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Math Review

First.

Blow ot light bulbs. Psychokinesis ?

- A TV host turns to the main camera and with a

serious, coaxing air looks the viewer straight in

the eye and says - Go ahead! Turn on 5 or 6 lights around you and

see what happens - Then he turns to the medium and says
- Do you really think you can do it ?
- After hesitating a few moments, the medium

replies - I hope I have enough concentration this evening,

but the conditions are not ideal. To produce

long-distance phenomena like this I usually spend

a few days in complete and utter solitude, after

rigorous fasting. (if she fails the public will

blame the circumstances, not her abilities ) - The medium does not fail ! Light bulbs do blow

out in the homes of viewers of this program, and

over 1000 viewers call the TV station to testify

! - The medium has successfully focused her spiritual

power on the material world and blown out light

balls far away! - Amazing, right ?

The Man

Lets examine this a little more closely

- Suppose 1 million people were watching the show
- ? 5 or 6 million light bulbs were on for an hour

or more - Assume , considering economics, 2 million light

bulbs were on for 1 hour - On average a light bulb lasts 1,000 hours
- Among the light bulb installed at random by

viewers, there is no reason to think that they

tend to be very old or very new - Among the 2 million, there are
- 2000 with 1 hour of life used, 2000 with 2 hours

of life used, 2000 with 3 hours 2000 with 999

hours of life used, 2000 with 1000 hours of life

used - Thus, during a 1-hour show , those last 2000

bulbs will reach the end of their life span and

burn out

What, Youre a Scorpio too ?

- Thats amazing !! WOW
- Whats the probability that at least 2 people in

any party have the same birthday? (month and day)

The Birthday Problem

- What is the probability that at least two people

in this class share the same birthday?

Assumptions

- Only 365 days each year.
- Birthdays are evenly distributed throughout the

year, so that each day of the year has an equal

chance of being someones birthday.

Take group of 5 people.

Let A event no one in group shares same

birthday. Then AC event at least 2 people share

same birthday. P(A) 364/365 363/365

362/365 361/365 360/365 0.973 P(AC) 1

- 0.973 0.027 That is, about a 3 chance that

in a group of 5 people at least two people share

the same birthday.

Take group of 23 people.

Let A event no one in group shares same

birthday. Then AC event at least 2 people share

same birthday. P(A) 364/365 363/365

342/365 0.493 P(AC) 1 - 0.493 0.507 That

is, about a 50 chance that in a group of 23

people at least two people share the same

birthday.

Take group of 50 people.

Let A event no one in group shares same

birthday. Then AC event at least 2 people share

same birthday. P(A) 364/365 363/365

315/365 0.03 P(AC) 1 - 0.03 0.97 That is,

virtually certain that in a group of 50 people

at least two people share the same birthday.

Premonition?

- Youre peacefully lying in your bed.
- It is 604 in the morning, and youre hardly

awakened when youre struck by the thought of

your cousin, whom you havent seen for years and

whom you havent thought of for a long time

either. - Now at 608 the damn phone rings and you pick it

up, only to hear the sad news - Your cousin has died !
- Here is the long-awaited proof that premonition

is for real !!

Premonition--Debunked

- Put the question like this
- What is the probability that , having thought

about a person, we will somehow learn in the next

5 minutes, purely by coincidence and without any

paranormal influence, that that person has died

? - We need to know 2 things

Consider

- 1. The number of people whose death comes to our

attention during say 1 year. - 2. The number of times one thinks of these people

during the same period. - 1. Assumption you know 10 people whose death you

learn over a 1 year period - 2. Assumption you think of each of those people

a single time over the 1 year period.

Consider

- 1 particular person among the 10.
- 1 year has 105,120 five min. intervals
- The chance that well be informed of his death

during that 5 min. interval is - 1 in 105,120 (small!)
- What about the other 9 people ?
- For each of them the probability of
- having the thought then learning of their death

is 1 in 105,120 - Addition rule
- P(having the thought then learning of their

death of any of the 10) 1/10,512 (still

small)

Hate to tell youbut

- There is nothing unique about you in this respect

- There are about 250,000,000 people in the US
- So the thought - notification connection must

occur each year to about - 1/10,512 x 250,000,000 23,782 people
- So, by chance alone there are 65 cases like this

each day in the US !

Statistics

?

