ECS 526 Statistics for Engineers & Scientists - Suny …web.cortland.edu/matresearch/FinProjEcsG203.pdf · ECS 526 – Statistics for Engineers & Scientists ... Reliability The data

ECS 526 – Statistics for Engineers & Scientists

Final Project Presentation

Fall- 2003

Mithun Sheth

Vivek Mehta

Nawaz Zahed

Swapana Reddy

Jayeda Khatun

Problem Statement

Problem Statement: - Boca Industrial tool Manufacturers and suppliers

located in Chicago supplies ball bearings mainly to automobile industry.

It faces local as well as global competition. Precision bearing

manufacturers of New York one of the major competitors of Boca

Industries supplies ball bearings to a list of car manufacturers. Global

motors get the competitive price from both of them and have to decide

on the best quality supplier. Considering that both the companies have

given almost the same price for the same quality of bearings. Global

Motors has decided to send a group of quality engineers to both

Manufacturers plants, to decide on which company manufactures

bearings of superior quality. The parameters of consideration are set by

Global Motors as Strength Pressure and Temperature.

Statistical Restatement

Statistical restatement of the problem means

translating the verbal problem statement into a

statistical problem statement, keeping in mind

the specific analysis process to be used for

analyzing the problem.

Approach to the Problem

• Exploratory Data Analysis (EDA)

• Assumptions EDA

• Data Collection and Discussion

• Organization and Data Analysis

• Basic Statistic

• Relationship among and within the data sets will also be

investigated

• Hypothesis Testing and Confidence Interval Derivation

• Modeling: Regression Analysis, ANOVA

• Quality Tools – Run Charts

• Conclusion and Recommendation to Management

Exploratory Data Analysis (EDA)

Objectives:-

•To Verify the Normality of the data sets

•To have an Initial Inferences from the collected

Data

•To check for any trend if present in the data set.

EDA: Data Collection and Source

Reliability

The data used here is collected from the NIST web site

given under the Plastic Data Set.

http://www.itl.nist.gov/div898/wrkshops/stats_sci/reg/plasti

c1.dat

We choose 100 Random data points from the data

presented and divided it into Two for our Problem.

Source Reliability:-

NIST is a well known and legitimate source as major

standards are set by NIST. So the data obtained is highly

reliable.

http://www.itl.nist.gov/div898/wrkshops/stats_sci/reg/plastic1.dat















Variable N Mean Median TrMean StDev SE Mean

Str Boca 50 29.922 30.400 29.748 6.942 0.982

Str Prec 50 29.00 27.15 28.85 7.42 1.05

Temp Boc 50 248.20 250.00 248.18 27.01 3.82

Temp Pre 50 247.80 240.00 247.50 31.77 4.49

Prs Boca 50 14.680 15.000 14.636 3.172 0.449

Pres Pre 50 15.080 16.000 15.091 3.361 0.475

Variable Minimum Maximum Q1 Q3

Str Boca 18.600 44.700 24.200 34.275

Str Prec 16.30 43.50 23.43 36.20

Temp Boc 200.00 300.00 227.50 270.00

Temp Pre 200.00 300.00 220.00 280.00

Prs Boca 10.000 20.000 12.000 18.000

Pres Pre 10.000 20.000 12.000 18.000

Basic Statistical Analysis Observations

Scatter Plot of Temperature (Boca) Vs. Strength (Boca) - Str Boca- - x x - 2 x x 40+ x - - x x - x 3 x x - x x x 2 x 30+ x x x x 3 - x 2 x x - - x x 2 3 - 2 x 20+ 3 x x - 2 - ----+---------+---------+---------+---------+---------+--Temp Boc 200 220 240 260 280 300

Scatter Plot of Pressure (Boca) Vs. Strength (Boca) Str Boca- - x x - 2 x x 40+ x - - x x - x 2 2 x - x x x x 2 30+ x x x 4 - 3 2 - - x x 4 x - 2 x 20+ x 2 2 - 2 - ----+---------+---------+---------+---------+---------+--Prs Boca 10.0 12.0 14.0 16.0 18.0 20.0

