Engineering Probability and Statistics - SE-205 -Chap 1

By S. O. Duffuaa

Course Objectives



Introduce the students to basic probability and statistics and demonstrate its wide application in the area of Systems Engineering.

Main Course Outcomes



Students should be able to perform

: • • • • • •

Summarize and present data Describe probability distributions Compute probabilities using density/mass functions Conduct interval estimation Make inference about populations Use statistical package/Minitab

Text Book and References

  “Applied Statistics and Probability for Engineers “ by D. C. Montgomery and Runger, 1994.

“Probability and Statistics for Engineers and Scientists” 5 th by Walpole and Mayers.

 Statistics by Murry Speigel

Course Policy

     Home-works and attendance 15% Quizzes 15% Exam1 20% Exam II Final Exam 20% 30%

SE- 205 Place in SE Curriculum

 

Central Course

•

Prerequisite for 7 SE courses SE 303, SE 320, SE 323, SE 325, SE 447, SE 480, SE 463 and may be others. See SE Curriculum Tree

Engineering Problem Solving

 Develop clear and concise problem description  Identify the important factors in the problem.

 Propose a model for the problem  Conduct appropriate experimentation  Refine the model

Engineering Problem Solving

  Validate the solution Conclusion and recommendations

Statistics

•

Science of data collection, summarization, presentation and analysis for better decision making.

• • • •

How to collect data ?

How to summarize it ?

How to present it ?

How do you analyze it and make conclusions and correct decisions ?

Role of Statistics

 Many aspects of Engineering deals with data – Product and process design  Identify sources of variability  Essential for decision making

Data Collection

 • • Observational study Observe the system Historical data  The objective is to build a system model usually called empirical models  Design of experiment • Plays key role in engineering design

Data Collection

 Sources of data collection:  Observation of something you can’t control (observe a system or historical data)  Conduct an experiment  Surveying opinions of people in social problem



Divided into :

Statistics

•

Descriptive Statistics

•

Inferential Statistics

Forms of Data Description



Point summary

 

Tabular format Graphical format



Diagrams

Point Summary

1) Central tendency measures • Sample Mean x =  x i /n • Population Mean(µ) • Median --- Middle value • Mode --- Most frequent value • Percentile

Point Summary

2) Variability measures • • • Range = Max x i - Min x i Variance = V = S 2 =  (x i – x ) 2 / n-1 also =  (x i 2 ) – {[(  n -1 x i ) 2 ]/n} Standard deviation = S S = Square root (V) • Coefficient of variation = S/ x • Inter-quartile range (IQR)

Diagrams: Dot Diagram

 A diagram that has on the x-axis the points plotted : Given the following grades of a class: 50, 23, 40, 90, 95, 10, 80, 50, 75, 55, 60, 40.

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0 50 100

Dot Diagram

 A diagram that has on the x-axis the points plotted : Given the following grades of a class: 50, 23, 40, 90, 95, 10, 80, 50, 75, 55, 60, 40.

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.

.

.

0 50 100

Graphical Format

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Time Frequency Plot

The Time Frequency Plot tells the following : 1) The Center of Data 2) The Variability 3) The Trends or Shifts in the data •

Control Chart

11 y 10 9 8 7 6 5 15 14 13 12 0

Time Frequency Plot

10 20 30 Observation number 40 50

Time Frequency Plot

11 y 10 9 8 7 6 5 15 14 13 12 0 10 20 30 Observation number 40 50

Control Charts

105 Upper control limit = 100.5

95 x = 91.50

85 Lower control limit = 82.54

75 0 10 20 Observation number 30

Control Charts

• Central Line = Average ( X ) • Lower Control Limit (LCL)= X – 3S • Upper Control Limit (UCL)= X + 3S

Lecture Objectives



Sample and population



Random sample



Present the following:



Stem-leaf diagram

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The frequency distribution

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Histogram

Population and Sample

 Population is the totality of observations we are concerned with.

 Example: All Engineers in the Kingdom,  All SE students etc.

Sample : Subset of the population 50 Engineers selected at random, 10 SE students selected at random.

Mean and Variance

 Sample mean X-bar  Population mean µ  Sample variance S 2  Population variance σ 2

Stem-And –Leaf Diagram

 Each number x i is divided into

two parts the stem consisting of one or two leading digits



The rest of the digits constitute the leaf.



