1Prof. Indrajit Mukherjee, School of Management, IIT Bombay
•Supplies the data to confirm a hypothesis that two variables are related• Provides both a visual and statistical means to test the strength of a relationship• Provides a good follow-up to cause and effect diagrams
Scatter Diagram
*
*
**
**
2Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Mathematical Model Driven Quality Decisions
Y = F(x)
• Independent variables• Inputs, and In-Process Variables• Cause• Problem• Control• Input Conditions
• Dependent variable (s)• Output (s)
• Effect (s)• Symptom• Monitor• Response
Y X= X1 . . . X N
3Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Scatter PlotY
X
Dependent Variable (Output)
Independent Variable (Inputs)
4Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Scatter Diagram Example
volumeper day
costper day
23 125
26 140
29 146
33 160
38 167
42 170
50 188
55 195
60 200
5Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Scatter plot Examples
Y
XX
X X
Y Y
YLinear Relationships Curvilinear Relationships
6Prof. Indrajit Mukherjee, School of Management, IIT Bombay
X
Y
X
Y
X
Y
X
Y
Scatter plot ExamplesStrong Relationships Weak Relationships
(Continued)
7Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Scatter plot Examples
X
Y
X
Y
No Relationship(Continued)
8Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Types of Correlation
Positive Correlation Negative Correlation No Correlation
9Prof. Indrajit Mukherjee, School of Management, IIT Bombay
-3 -2 -1 0 1 2 3 0
6
5
43
2
1
X
Y
A Nonlinear Relationship for Which r = 0
10Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Calculation Example
TreeHeight
TrunkDiameter
yi xi xiyi yi2 xi2
35 8 280 1225 64
49 9 441 2401 81
27 7 189 729 49
33 6 198 1089 36
60 13 780 3600 169
21 7 147 441 49
45 11 495 2025 121
51 12 612 2601 144
321 73 3142 14111 713
11Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Excel OutputExcel Correlation OutputTools / data analysis / correlation….
Correlation betweenTree Height and Trunk Diameter
12Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Explaining Attitude Towardthe City of Residence
Respondent number
Attitude toward the city
Duration of the residence
1 6 102 9 123 8 124 3 45 10 126 4 67 5 88 2 29 11 1810 9 911 10 1712 2 2
13Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Simple Linear Regression
Simple Linear Regression Describes the linear relationship between a Predictor variable, plotted on the x-axis, and a response variable, plotted on the y-axis
Resp
onse
PredictorDe
pend
ent V
aria
ble
(Y)
Independent Variable (X)
14Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Types of Regression ModelsRegression
Models
Simple Multiple
Linear Non-LinearLinearNon-
Linear
X=1Variable
X≥2Variables
15Prof. Indrajit Mukherjee, School of Management, IIT Bombay
• Only one independent variable, x
• Relationship between x and y is described by a linear function
• Changes in y are assumed to be caused by changes in x
Simple Linear Regression Model
16Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Parameter Estimation Table [Volume Sales /month(Y) vs. Advertising/month (X)]
1 1 1 1 1
2 1 4 1 2
3 2 9 4 6
4 2 16 4 8
5 4 25 16 20
15 10 55 26 37
iX iY2iX
2iY i iX Y
17Prof. Indrajit Mukherjee, School of Management, IIT Bombay
observation number Hydrocorban number purity1 0.99 90.012 1.02 89.053 1.05 91.434 1.29 93.745 1.46 96.736 1.36 94.457 0.87 87.598 1.23 91.779 1.55 99.42
10 1.4 93.6511 1.19 93.5412 1.15 92.5213 0.98 90.5614 1.01 89.5415 1.11 89.8516 1.2 90.3917 1.26 93.2518 1.32 93.4119 1.43 94.9820 0.95 87.33
Empirical ModelsTable Oxygen and hydrocarbon levels
18Prof. Indrajit Mukherjee, School of Management, IIT Bombay
observation number
Hydrocorban number purity predicted value residual
1 0.99 90.01 89.069009 0.9409912 1.02 89.05 89.51836 -0.4681363 1.05 91.43 91.464353 -0.0343534 1.29 93.74 93.560279 0.1797215 1.46 96.73 96.105332 0.6246686 1.36 94.45 94.608242 -0.1582427 0.87 87.59 87.272501 0.3174998 1.23 91.77 92.662025 -0.8920259 1.55 99.42 97.452713 1.967287
10 1.4 93.65 95.207078 -1.55707811 1.19 93.54 92.063189 1.47681112 1.15 92.52 91.614062 0.90593813 0.98 90.56 88.9193 1.640714 1.01 89.54 89.368427 0.17157315 1.11 89.85 90.865571 -1.01551716 1.2 90.39 92.212898 -1.82289817 1.26 93.25 93.111152 0.13884818 1.32 93.41 94.009406 -0.59940619 1.43 94.98 95.656205 -0.67620520 0.95 87.33 88.470173 -1.140173
Adequacy of the Regression Model
19Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Sample Data for House Price Model
House Price in Rs.1000’s ( Y) Square Feet(x)245 1400312 1600279 1700308 1875199 1100219 1550405 2350324 2450319 1425255 1700
20Prof. Indrajit Mukherjee, School of Management, IIT Bombay
The DataData on sales of breadstick baskets andmargaritas for 25 weeks are shown below.