Probability

Science of data

Science of chance, uncertainties

collecting, processing, presentation,

analyzing interpretation of data

what is possible , what is probable

numbers with context

mathematical formulas

Statistics

- Data Collection
- Summarizing Data
- Interpreting Data
- Drawing Conclusions from Data

Data Categories

Data

Quantitative (numerical)

Qualitative (categorical)

Qualitative Data

- Ideas
- Opinions
- Categorical Evaluation
- Examples
- Color Preference
- Favored Political Candidate
- Quality Evaluation - Defective of non-defective

Quantitative Data

Annual Income Football Attendance Interest

Rates Dow Jones Industrials Average Number of

Defective Parts in a Shipment Number of Late

Deliveries Last Month Percentage of Satisfied

Customers

Discrete

Continuous

Data Collection

- Designing experiments
- Does adjusting the oxygen-fuel ratio in an

automotive fuel injection system improve emission

quality? - Observational studies
- Polls - Bushs (dis) approval rating

Time for some definitions

Population

- The set of data (numerical or otherwise)

corresponding to the entire collection of units

about which information is sought

Population Examples

- Air QualityValues from all sampling devices in

the country - Unemployment - Status of ALL employable people

(employed, unemployed) in the U.S. - SAT Scores - Math SAT scores of EVERY person that

took the SAT during 2002 - Responses of ALL currently enrolled underage

college students as to whether they have consumed

alcohol in the last 24 hours

Population Examples cont.

- Again Population Defined
- The Collection of All Items of Interest

(Universe) - All People Living in Georgia
- All HP Laser-jet Printers Sold in 2001
- All Accounts Receivable Balances
- All Homeowners in Atlanta
- over 35 years old
- employed
- married
- 2 or more children

Sample

- A subset of the population data that are actually

collected in the course of a study.

Sample Examples

- Air QualityValues from samples at Midwestern

urban sites during July - Unemployment - Status of the 1000 employable

people interviewed. - SAT Scores - Math SAT scores of 20 people that

took the SAT during 2002 - Responses of 538 currently enrolled underage

college students as to whether they have consumed

alcohol in the last 24 hours

Population vs. Sample

Population

Sample

Samples

- Again Sample Defined
- A Subset of a population.
- A Representative Sample
- Has the characteristics of the population
- Census - A Sample that Contains all Items in the

Population

WHO CARES?

- In most studies, it is difficult to obtain

information from the entire population. We rely

on samples to make estimates or inferences

related to the population.

Types of Statistical Analysis

- Descriptive Statistics
- Graphical Tools
- Numerical Measures
- Inferential Statistics
- Populations
- Samples
- Probability
- Linking Descriptive and Inferential Statistics

Statistical Inference

Drawing Conclusions (Inferences) about a

Population Based on an examination of a Sample

taken from the population

Statistical Inference Examples

- Nielson TV Ratings
- Gallup and Harris Polls
- Market Research
- Financial Auditing
- Opinion Surveys

Review of Descriptive Stats.

- Descriptive Statistics are used to present

quantitative descriptions in a manageable form. - This method works by reducing lots of data into a

simpler summary. - Example
- Batting Average in baseball
- Creightons Grade Point System

Univariate Analysis

- This is the examination across cases of one

variable at a time. - Frequency distributions are used to group data.
- One may set up margins that allow us to group

cases into categories. - Examples include
- age categories
- price categories
- temperature categories
- concentration categories

Distributions

- Two ways to describe a univariate distribution
- a table
- a graph (histogram, bar chart)

Distributions (cont)

- Distributions may also be displayed using

percentages. - For example One could use percentages to

describe the - percentage of people under the poverty level
- over a certain age
- over a certain score on a standardized test
- days with a AQI 100

Distributions (cont.)

A Frequency Distribution Table

Category Percent Under 35 9 36-45 21 46-55 45 56-

65 19 66 6

Distributions (cont.)

A Histogram

Central Tendency

- An estimate of the center of a distribution
- Three different types of estimates
- Mean
- Median
- Mode

Mean

- The most commonly used method of describing

central tendency. - One basically totals all the results and then

divides by the number of units or n of the

sample. - Example The ATS-542 Homework mean was determined

by the sum of all the scores divided by the

number of students turning in the HW.

Working Example (mean)

- Lets take the set of scores 15,20,21,20,36,15,

25,15 - The Mean would be 167/820.875

Median

- The median is the score found at the exact middle

of the set. - One must list all scores in numerical order, and

then locate the score in the center of the

sample. - Example if there are 500 scores in the list,

score 250 would be the median. - This is useful in weeding out outliers.

Working Example (median)

- Lets take the set of scores 15,20,21,20,36,15,

25,15 - First line up the scores.
- 15,15,15,20,20,21,25,36
- The middle score falls at 20. There are 8 scores

and score 4 and 5 represent the halfway point.

Mode

- The mode is the most repeated score in the set of

results. - Lets take the set of scores 15,20,21,20,36,15,

25,15 - Again we first line up the scores
- 15,15,15,20,20,21,25,36
- 15 is the most repeated score and is therefore

labeled the mode.

