Yi-)2 - b Σ(Xi-)(Yi-)	SSE/(n-2)
Total	n-1	Σ(Yi-)2

Machine Formula: The initial calculations for a and b can be simplified when these calculation are performed with a computer program:

b = (ΣXY * (ΣX*ΣY/n)) / ( ΣX² * ((ΣX) ² /n))
a =  - b *

A summary of linear regression from the American Statistical Association

To double-check your solution, compare with these results