Example if the data is 126 then 12 is stem and 6 is the leaf. What is the stem and leaf for 76

Data Table 1.1 Compressive Strength of 80 Aluminum Lithium Alloy

105 221 183 186 121 181 180 143 97 154 153 174 120 168 167 141 245 228 174 199 181 158 176 110 163 131 154 115 160 208 158 133 207 180 190 193 194 133 156 123 134 178 76 167 184 135 229 146 218 157 101 171 165 172 158 169 199 151 142 163 145 171 148 158 160 175 149 87 160 237 150 135 196 201 200 176 150 170 118 149

Stem-And-Leaf f

Stem leaf frequency 7 6 8 7 9 7 10 5 1 11 5 0 8 12 1 0 3 3 13 4 1 3 5 3 5 6 14 2 9 5 8 3 1 6 9 2 3 8 1 1 1 15 4 7 1 3 4 0 8 8 6 8 0 8 12 16 3 0 7 3 0 5 0 8 7 9 10 17 8 5 4 4 1 6 2 1 0 6 10 18 0 3 6 1 4 1 0 7 19 9 6 0 9 3 4 6 20 7 1 0 8 21 8 22 1 8 9 4 1 3 23 7 1 24 5 1

Number of Stems Considerations

Stem Leaf 6 1 3 4 5 5 6 7 0 1 1 3 5 7 8 8 9 8 1 3 4 4 7 8 8 9 2 3 5

Stem 6L 6U 7L 7U 8L 8U 9L 9U

Stem number considerations

leaf 1 3 4 5 5 6 0 1 1 3 5 7 8 8 1 3 4 4 7 8 8 2 3 5



Number of Stems

Between 20 and 5  Roughly  n where n number of data points

Percentiles

 Pth percentile of the data is a value where at least P% of the data takes on this value or less and at least (1-P)% of the data takes on this value or more.

   Median is 50 th percentile. ( Q 2 ) First quartile Q 1 is the 25 th percentile.

Third quartile Q 3 is the 75 th percentile.

Percentile Computation : Example

Data : 5, 7, 25, 10, 22, 13, 15, 27, 45, 18, 3, 30 Compute 90 th percentile.

1. Sort the data from smallest to largest 3, 5, 7, 10, 13, 15, 18, 22, 25, 27, 30, 45 2. Multiply 90/100 x 12 = 10.8 round it to to the next integer which is 11.

Therefore the 90 th which is 30. percentile is point # 11

Percentile Computation : Example

 If the product of the percent with the number of the data came out to be a number. Then the percentile is the average of the data point corresponding to this number and the data point corresponding to the next number.

 Quartiles computation is similar to the percentiles.

 Pth percentile = (P/ 100)*n = r double (round it up & take its rank) (r)  integer (take Avg. of its rank & # after) Inter-quartile range = Q 3 – Q 1  Frequency Distribution Table : 1) # class intervals (k) = 5 < k < 20 k ~  n 2) The width of the intervals (W) = Range/k = (Max-Min) /  n

Class Interval (psi) 70 ≤

x

< 90 90 ≤

x

< 110 110 ≤

x

< 130 130 ≤

x

< 150 150 ≤

x

< 170 170 ≤

x

<1 90 190 ≤

x

< 210 210 ≤

x

< 230 230 ≤

x

< 250 || Tally (# data in this interval ) ||| |||| | |||| || |||| |||| |||| |||| |||| |||| |||| || |||| |||| |||| || |||| |||| Frequency 2 3 6 4 2 14 22 17 10 Relative Frequency = (Frequency/ n) 0.0250

Cumulative Relative Frequency 0.0250

0.0375

0.0750

0.0625

0.1375

0.1750

0.2750

0.2125

0.1250

0.0500

0.0250

0.3125

0.5875

0.8000

0.9250

0.9750

1.0000

25 20 15 10 5 0 70 90 110 130 150 170 190 210 230 250 Compressive Strength (psi)

30 20 10 0 90 80 70 60 50 40 1 Strength 100 150 200 250

Histogram

: is the graph of the frequency distribution table that shows class intervals V.S. freq. or (Cumulative) Relative freq.

Whisker extends to smallest data point within 1.5 interquartile ranges from first quartile Whisker extends to largest data point within 1.5 interquartile ranges from third quartile First Quartile Second Quartile Third Quartile Extreme Outliers 1.5 IQR Outliers 1.5 IQR IQR Outliers 1.5 IQR Extreme Outliers 1.5 IQR

100 150 200 250 Strength

120 110 100 90 80 70 1 2 Plant 3