Intro to Simple Linear Regression
C1
C2
C1
C2
C1
C2
C1
C2
x
y
x
y
x
y
x
y
1
80
399
:
:
:
2
30
121
8
80
352
3
50
221
9
100
4
90
376
10
5
70
361
6
60
7
:
:
:
14
20
113
353
15
110
50
157
16
11
40
160
224
12
70
120
546
13
:
:
:
:
:
:
20
110
421
435
21
30
273
100
420
22
90
468
17
30
212
23
40
244
252
18
50
268
24
80
342
90
389
19
90
377
25
70
323
:
:
:
:
:
:
Example
The Toluca Company manufactures refrigeration equipment as well as many replacement parts.
In the past, one of the replacement parts has been produced periodically in lots of varying sizes.
When a cost improvement program was undertaken, company officials wished to determine the
optimum lot size for producing this part. The production of this part involves setting up the
production process (which must be done no matter what is the lot size), and machining and
assembly operations. One key input for the model to ascertain the optimum lot size was the
relationship between lot size and labor hours required to produce the lot. To determine this
relationship, data on lot size (# of lots) and work hours for 25 recent production runs were
utilized. All other production conditions were stable during the period in which the 25 runs were
made and were expected to continue to be stable during the period in which the cost
improvement program is to be conducted.
Descriptive Statistics
Variable
x
y
N
25
25
Mean
70.00
312.3
Median
70.00
342.0
TrMean
70.00
310.8
Variable
x
y
Minimum
20.00
113.0
Maximum
120.00
546.0
Q1
45.00
222.5
Q3
90.00
394.0
Intro to Simple Linear Regression
StDev
28.72
113.1
SE Mean
5.74
22.6
1
15
20
Percent
Percent
10
10
5
0
0
75
125
175
225
275
325
375
425
475
525
575
0
50
y
100
x
> Stat > Regression > Fitted line plot
Linear
> Type
Regression Plot
workhours = 62.3659 + 3.57020 lotsize
S = 48.8233
R-Sq = 82.2 %
R-Sq(adj) = 81.4 %
550
500
workhours
450
400
350
300
250
200
150
100
20
30
40
50
60
70
80
90
100
110
120
lotsize
Intro to Simple Linear Regression
2
> Stat > Regression > Regression > Options>
Prediction intervals for new observations:
[enter appropriate X values]
Regression Analysis
b0
The regression equation is
y = 62.4 + 3.57 lotsize
SE(b0)
For H0: 0 = 0
Predictor
Coef
SE Coef
T
P
Constant
62.37
26.18
2.38
0.026
lotsize
3.5702
0.3470
10.29
0.000
b1
For H0: 1 = 0
SE(b1)
S = 48.82
For HA: 0  0
R-Sq = 82.2%
R-Sq(adj) = 81.4%
For HA:  1  0
Analysis of Variance
Source
DF
SS
MS
F
P
1
252378
252378
105.88
0.000
Residual Error
23
54825
2384
Total
24
307203
Regression
Total squared
residuals
Total squared
variability in y
Unusual Observations (n – 2) = error degrees of freedom
Obs
x
y
Fit
SE Fit
Residual
21
30
273.00
169.47
16.97
103.53
St Resid
2.26R
R denotes an observation with a large standardized residual
> Options > Prediction intervals for new observations: 75 [was entered for X]
Predicted Values for New Observations
New Obs
1
Fit
SE Fit
330.13
9.92
95.0% CI
(
Predictors for New Observations
309.61, 350.65)
95.0% PI
(
227.07, 433.19)
Values of
SE  yˆ 
yˆ  62.37  3.5702(75)
New Obs
1
lotsize
75.0
Intro to Simple Linear Regression
3
The three most-seen residual plots.
Normal Probability Plot of the Residuals
Histogram of the Residuals
(response is workhour)
(response is workhour)
2
6
5
Normal Score
Frequency
1
4
3
2
0
-1
1
0
-2
-80
-60
-40
-20
0
20
40
60
80
-100
100
0
100
Residual
Residual
Residuals Versus the Fitted Values
(response is workhour)
Residual
100
0
-100
100
200
300
400
500
Fitted Value
> Options > Prediction intervals for new observations: 105 [was entered for X]
Predicted Values for New Observations
New Obs
1
Fit
437.24
SE Fit
15.58
(
95.0% CI
405.00, 469.47)
(
95.0% PI
331.22, 543.26)
Values of Predictors for New Observations
New Obs
1
lotsize
105
Intro to Simple Linear Regression
4