Breadstick
week orders margaritas1 860 13302 850 13503 800 12904 850 13505 880 13606 780 12507 815 12758 780 12509 750 1160
10 710 114011 740 114012 675 108013 720 114014 730 115015 645 102016 650 100017 730 120018 870 138019 890 139020 910 138021 940 140022 830 125023 840 125024 815 124525 800 1250
21Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Year Income(X) Retail sales(Y)1 9098 54922 9138 55403 9094 53054 9282 55075 9229 54186 9347 53207 9525 55388 9756 56929 10282 5871
10 10662 615711 11019 634212 11307 590713 11432 612414 11449 618615 11697 622416 11871 649617 12018 671818 12523 692119 12053 647120 12088 639421 12215 655522 12494 6755
Check This Regression Analysis in Excel
22Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Observation number
Pull strength
wire length
Die height
Observation number
Pull strength
wire length
Die height
1 9.95 540 585 600 400 205 360
2 24.45 8 110 15 21.65 4 205
3 31.75 11 120 16 17.89 4 400
4 35 10 550 17 69 20 600
5 25.02 8 295 18 10.3 1 585
6 16.86 4 200 19 34.93 10 540
7 14.38 2 375 20 46.59 15 250
8 9.6 2 52 21 44.88 15 290
9 24.35 9 100 22 54.12 16 510
10 27.5 8 300 23 56.63 17 590
11 17.08 4 412 24 22.13 6 100
12 37 11 400 25 21.15 5 400
13 41.95 12 500
Multiple Linear Regression Models
Example
23Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Multiple Linear Regression Models
Least Squares Estimation of the ParametersThe method of least squares may be used to estimate the regression Coefficient, in the multiple regression model, equation suppose that n>k Observations are available, and let xij denote the ith observation or level of variable xj, the observations are(xi1,xi2,…,xik,yi), i=1,2,...,n and n>kIt is customary to present the data for multiple regression in a table such as table. Table data for multiple regression
y x1 x2 … xk
Y1 x11 x12 … x1k
Y2 x21 x22 … x2k
…
yn xn1 xn2 … xnk
24Prof. Indrajit Mukherjee, School of Management, IIT Bombay
y x1 x2 … xk
Y1 x11 x12 … x1k
Y2 x21 x22 … x2k
…
yn xn1 xn2 … xnk
0 1 1 2 21 1 1 1
ˆ ˆ ˆ ˆ...n n n n
i i k ik ii i i i
n x x x y
2
0 1 1 1 2 1 2 1 11 1 1 1 1
ˆ ˆ ˆ ˆ...n n n n n
i i i i k i ik i ii i i i i
x x x x x x x y
2
0 1 1 2 21 1 1 1 1
ˆ ˆ ˆ ˆ...n n n n n
ik ik i ik i k i k ik ii i i i i
x x x x x x x y
Table data for multiple regression
25Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Hypothesis Tests in Multiple Linear Regression
Test for Significance of Regression
Source of variation
Sum of squares Degrees of freedom
Mean square F0
regression SSR k MSR MSR/MSE
Error or residual SSE n-p MSE
total SST n-1
2 2 2
1 1 1
ˆ ˆ( ) ( ) ( )n n n
i i ii i i
y y y y y y
T R ESS SS SS
0
// 1R
E
SS kF
SS n k
26Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Source of variation
Sum of squares
Degrees of freedom
Mean square F0
regression SSR k MSR MSR/MSE
