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
Y = response, dependent variable
x = regressor, independent variable, predictor
EGR Ch. 11

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
EGR Ch. 11

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
EGR Ch. 11

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 = ______________________________
EGR Ch. 11

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.
EGR Ch. 11

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)
EGR Ch. 11

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
EGR Ch. 11

8. Or … Using Excel EGR 252 - Ch. 11 yhat = intercept + “variable name”*x
standard error = std dev of the residuals
R2 = proportion of the variation in y that is explained by the regression line
Note the confidence interval … we can also draw a confidence interval around our predictions.
EGR Ch. 11