Scatter Plot of Pressure (Boca) Vs. Temperature (Boca) Temp Boc- x x - - x x 280+ 2 x x x - 2 x x - - x 2 x 5 - x 2 2 x 245+ - 2 x 2 2 x - x x - - x x 2 210+ x 3 x - x 2 - - ----+---------+---------+---------+---------+---------+--Prs Boca 10.0 12.0 14.0 16.0 18.0 20.0

Scatter Plot of Temperature (Precision) Vs. Strength (Precision) Str Prec- - x - x 2 40+ x x - 2 2 - x x x - x - x x x 30+ x x x x - 2 x x - x 2 x x x x - 2 2 x 2 - x 3 x 20+ x x - x 2 - x - ----+---------+---------+---------+---------+---------+--Temp Pre 200 220 240 260 280 300

Scatter Plot of Temperature (Precision) Vs. Strength (Precision) Str Prec- - x - 2 x 40+ x x - 2 2 - x 2 - x - x x x 30+ x x x x - 2 x x - 2 2 x x x - x 4 2 - x 3 x 20+ x x - x x x - x - ----+---------+---------+---------+---------+---------+--Pres Pre 10.0 12.0 14.0 16.0 18.0 20.0

Scatter Plot of Temperature (Precision) Vs. Pressure (Precision)

- x x 2 x x x Pres Pre- - - 2 x 3 x x x 17.5+ - - x x 2 2 2 x x 2 - - 14.0+ x x 2 x 2 - - - x x x x x x - 10.5+ - x 3 2 x 2 - ----+---------+---------+---------+---------+---------+--Temp Pre 200 220 240 260 280 300

200 210 220 230 240 250 260 270 280 290 300

20

30

40

Temp Boca

Str

Boca

10 12 14 16 18 20

20

30

40

Prs BocaS

tr B

oca

Box Plots (Boca Manufacturers)

10 12 14 16 18 20

15

25

35

45

Pres Prec

Str

Pre

c

200 210 220 230 240 250 260 270 280 290 300

15

25

35

45

Temp Prec

Str

Pre

cBox Plots (Precision Bearings)

15 25 35 45

0

1

2

3

4

5

6

7

Str Boca

Fre

qu

en

cy

Histogram of Str Boca, w ith Normal Curve

200 210 220 230 240 250 260 270 280 290 300

0

1

2

3

4

5

6

7

8

9

Temp Boca

Fre

qu

en

cy

Histogram of Temp Boca, w ith Normal Curve

10 11 12 13 14 15 16 17 18 19 20

0

5

10

Prs Boca

Fre

qu

en

cy

Histogram of Prs Boca, w ith Normal Curve

Histograms With Normal Curve for Boca Manufacturers

15 25 35 45

0

1

2

3

4

5

6

7

Str Prec

Fre

qu

en

cy

Histogram of Str Prec, w ith Normal Curve

200 210 220 230 240 250 260 270 280 290 300

0

5

10

Temp Prec

Fre

qu

en

cy

Histogram of Temp Prec, w ith Normal Curve

10 11 12 13 14 15 16 17 18 19 20

0

5

10

Pres Prec

Fre

qu

en

cy

Histogram of Pres Prec, w ith Normal Curve

Histograms With Normal Curve for Precision Bearings

Normal Probability Plot for Boca Manufacturers

Normal Probability Plot for Precision Bearing

18 22 26 30 34 38 42

95% Confidence Interval for Mu

27 28 29 30 31 32

95% Confidence Interval for Median

Variable: Str Boca

A-Squared:P-Value:

MeanStDevVarianceSkewnessKurtosisN

Minimum1st QuartileMedian3rd QuartileMaximum

27.9491

5.7990

27.4343

0.5120.187

29.9220 6.9422

48.19400.280528-6.8E-01

50

18.600024.200030.400034.275044.7000

31.8949

8.6509

32.3313

Anderson-Darling Normality Test


95% Confidence Interval for Sigma


Descriptive Statistics

200 220 240 260 280 300


240 250 260


Variable: Temp Boca

A-Squared:P-Value:



