STAT%20551%20PROBABILITY%20AND%20STATISTICS%20I

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STAT 551PROBABILITY AND STATISTICS I

• INTRODUCTION

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WHAT IS STATISTICS?

• Statistics is a science of collecting data,
  organizing and describing it and drawing
  conclusions from it. That is, statistics is a way
  to get information from data. It is the science
  of uncertainty.

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WHAT IS STATISTICS?

• A pharmaceutical CEO wants to know if a new drug
  is superior to already existing drugs, or
  possible side effects.
• How fuel efficient a certain car model is?
• Is there any relationship between your GPA and
  employment opportunities?
• Actuaries want to determine risky customers for
  insurance companies.

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STEPS OF STATISTICAL PRACTICE

• Preparation Set clearly defined goals, questions
  of interests for the investigation
• Data collection Make a plan of which data to
  collect and how to collect it
• Data analysis Apply appropriate statistical
  methods to extract information from the data
• Data interpretation Interpret the information
  and draw conclusions

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STATISTICAL METHODS

• Descriptive statistics include the collection,
  presentation and description of numerical data.
• Inferential statistics include making inference,
  decisions by the appropriate statistical methods
  by using the collected data.
• Model building includes developing prediction
  equations to understand a complex system.

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BASIC DEFINITIONS

• POPULATION The collection of all items of
  interest in a particular study.

• SAMPLE A set of data drawn from the population
• a subset of the population available for
  observation

• PARAMETER A descriptive measure of the
• population, e.g., mean

• STATISTIC A descriptive measure of a sample

• VARIABLE A characteristic of interest about each
  
• element of a population or
  sample.

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EXAMPLE

• Population Unit
  Sample Variable
• All students currently Student Any
  department GPA
• enrolled in school
  Hours of works per
• 
  week
• All books in library Book
  Statistics Books Replacement cost
• Frequency of check out
• Repair needs
• All campus fast food Restaurant Burger King
  Number of employees
• restaurants Seating capacity
• Hiring/Not hiring

Note that some samples are not representative of
population and shouldnt be used to draw
conclusions about population. In the first
example, some students from all (or almost all)
departments would constitute a better sample.
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How not to run a presidential poll

• For the 1936 election, the Literary Digest picked
  names at random out of telephone books in some
  cities and sent these people some ballots,
  attempting to predict the election results,
  Roosevelt versus Landon, by the returns. Now,
  even if 100 returned the ballots, even if all
  told how they really felt, even if all would
  vote, even if none would change their minds by
  election day, still this method could be (and
  was) in trouble They estimated a conditional
  probability, used part of the American population
  which had phones, that part was not typical of
  the total population. Dudewicz Mishra, 1988

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STATISTIC

• Statistic (or estimator) is any function of a
  r.v. of r.s. which do not contain any unknown
  quantity. E.g.
• are statistics.
• are NOT.
• Any observed or particular value of an estimator
  is an estimate.

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RANDOM VARIABLES

• Variables whose observed value is determined by
  chance
• A r.v. is a function defined on the sample space
  S that associates a real number with each outcome
  in S.
• Rvs are denoted by uppercase letters, and their
  observed values by lowercase letters.
• Example Consider the random variable X, the
  number of brown-eyed children born to a couple
  heterozygous for eye color (each with genes for
  both brown and blue eyes). If the couple is
  assumed to have 2 children, X can assume any of
  the values 0,1, or 2. The variable is random in
  that brown eyes depend on the chance inheritance
  of a dominant gene at conception. If for a
  particular couple there are two brown-eyed
  children, we have x2.

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COLLECTING DATA

• Target Population The population about which we
  want to draw inferences.
• Sampled Population The actual population from
  which the sample has been taken.

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SAMPLING PLAN

• Simple Random Sample (SRS) All possible members
  are equally likely to be selected.
• Stratified Sampling Population is separated
  into mutually exclusive sets (strata) and then
  sample is drawn by using simple random samples
  from each strata.
• Convenience Sample It is obtained by selecting
  individuals or objects without systematic
  randomization.

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EXAMPLE

• A politician who is running for the office of
  mayor of a city with 25,000 registered voters
  runs a survey. In the survey, 48 of the 200
  registered voters interviewed say they plan to
  vote for her.
• What is the population of interest?
• What is the sample?
• Is the value 48 a parameter or a statistic?

The political choices of the 25,000 registered
voters
The political choices of the 200 voters
interviewed
Statistic
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EXAMPLE

• A manufacturer of computer chips claims that less
  than 10 of his products are defective. When 1000
  chips were drawn from a large production run,
  7.5 were found to be defective.
• What is the population of interest?
• What is the sample?
• What is parameter?
• What is statistic?
• Does the value 10 refer to a parameter or a
  statistics?
• Explain briefly how the statistic can be used to
  make inferences about the parameter to test the
  claim.

