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Math Review

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Title: Math Review


1
Math Review

2
First.
3
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 ?

4
The Man
5
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

6
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)

7
The Birthday Problem
  • What is the probability that at least two people
    in this class share the same birthday?

8
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.

9
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.
10
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.
11
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.
12
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 !!

13
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

14
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.

15
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)

16
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 !

17
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
18
Statistics
  • Data Collection
  • Summarizing Data
  • Interpreting Data
  • Drawing Conclusions from Data

19
Data Categories
Data
Quantitative (numerical)
Qualitative (categorical)
20
Qualitative Data
  • Ideas
  • Opinions
  • Categorical Evaluation
  • Examples
  • Color Preference
  • Favored Political Candidate
  • Quality Evaluation - Defective of non-defective

21
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
22
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

23
Time for some definitions
24
Population
  • The set of data (numerical or otherwise)
    corresponding to the entire collection of units
    about which information is sought

25
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

26
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

27
Sample
  • A subset of the population data that are actually
    collected in the course of a study.

28
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

29
Population vs. Sample
Population
Sample
30
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

31
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.

32
Types of Statistical Analysis
  • Descriptive Statistics
  • Graphical Tools
  • Numerical Measures
  • Inferential Statistics
  • Populations
  • Samples
  • Probability
  • Linking Descriptive and Inferential Statistics

33
Statistical Inference
Drawing Conclusions (Inferences) about a
Population Based on an examination of a Sample
taken from the population
34
Statistical Inference Examples
  • Nielson TV Ratings
  • Gallup and Harris Polls
  • Market Research
  • Financial Auditing
  • Opinion Surveys

35
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

36
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

37
Distributions
  • Two ways to describe a univariate distribution
  • a table
  • a graph (histogram, bar chart)

38
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

39
Distributions (cont.)
A Frequency Distribution Table
Category Percent Under 35 9 36-45 21 46-55 45 56-
65 19 66 6
40
Distributions (cont.)
A Histogram
41
Central Tendency
  • An estimate of the center of a distribution
  • Three different types of estimates
  • Mean
  • Median
  • Mode

42
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.

43
Working Example (mean)
  • Lets take the set of scores 15,20,21,20,36,15,
    25,15
  • The Mean would be 167/820.875

44
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.

45
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.

46
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.

47
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

48
Dispersion
  • Two estimates
  • Range
  • Standard Deviation
  • Standard Deviation is more accurate/detailed,
    because an outlier can greatly extend the range

49
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.

50
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.

51
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.

52
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.

53
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

54
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.

55
Describing Data Graphically
56
Tools for Describing Data
  • Graphical Tools
  • Pie Charts
  • Bar Charts
  • Histograms
  • Stem and Leaf Diagrams
  • Trend Charts
  • Many Variations of the above......

57
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

58
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
59
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

60
Frequency Distribution
A table that divides the data into classes and
shows the number of observed values that fall
into each class.
61
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
62
Forming a HistogramOn-Time Delivery Example
25 20 15 10 0
Frequency
x
0 1 2 3 4
Days Late
63
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.
64
Relative Frequency HistogramOn-Time Delivery
Example
.675 .50 .375 .25 0
Relative Frequency
x
0 1 2 3 4
Days Late
65
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
66
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

67
Concentrations PPM
Raw Data
68
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

69
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

70
Develop a Grouped Data Frequency Distribution -
Form a Data Array
Low
Sorted
High
71
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
72
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
73
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
74
Class Mark(Midpoint)
Midpoint lower limit .50 (Class
Interval) For first class Midpoint 0.00
.50(12.50)
6.25
75
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
76
Frequency Polygon
Frequency
Concentrations PPM
77
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
78
Histogram
Concentrations
79
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

80
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

81
Amplifier Data Arrayn120already sorted
Low
High
82
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!
83
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
84
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
85
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

86
Other Graphical Tools
  • Bar Charts
  • Pie Charts
  • Trend Charts
  • Quality Control Charts
  • Stem and Leaf Diagrams
  • Dot Plots
  • Others

87
Bar Charts
A graphical tool used to represent qualitative
data. Typically used when the available data
are in a summary form already.
88
Bar Chart ExampleForecasted Total Returns
Percent Return
89
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90
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91
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92
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
93
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94
Line Chart (Trend Chart)
95
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96
Line Charts(Figure 2-25)
Profit and sales going in opposite directions
97
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)

98
Scatter Plot Example
99
Scatter Plot Example
100
Other Data Displays
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