240.525

22.559

240.000

0.5780.126

248.200 27.006

729.347-8.6E-02-7.5E-01

50

200.000227.500250.000270.000300.000

255.875

33.654

260.000






10 12 14 16 18 20


14 15 16


Variable: Prs Boca

A-Squared:P-Value:



13.7787

2.6493

14.0000

1.5870.000

14.6800 3.1716

10.0588-1.1E-01-1.22682

50

10.000012.000015.000018.000020.0000

15.5813

3.9522

16.0000






Descriptive Statistics Boca Manufacturers

18 24 30 36 42


24.5 25.5 26.5 27.5 28.5 29.5 30.5 31.5


Variable: Str Prec

A-Squared:P-Value:



26.8919

6.2020

24.9015

0.8630.025

29.0020 7.4246

55.12510.392485-9.1E-01

50

16.300023.425027.150036.200043.5000

31.1121

9.2521

30.5657






200 220 240 260 280 300


230 240 250 260


Variable: Temp Prec

A-Squared:P-Value:



238.771

26.539

230.000

1.3380.002

247.800 31.770

1009.350.312876-1.14400

50

200.000220.000240.000280.000300.000

256.829

39.590

260.000






10 12 14 16 18 20


14 15 16


Variable: Pres Prec

A-Squared:P-Value:



14.1247

2.8080

14.0000

1.4140.001

15.0800 3.3615

11.2996-1.7E-01-1.14988

50

10.000012.000016.000018.000020.0000

16.0353

4.1889

16.0000






Descriptive Statistics Precision Bearing

From all the data analysis done earlier we can see the following

trend.

Mean

(Precision)

Std. Dev

(Precision)

Mean

(Boca)

Std Dev

(Boca)

Strength 29.00 7.42 29.22 6.942

Temperature 247.8 31.77 248.2 27.01

Pressure 15.08 3.361 14.68 3.172

EDA (Graphical) Summary/Observations

Hypothesis Testing

Approach

Test for variances

Test for Means

F-TEST FOR CHECKING THE EQUALITY OF VARIANCES

STRENGTH:

Hypothesis:

Ho: 1s2 - 2s2 = 0, 1s2 /2s2 = 1

H1: 1s2 > 2s2 or 1s2 < 2s2

F = (s1)2/(s2)2 = 0.9960

Critical Value : 2.16 and 0.4630

Decision:

Since the Test Statistic Value is > i.e. in the acceptance region. Null Hypothesis

is accepted.

Hence, we can conclude that the variances of two strength samples (Boca &

Precision) are equal.

19,19,05.0f 19,19,05.0f

TEMPERATURE:

Hypothesis:

Ho: 1s2 - 2s2 = 0, 1s2 /2s2 = 1

H1: 1s2 > 2s2 or 1s2 < 2s2

F = (s1)2/(s2)2 = 0.9657

Critical Value : 2.16 and 0.4630

Decision:

Since the Test Statistic Value is > i.e. in the acceptance region. Null

Hypothesis is accepted.

Hence, we can conclude that the variances of two strength samples (Boca

& Precision) are equal.

PRESSURE:

Hypothesis: Ho: 1s2 - 2s2 = 0, 1s2 /2s2 = 1 H1: 1s2 > 2s2 or 1s2 < 2s2 F = (s1)2/(s2)2 =1.047 Critical Value : 2.16 and 0.4630 Decision: Since the Test Statistic Value is > i.e. in the acceptance region. Null Hypothesis is accepted. Hence, we can conclude that the variances of two strength samples (Boca & Precision) are equal.