The complete production run for the computer
chips
1000 chips
Proportion of the all chips that are defective
Proportion of sample chips that are defective
Parameter
Because the sample proportion is less than 10,
we can conclude that the claim may be true.
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DESCRIPTIVE STATISTICS

• Descriptive statistics involves the arrangement,
  summary, and presentation of data, to enable
  meaningful interpretation, and to support
  decision making.
• Descriptive statistics methods make use of
• graphical techniques
• numerical descriptive measures.
• The methods presented apply both to
• the entire population
• the sample

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Types of data and information

• A variable - a characteristic of population or
  sample that is of interest for us.
• Cereal choice
• Expenditure
• The waiting time for medical services
• Data - the observed values of variables
• Interval and ratio data are numerical
  observations (in ratio data, the ratio of two
  observations is meaningful and the value of 0 has
  a clear no interpretation. E.g. of ratio data
  weight e.g. of interval data temp.)
• Nominal data are categorical observations
• Ordinal data are ordered categorical observations

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Types of data examples
Examples of types of data Examples of types of data
Quantitative Quantitative
Continuous Discrete
Blood pressure, height, weight, age Number of children Number of attacks of asthma per week
Categorical (Qualitative) Categorical (Qualitative)
Ordinal (Ordered categories) Nominal (Unordered categories)
Grade of breast cancer Better, same, worse Disagree, neutral, agree Sex (Male/female) Alive or dead Blood group O, A, B, AB
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Types of data analysis

• Knowing the type of data is necessary to properly
  select the technique to be used when analyzing
  data.
• Types of descriptive analysis allowed for each
  type of data
• Numerical data arithmetic calculations
• Nominal data counting the number of observation
  in each category
• Ordinal data - computations based on an ordering
  process

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Types of data - examples
Numerical data
Nominal
Age - income 55 75000 42 68000 . . . .
Person Marital status 1 married 2 single 3 sin
gle . . . .
Weight gain 10 5 . .
Computer Brand 1 IBM 2 Dell 3 IBM . . . .
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Types of data - examples
Numerical data
Nominal data
A descriptive statistic for nominal data is the
proportion of data that falls into each
category.
Age - income 55 75000 42 68000 . . . .
Weight gain 10 5 . .
IBM Dell Compaq Other Total 25
11 8 6 50
50 22 16 12
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Cross-Sectional/Time-Series/Panel Data

• Cross sectional data is collected at a certain
  point in time
• Test score in a statistics course
• Starting salaries of an MBA program graduates
• Time series data is collected over successive
  points in time
• Weekly closing price of gold
• Amount of crude oil imported monthly
• Panel data is collected over successive points in
  time as well

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Differences
Cross-sectional Time series Panel
Change in time Cannot measure Can measure Can measure
Properties of the series No series Long usually just one or a few series Short hundreds of series
Measurement time Measurement only at one time point even if more than one time point, samples are independent from each other Usually at regular time points (all series are taken at the same time points and time points are equally spaced) Varies
Measurements Response(s) time-independent covariates Response(s) time usually no covariate Response(s) time time-dependent and independent covariates
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GAMES OF CHANCE

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COUNTING TECHNIQUES

• Methods to determine how many subsets can be
  obtained from a set of objects are called
  counting techniques.

FUNDAMENTAL THEOREM OF COUNTING If a job
consists of k separate tasks, the i-th of which
can be done in ni ways, i1,2,,k, then the
entire job can be done in n1xn2xxnk ways.
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THE FACTORIAL

• number of ways in which objects can be permuted.
• n! n(n-1)(n-2)2.1
• 0! 1, 1! 1
• Example Possible permutations of 1,2,3 are
  1,2,3, 1,3,2, 3,1,2, 2,1,3, 2,3,1,
  3,2,1. So, there are 3!6 different
  permutations.

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COUNTING

• Partition Rule There exists a single set of N
  distinctly different elements which is
  partitioned into k sets the first set containing
  n1 elements, , the k-th set containing nk
  elements. The number of different partitions is

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COUNTING

• Example Lets partition 1,2,3 into two sets
  first with 1 element, second with 2 elements.
• Solution
• Partition 1 1 2,3
• Partition 2 2 1,3
• Partition 3 3 1,2
• 3!/(1! 2!)3 different partitions

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Example

• How many different arrangements can be made of
  the letters ISI?
• 1st letter 2nd letter 3rd letter

I
I
S
I
S
S
I
I
N3, n12, n21 3!/(2!1!)3
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Example

• How many different arrangements can be made of
  the letters statistics?
• N10, n13 s, n23 t, n31 a, n42 i, n51 c

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COUNTING

• Ordered, without replacement
• Ordered, with replacement
• 3. Unordered, without replacement
• 4. Unordered, with replacement

(e.g. picking the first 3 winners of a
competition)
(e.g. tossing a coin and observing a Head in the
k th toss)
(e.g. 6/49 lottery)
(e.g. picking up red balls from an urn that has
both red and green balls putting them back)
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PERMUTATIONS

• Any ordered sequence of r objects taken from a
  set of n distinct objects is called a permutation
  of size r of the objects.

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COMBINATION

• Given a set of n distinct objects, any unordered
  subset of size r of the objects is called a
  combination.

Properties
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COUNTING
Number of possible arrangements of size r from n objects Number of possible arrangements of size r from n objects
Without Replacement With Replacement
Ordered
Unordered
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EXAMPLE

• How many different ways can we arrange 3 books
  (A, B and C) in a shelf?
• Order is important without replacement
• n3, r3 n!/(n-r)!3!/0!6, or

Possible number of books for 1st place in the shelf Possible number of books for 2nd place in the shelf Possible number of books for 3rd place in the shelf
3 x 2 x 1
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EXAMPLE, cont.

• How many different ways can we arrange 3 books
  (A, B and C) in a shelf?
• 1st book 2nd book 3rd book

A
B
C
C
B
A
B
C
C
A
C
A
B
A
B
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EXAMPLE

• Lotto games Suppose that you pick 6 numbers out
  of 49
• What is the number of possible choices
• If the order does not matter and no repetition is
  allowed?
• If the order matters and no repetition is
  allowed?