Central Tendency

- If the distribution is normal (i.e.,

bell-shaped), the mean, median and mode are all

equal - In our analyses, well use the mean

Dispersion

- Two estimates
- Range
- Standard Deviation
- Standard Deviation is more accurate/detailed,

because an outlier can greatly extend the range

Range

- The range is used to identify the highest and

lowest scores. - Lets take the set of scores 15,20,21,20,36,15,

25,15 - The range would be 15-36. This identifies the

fact that 21 points separates the highest to the

lowest score.

Standard Deviation

- The Standard Deviation is a value that shows the

relation that individual scores have to the mean

of the sample. - If scores are said to be standardized to a normal

curve then there are several statistical

manipulations that can be performed to analyze

the data set.

Standard Dev. (cont)

- Assumptions may be made about the percentage of

scores as they deviate from the mean. - If scores are normally distributed, then one can

assume that approximately 69 of the scores in

the sample fall within one standard deviation of

the mean. Approximately 95 of the scores would

then fall within two standard deviations of the

mean.

Standard Dev. (cont)

- The standard deviation calculates the square root

of the sum of the squared deviations from the

mean of all the scores divided by the number of

scores. - This process accounts for both positive and

negative deviations from the mean.

Working Example (stand. dev.)

- Lets take the set of scores 15,20,21,20,36,15,

25,15 - The mean of this sample was found to be 20.875.

Round up to 21. - Again we first line up the scores
- 15,15,15,20,20,21,25,36.
- 21-156, 21-156, 21-156,21-201,21-201,

21-210, 21-25-4, 21-36-15

Working Ex. (Stan. dev. cont)

- Square these values.
- 36,36,36,1,1,0,16,225
- Total these values. 351.
- Divide 351 by 8 43.8
- Take the square root of 43.8 6.62
- 6.62 is your Standard Deviation.

Describing Data Graphically

Tools for Describing Data

- Graphical Tools
- Pie Charts
- Bar Charts
- Histograms
- Stem and Leaf Diagrams
- Trend Charts
- Many Variations of the above......

Analyzing Quantitative DataOn-Time Delivery

Example

- Variable x Number of days Delivery is Late
- (Each data point represents one shipment.)
- Raw Data
- 0 2 3 4 1 0 0 1
- 3 0 3 1 1 0 0 0
- 2 2 0 0 0 1 2 0
- 4 1 0 1 0 0 0 1
- 1 0 0 0 0 1 3 1
- N 40 shipments

Organizing the DataStep 1

Form a Data Array Sort the data in numerical

order

Raw Data 0 2 3 4 1 0 0 1 3 0 3 1 1 0 0 0 2 2

0 0 0 1 2 0 4 1 0 1 0 0 0 1 1 0 0 0 0 1 3 1

Data Array

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1

1 1 1 1 1 2 2 2 2 3 3 3 3 4 4

Low

High

Organizing the DataStep 2Construct a Frequency

Distribution

- Ungrouped Frequency Distribution
- When the variable has only a few different values
- Number of data values may be high or low
- Grouped Data Frequency Distribution
- When the variable has more than a few different

values - Number of data values is high

Frequency Distribution

A table that divides the data into classes and

shows the number of observed values that fall

into each class.

Frequency DistributionOn-Time Delivery Example

Use ungrouped Frequency Distribution since the

variable takes on only a few different values.

Data Array

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1

1 1 1 1 1 2 2 2 2 3 3 3 3 4 4

Low

High

x Frequency

0 19 1 11 2 4 3 4 4 2

Frequency Distribution

N 40 values

Forming a HistogramOn-Time Delivery Example

25 20 15 10 0

Frequency

x

0 1 2 3 4

Days Late

Relative Frequency DistributionOn-Time Delivery

Example

x Frequency Relative Frequency

0 19 19/40 .475 1 11

11/40 .275 2 4

4/40 .100 3 4

4/40 .100 4 2

2/40 .050

40

1.000

Relative Frequency Distributions are useful for

comparing two or more data sets which have

different volumes of data.

Relative Frequency HistogramOn-Time Delivery

Example

.675 .50 .375 .25 0

Relative Frequency

x

0 1 2 3 4

Days Late

Cumulative Frequency DistributionOn-Time

Delivery Example

X F CF

Cumulative Frequency Histogram

0 19 19 1 11 30 2 4

34 3 4 38 4 2 40

40

Type of Frequency Distributions

- Ungrouped Frequency Distribution
- When the variable has only a few different values
- Number of data values may be high or low
- Grouped Data Frequency Distribution
- When the variable has more than a few different

values - Number of data values is high

Concentrations PPM

Raw Data

Grouped Data Frequency Distribution

- Class - data category
- Frequency - number of items in each class
- Class limits - boundaries of each class
- Class interval - width of each class
- difference between lower limits of a class and

the preceding class lower limit - Class Mark - midpoint of the class

Guidelines for Grouped Frequency Distributions

- Mutually Exclusive Classes - no overlap
- All inclusive - a place for each data point
- Equal width classes (if possible)
- 5-12 classes (rule of thumb)
- Try to have round numbered class limits
- Avoid open-ended classes