POOLED T-TEST FOR CHECKING THE EQUALITY OF MEANS

STRENGTH:

Hypothesis: Ho: 1s - 2s = 0 H1: 1s - 2s > 0 Test Statistic: 0.4369 Critical Value : 1.303 Rejection Region : >1.303 Decision: Since the Test Statistic value is <, Null Hypothesis is accepted

POOLED T-TEST FOR CHECKING THE EQUALITY OF MEANS TEMPERATURE Hypothesis: Ho: 1s - 2s = 0 H1: 1s - 2s > 0

Test Statistic: 0.7762 Critical Value : 1.303 Rejection Region : >1.303 Decision: Since the Test Statistic value is <, Null Hypothesis is accepted

POOLED T-TEST FOR CHECKING THE EQUALITY OF MEANS PRESSURE: Hypothesis: Ho: 1s - 2s = 0 H1: 1s - 2s > 0 Test Statistic: 0.5901 Critical Value : 1.303 Rejection Region : >1.303 Decision: Since the Test Statistic value is <, Null Hypothesis is accepted

Regression and Correlation

Best Subset

Best Subsets Regression: Str Boca versus Temp Boca, Prs Boca: Response is Str Boca

T

e P

m r

p s

B B

o o

Mallows c c

Vars R-Sq R-Sq(adj) C-p S a a

1 71.5 70.9 194.2 3.7434 X

1 31.0 29.6 535.9 5.8261 X

2 94.4 94.2 3.0 1.6734 X X

Best Subset(Continued)

Best Subsets Regression: Str Prec versus Temp Prec, Pres Prec Response is Str Prec

T P

e r

m e

p s

P P

r r

Mallows e e

Vars R-Sq R-Sq(adj) C-p S c c

1 72.2 71.6 299.1 3.9562 X

1 35.2 33.9 757.7 6.0371 X

2 96.2 96.1 3.0 1.4754 X X

Multiple Regression

Regression Analysis: Str Boca versus Temp Boca, Prs Boca The regression equation is

Str Boca = - 5.67 + 0.206 Temp Boca - 1.05 Prs Boca

Predictor Coef SE Coef T P

Constant -5.668 2.575 -2.20 0.033

Temp Boca 0.205643 0.008892 23.13 0.000

Prs Boca -1.05246 0.07572 -13.90 0.000

S = 1.67337 R-Sq = 94.4% R-Sq(adj) = 94.2%

PRESS = 149.508 R-Sq(pred) = 93.67%

Analysis of Variance

Source DF SS MS F P

Regression 2 2229.9 1114.9 398.17 0.000

Residual Error 47 131.6 2.8

Total 49 2361.5

Source DF Seq SS

Temp Boca 1 1688.9

Prs Boca 1 541.0

Unusual Observations

Temp

Obs Boca Str Boca Fit SE Fit Residual St Resid

1 240 30.700 26.846 0.264 3.854 2.33R

6 260 34.500 30.959 0.281 3.541 2.15R

27 210 20.500 16.467 0.555 4.033 2.55R

50 300 44.700 41.290 0.516 3.410 2.14R

R denotes an observation with a large standardized residual.

Residual

Pe

rce

nt

420-2-4

99

90

50

10

1

Fitted Value

Re

sid

ua

l

403020

4

2

0

-2

Residual

Fre

qu

en

cy

43210-1-2-3

16

12

8

4

0

Observation Order

Re

sid

ua

l

50454035302520151051

4

2

0

-2

Normal Probability Plot of the Residuals Residuals Versus the Fitted Values

Histogram of the Residuals Residuals Versus the Order of the Data

Residual Plots for Str Boca

Multiple Regression (Continued)

Regression Analysis: Str Prec versus Temp Prec, Pres Prec The regression equation is

Str Prec = - 0.13 + 0.184 Temp Prec - 1.09 Pres Prec

Predictor Coef SE Coef T P

Constant -0.127 2.024 -0.06 0.950

Temp Prec 0.183966 0.006688 27.51 0.000

Pres Prec -1.09137 0.06321 -17.27 0.000

S = 1.47537 R-Sq = 96.2% R-Sq(adj) = 96.1%

PRESS = 115.737 R-Sq(pred) = 95.72%

Analysis of Variance

Source DF SS MS F P

Regression 2 2598.8 1299.4 596.96 0.000

Residual Error 47 102.3 2.2

Total 49 2701.1

Source DF Seq SS

Temp Prec 1 1949.9

Pres Prec 1 649.0

Unusual Observations

Temp

Obs Prec Str Prec Fit SE Fit Residual St Resid

9 260 40.500 36.791 0.383 3.709 2.60R

16 230 30.500 26.906 0.254 3.594 2.47R

17 200 20.500 17.022 0.406 3.478 2.45R

18 230 28.400 31.272 0.413 -2.872 -2.03R

R denotes an observation with a large standardized residual.

