1. Simple Linear Regression
Often we want to understand the relationships among variables, e.g.,
SAT scores and college GPA
car weight and gas mileage
amount of a certain pollutant in wastewater and bacteria growth in local streams
number of takeoffs and landings and degree of metal fatigue in aircraft structures
Simplest relationship 
Y = α + βx

2. Example
The owner of a small harness race track in Florida is interested in understanding the relationship between attendance at the track and the total amount bet each night. The data for a two-week period (10 racing nights) is as follows:
Attendance, x
Amount Bet ($000), Y
117
2.07
128
2.8
122
3.14
119
2.26
131
3.4
135
3.89
125
2.93
120
2.66
130
3.33
127
3.54

3. Estimating the Regression Coefficients
Method of Least Squares
Determine a and b (estimates for α and β) so that the sum of the squares of the residuals is minimized.)
Steps:
Calculate b using
and a using

4. For Our Example b = _______________________________________
Night
Attendance, x
Amount Bet, Y
xiyi
xi2 1
117
2.07
242.19
13689 2
128
2.8
358.4
16384 3
122
3.14
383.08
14884 4
119
2.26
268.94
14161 5
131
3.4
445.4
17161 6
135
3.89
525.15
18225 7
125
2.93
366.25
15625 8
120
2.66
319.2
14400 9
130
3.33
432.9
16900 10
127
3.54
449.58
16129
TOTAL
1254
30.02
157558
b =((10* )-(1254*30.02))/((10*157558)-1254^2) =
a = (30.02/10) – *(1254/10) =
b = _______________________________________
a = ______________________________

5. What does this mean?
We can draw the regression line that describes the relationship between attendance and amount bet:
We can also predict amount bet based on attendance.

6. How good is our prediction?
Estimating the variance:
Coefficient of determination, R2
a measure of the “quality of fit,” or the proportion of the variability explained by the fitted model.
SSE = sum(residuals2)=
s2 = SSE/8 =
SST = Σ(Yi - Y)2 =
R2 = 1-(SSE/SST) = 1-(0.639/2.945)=
(see next page)

7. Calculations … Night Attendance, x Amount Bet, Y xiyi xi2 yhat
residuals2 1
117
2.07
242.19
13689 2
119
2.26
268.94
14161 3
120
2.66
319.2
14400 4
122
3.14
383.08
14884 5
125
2.93
366.25
15625
2.9673 6
127
3.54
449.58
16129 7
128
2.8
358.4
16384
0.0408 8
130
3.33
432.9
16900 9
131
3.4
445.4
17161
0.1584 10
135
3.89
525.15
18225
TOTAL
1254
30.02
157558
(Y - Y)2

8. Or … Using Excel
Note the confidence interval … we can also draw a confidence interval around our predictions.