Develop a Grouped Data Frequency Distribution -

Form a Data Array

Low

Sorted

High

Forming the Class Limits

Class Interval High Value - Low Value

number of

classes

Try 6 classes

Class Interval 74.95 - 0.97

12.33

6

round to nicer interval -- 12.50

Class Limits

Classes

All Inclusive Mutually Exclusive Equal

Width No Open-Ended Classes

0.00 and under 12.50 12.50 and under

25.00 25.00 and under 37.50 37.50 and under

50.00 50.00 and under 62.50 62.50 and under 75.00

Frequency DistributionConcentrations

Classes Frequency

0.00 and under 12.50

38 12.50 and under 25.00

14 25.00 and under 37.50

4 37.50 and under 50.00

2 50.00 and under 62.50

1 62.50 and under 75.00 5

64

Class Mark(Midpoint)

Midpoint lower limit .50 (Class

Interval) For first class Midpoint 0.00

.50(12.50)

6.25

Frequency Distribution With Midpoints

Classes Frequency

Midpoint

0.00 and under 12.50 38

6.25 12.50 and under 25.00 14

18.75 25.00 and under 37.50

4 31.25 37.50 and under

50.00 2 43.75 50.00

and under 62.50 1

56.25 62.50 and under 75.00 5

68.75

64

Frequency Polygon

Frequency

Concentrations PPM

Cumulative Frequency DistributionConcentrations

Example

Classes Frequency

Midpoint Cumulative Freq.

0.00 and under 12.50 38

6.25 38 12.50 and

under 25.00 14 18.75

52 25.00 and under 37.50

4 31.25

56 37.50 and under 50.00

2 43.75

58 50.00 and under 62.50 1

56.25 59 62.50

and under 75.00 5

68.75 64

64

Histogram

Concentrations

Histograms

- A Graphical Summary of Variation in a Set of Data
- Key Concepts
- Generated data will show variation because of

many factors - process equipment, materials, people,

environment, etc. - The variation will display a pattern

(distribution) - Patterns are hard to see in data tables
- Histograms make it easier to see patterns

Cable TV Amplification Example

- Amplifiers made to boost cable TV signals (Gain)
- Complaints about weak signals in outlying areas
- Amplifiers are the prime suspect
- Specifications
- nominal (average) gain is 10 units
- Amplifiers to provide between 7.75 and 12.25

units gain. - Tests conducted on 120 amplifiers

Amplifier Data Arrayn120already sorted

Low

High

First Pass Conclusion

Specifications Gain 7.75 ------ 12.25

Since all 120 amplifiers tested fall between 7.8

and 11.7 the problem cant be the

amplifiers. They all meet specifications!

The Frequency DistributionAmplifier Test Data

Class Frequency

Relative Frequency 7.75 - 8.25 24 .20 8.26 -

8.75 28 .23 8.76 - 9.25 26 .22 9.26 -

9.75 19 .16 9.76 - 10.25 12 .10 10.26 -

10.75 7 .06 10.76 - 11.25 2 .02 11.26 -

11.75 2 .02

120

Frequency Histogram

30 25 20 15 10 5 0

Nominal Specification 10.0 gain

Frequency

7.75 8.25 8.75 9.25

9.75 10.25 10.75 11.25

11.75

Gain

Amplifier ExampleNew Conclusions

- Distribution of gains is not evenly spread around

the nominal target - All amplifiers do operate within specifications
- Most amplifiers provide gains below nominal

target of 10 units 85 percent - There is a wide variation in performance of

individual amplifiers in the test - By random assignment it would be possible to get

a series of below target amplifiers, thus

generating a weak signal - The company needs to focus on why the amplifiers

are not spread more evenly around the target of 10

Other Graphical Tools

- Bar Charts
- Pie Charts
- Trend Charts
- Quality Control Charts
- Stem and Leaf Diagrams
- Dot Plots
- Others

Bar Charts

A graphical tool used to represent qualitative

data. Typically used when the available data

are in a summary form already.

Bar Chart ExampleForecasted Total Returns

Percent Return

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Pie Charts

T o t a l F e d e r a l F u n d s ( O u t

l a y s ) 1, 4 3 8 B i l l i o n

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Line Chart (Trend Chart)

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Line Charts(Figure 2-25)

Profit and sales going in opposite directions

Scatter DiagramsDependent and Independent

Variables

- A dependent variable is one whose values are

thought to be a function of the values of another

variable. (y-axis) - An independent variable is one whose values are

thought to impact the values of the dependent

variable. (x-axis)

Scatter Plot Example

Scatter Plot Example

Other Data Displays