Residual

Pe

rce

nt

420-2-4

99

90

50

10

1

Fitted Value

Re

sid

ua

l

403020

4

2

0

-2

Residual

Fre

qu

en

cy

3210-1-2-3

12

9

6

3

0

Observation Order

Re

sid

ua

l

50454035302520151051

4

2

0

-2

Normal Probability Plot of the Residuals Residuals Versus the Fitted Values

Histogram of the Residuals Residuals Versus the Order of the Data

Residual Plots for Str Prec

Correlation

Correlations for Boca bearings : Str Boca, Temp Boca, Prs Boca

Str Boca Temp Boca

Temp Boca 0.846

0.000

Prs Boca -0.557 -0.095

0.000 0.512

Cell Contents: Pearson correlation

P-Value

Correlation for Prec bearing : Str Prec, Temp Prec, Pres Prec

Str Prec Temp Prec

Temp Prec 0.850

0.000

Pres Prec -0.594 -0.126

0.000 0.382

Cell Contents: Pearson correlation

P-Value

ANOVA Analysis

One-way ANOVA: Str Boca versus Temp Boca Analysis of Variance for Str Boca Source DF SS MS F P Temp Boc 10 1837.8 183.8 13.69 0.000 Error 39 523.7 13.4 Total 49 2361.5 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ----+---------+---------+---------+-- 200 3 23.200 1.249 (---*---) 210 5 19.820 1.028 (--*--) 220 4 24.025 3.999 (---*---) 230 2 28.750 2.051 (----*----) 240 8 27.825 4.543 (--*-) 250 6 27.550 3.634 (--*--) 260 9 32.222 2.632 (-*--) 270 4 36.825 5.301 (---*---) 280 5 37.360 4.401 (--*---) 290 2 38.050 5.445 (----*----) 300 2 42.850 2.616 (----*----) ----+---------+---------+---------+-- Pooled StDev = 3.664 20 30 40 50

One-way ANOVA: Str Prec versus Temp Prec Analysis of Variance for Str Prec Source DF SS MS F P Temp Pre 10 2054.3 205.4 12.39 0.000 Error 39 646.9 16.6 Total 49 2701.1 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev -+---------+---------+---------+----- 200 4 20.025 3.734 (----*----) 210 3 24.167 3.453 (-----*-----) 220 6 22.683 4.468 (---*----) 230 10 26.010 3.570 (---*--) 240 4 26.075 4.589 (-----*----) 250 4 26.525 2.975 (----*----) 260 4 32.425 7.182 (-----*----) 270 2 31.500 4.667 (------*-------) 280 2 33.200 4.243 (-------*------) 290 6 37.933 3.354 (---*----) 300 5 40.840 2.557 (----*----) -+---------+---------+---------+----- Pooled StDev = 4.073 16.0 24.0 32.0 40.0

One-way ANOVA: Str Boca versus Prs Boca Analysis of Variance for Str Boca Source DF SS MS F P Prs Boca 5 920.5 184.1 5.62 0.000 Error 44 1441.0 32.8 Total 49 2361.5 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---------+---------+---------+------- 10 9 35.933 6.282 (----*-----) 12 8 31.763 6.660 (----*-----) 14 8 29.588 7.095 (-----*-----) 16 10 32.000 3.929 (----*----) 18 12 24.708 5.350 (---*----) 20 3 21.800 2.081 (--------*---------) ---------+---------+---------+------- Pooled StDev = 5.723 21.0 28.0 35.0

One-way ANOVA: Str Prec versus Pres Prec Analysis of Variance for Str Prec Source DF SS MS F P Pres Pre 5 1022.2 204.4 5.36 0.001 Error 44 1678.9 38.2 Total 49 2701.1 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ---------+---------+---------+------- 10 9 35.956 6.298 (------*------) 12 6 32.817 7.050 (--------*-------) 14 7 31.014 6.189 (-------*-------) 16 12 27.725 7.313 (-----*-----) 18 9 23.100 4.987 (------*-----) 20 7 24.557 4.002 (-------*-------) ---------+---------+---------+------- Pooled StDev = 6.177 24.0 30.0 36.0

Conclusion:

• We can deduce that the Precision data is

somewhat more consistent than the Boca

data, for Temperature v/s Strength.