Error or residual
SSE n-p MSE
total SST n-1
Hypothesis Tests in Multiple Linear RegressionTest for Significance of Regression
Source of variation
Sum of squares
Degrees of freedom
Mean square F0
regression 5990.7712 2 2995.3856 572.17
Error or residual 115.1735
22 5.2352
total6105.9447
24
27Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Observation number Pull strength wire length Die height Observation number Pull strength wire length Die height
1 9.95 8.38 1.57 14 11.66 12.26 -0.60
2 24.45 25.60 -1.15 15 21.65 15.81 5.84
3 31.75 33.95 -2.20 16 17.89 18.25 -0.36
4 35 96.60 -1.60 17 69 64.67 4.33
5 25.02 27.91 -2.89 18 10.3 12.34 -2.04
6 16.86 15.75 1.11 19 34.93 36.47 -1.54
7 14.38 12.45 1.93 20 46.59 46.56 -0.03
8 9.6 8.40 1.20 21 44.88 47.06 -2.18
9 24.35 28.21 -3.86 22 54.12 52.56 1.56
10 27.5 27.98 -0.48 23 56.63 56.31 0.32
11 17.08 18.40 -1.32 24 22.13 19.98 2.15
12 37 37.46 -0.46 25 21.15 21.00 0.15
13 41.95 41.46 0.49
Multiple Linear Regression ModelsExample
28Prof. Indrajit Mukherjee, School of Management, IIT Bombay
observation Temp(X) Feed rate(X2) Viscosity(Y)1 80 8 22562 93 9 23403 100 10 24264 82 12 22935 90 11 23306 99 8 23687 81 8 22508 96 10 24099 94 12 2364
10 9 11 237911 397 13 244012 95 11 236413 100 8 240414 85 12 231715 86 9 230916 87 12 2328
Assignment (Contd)Table:-
29Prof. Indrajit Mukherjee, School of Management, IIT Bombay
year revenue number of offices Profit margin()1 3.92 7298 0.752 3.61 6855 0.713 3.32 6636 0.664 3.07 6506 0.615 3.06 6450 0.76 3.11 6402 0.727 3.21 6368 0.778 3.26 6340 0.749 3.42 6349 0.910 3.42 6352 0.8211 3.45 6361 0.7512 3.58 6369 0.7713 3.66 6546 0.7814 3.78 6672 0.8415 3.82 6890 0.7916 3.97 7115 0.717 4.07 7327 0.6818 4.25 7546 0.7219 4.41 7931 0.5520 4.49 8097 0.6321 4.7 8468 0.5622 4.58 9717 0.4123 4.69 8991 0.5124 4.71 9179 0.4725 4.78 9318 0.32
30Prof. Indrajit Mukherjee, School of Management, IIT Bombay
Neural Networks
• Neural Network: A collection of neurons which areinterconnected. The output of one connects to several others with different strength connections.
– Initially, neural networks have no knowledge. (Allinformation is learned from experience using thenetwork.)
Neuron 1
Neuron 2
Output fromNeuron2
Output fromNeuron 1
Input 2
Input 3
Input 1
31Prof. Indrajit Mukherjee, School of Management, IIT Bombay
observation numer
Surface finish RPM
Type of cutting tool
observation numer
Surface finish RPM
Type of cutting tool
1 45.44 225 302 11 33.5 224 416
2 42.03 200 302 12 31.23 212 416
3 50.01 250 302 13 37.52 248 416
4 48.75 245 302 14 37.13 260 416
5 47.92 235 302 15 34.7 243 416
6 47.79 237 302 16 33.92 238 416
7 52.26 265 302 17 32.13 224 416
8 50.52 259 302 18 35.47 251 416
9 45.58 221 302 19 33.49 232 416
10 44.78 218 302 20 32.29 216 416
Multiple Regression ModelingWhat to Do in Such Cases?-Check Book