• Both the data sets are equally consistent for

Pressure v/s Strength.

Quality Tools

Stat > Quality Tools > Run Char

Use the Run Chart command to look for evidence

of patterns in your process data. Run Chart plots

all of the individual observations versus the

subgroup number, and draws a horizontal

reference line at the median.

5.54.53.52.51.50.5

40

30

20

Subgroup Number

Str

Boca

0.90798

0.09202

3.00000

3.00000

2.00000

0.66874

0.33126

2.00000

3.40000

3.00000

Approx P-Value for Oscillation:

Approx P-Value for Trends:

Longest run up or down:

Expected number of runs:

Number of runs up or down:

Approx P-Value for Mixtures:

Approx P-Value for Clustering:

Longest run about median:


Number of runs about median:

Run Chart for Str Boca

5.54.53.52.51.50.5

40

30

20

Subgroup Number

Str

Prec

0.90798

0.09202

3.00000

3.00000

2.00000

0.66874

0.33126

3.00000

3.40000

3.00000











Run Chart for Str Prec

Run Chart of Strength of Boca and Precision

Conclusion : The Strength parameter in case of Boca Bearings is much more consistent than in case of Precision Bearings.

0.5 1.5 2.5 3.5 4.5 5.5

200

250

300

Subgroup Number

Tem

p B

oca











4.00000

3.40000

2.00000

0.74365

0.25635

3.00000

3.00000

2.00000

0.50000

0.50000

Run Chart for Temp Boca

0.5 1.5 2.5 3.5 4.5 5.5

200

250

300

Subgroup Number

Tem

p P

rec











2.00000

3.40000

3.00000

0.06332

0.93668

2.00000

3.00000

3.00000

0.09202

0.90798

Run Chart for Temp Prec

Run Chart for Temperature of Boca and Precision

Conclusion : The Temperature parameter in case of Boca Bearings is much more consistent than in case of Precision Bearings.

0.5 1.5 2.5 3.5 4.5 5.5

10

15

20

Subgroup Number

Prs B

oca











2.00000

3.40000

3.00000

0.06332

0.93668

2.00000

3.00000

2.00000

0.09202

0.90798

Run Chart for Prs Boca

0.5 1.5 2.5 3.5 4.5 5.5

10

15

20

Subgroup Number

Pres P

rec











3.00000

3.40000

2.00000

0.33126

0.66874

2.00000

3.00000

3.00000

0.09202

0.90798

Run Chart for Pres Prec

Run Chart for Pressure of Boca and Precision.

Conclusion : The Temperature parameter in case of Boca Bearings is much more consistent than in case of Precision Bearings.

Conclusion and Recommendations to the Management

• From the basic EDA plots of the data we can clearly observe that the

consistency of parameters as strength and temperature is better observed

in bearings manufactured by Boca Bearings than Precision Bearings.

• The Hypothesis testing part done on the data indicate that the data is more

or less equivalent (95% confidence interval) in terms of characteristics in

this case.

• The regression and correlation factors help us know the strength of

relationship between each of the parameters.

• ANOVA shows that temperature and strength relationship is more

consistent in Precision Bearings.

• In order to assess the quality of the bearings the Run charts are observed

to assess the quality of each bearings in case of both the companies.

This report thus concludes that Boca Bearings need to be considered

for usage since they have much more consistency than Precision

Bearings, and hence will help the company gain better products at the

same cost.

Documents

ECS 526 Statistics for Engineers & Scientists - Suny …web.cortland.edu/matresearch/FinProjEcsG203.pdf · ECS 526 – Statistics for Engineers & Scientists ... Reliability The data