Stat 115 Chapter 1: Basic Statistical Methods

• Stat 115: Basic Statistical Methods is covered in this material.
• Chapter 1 covers Preliminaries.
• The material is from the University of the Philippines School of Statistics.

Topics to Learn:

• Random Experiment, Sample Space and Probability
• Random Variable and its Distribution
• The Binomial Distribution
• The Normal Distribution
• The materials includes a Stat 114 Review

POPULATION vs SAMPLE

• Population data is referred to as {X1, X2, ..., XN}.
• Parameter, a summary measure describes a particular characteristic of the population computed using population data.
• population mean, μ = (Σ from i=1 to N, of Xi)/N
• population variance, σ² = (Σ from i = 1 to N, of (Xi-μ)² )/N
• Sample data is referred to as {X1, X2, ..., Xn}
• Statistic, a summary measure that describes a particular characteristic
• sample mean, X = (Σ from i = 1 to n, of Xi) / n
• sample variance, s² = Σ from i = 1 to n, of (Xi-X)² / n-1

Remarks

• Both the parameter and statistic are summary measures that are computed using data.
• The computed summary measure is a parameter if population data exists
• The computed summary measure is a statistic, if only sample data is all that exists
• In a statistical inquiry, the answer to the research problem is based on the parameter's value
• That parameter value describes the characteristic of interest of the population under study.
• This parameter’s value can only be computed using population data.
• The value of the parameter cannot computed if only sample data are available

Descriptive vs Inferential Statistics

• Descriptive Statistics methods relate to collecting, describing, and analyzing data.
• Conclusions about a larger group are not included.
• Inferential Statistics relate to sample data analysis leading to predictions or inferences about the population.

Inferential Statistics

• Inferential Statistics methods provides inference on the value of a parameter.
• The statistic value is computed using sample data, but so that inference on the parameter value of interest can be made.

Remarks on Errors in Statistics

• Inferences are based on partial information about the population.
• Conclusions will always be subject to some error.
• Probability and distribution theory, a background to understand the errors that can be made

Remarks on Probability Theory

• Probability theory was developed to give answers to questions on the systematic game outcome patterns.
• Professional gamblers adjusted bets to the "odds” of success using probability theory.
• Basic probability theory examples are die-throwing experiments and the selection in a deck of cards
• Important phenomena that are of interest to humans are similar to games of chance today.
• Predictability with certainty is impossible as it is impossible tell when such a phenomenon will occur.
• Behavior patterns of interest facilitates predictability of occurrence with a certain degree of confidence.

Definition of Random Experiment

• A random experiment is a process that can be repeated under similar conditions
• The outcome of a random experiment cannot be predicted with certainty beforehand.
• Some examples of random experiments include tossing a pair of dice, tossing a coin, selecting 5 cards from a well-shuffled deck, or selecting a sample of size n from a population of N.
• The number of process repetitions does not allow determination in advance what the next outcome will be.

Definition of Sample Space

• The sample space, denoted by Q, is the collection of all possible outcomes of a random experiment.
• A sample point is an element of a sample space.
• The sample space is a set because it is a collection of elements.
• In set theory, this set is referred to as the universal set since it contains all elements under consideration.

Specifying a Set

• Roster method specifies the set by listing down all the elements belonging in the set and enclosing them in braces.
• Rule method specifies elements satisfying a particular rule enclosed in braces.

Illustration Experiment Of Tossing A coin Twice.

• A random experiment of tossing a coin twice, with H specifying a Head and T a tail, in the sample space can be specified by the roster method.
• As an alternative to recording what comes up on the first and second tosses, counting the number of heads that come up in the two tosses can also be done.
• Description of the sample space is not unique as there are many ways to specify the collection of all possible outcomes of the experiment.
• The choice depends on characteristic of interest will facilitate assignment and computation of probabilities.

Simple random sampling

• Simple random sampling is an example of a probability sampling method.
• In simple random sampling without replacement (SRSWOR), all possible subsets of 𝑛 distinct elements selected from the 𝑁 elements of the population have the same selection chances.
• In simple random sampling with replacement (SRSWR), all possible ordered 𝑛-tuples (coordinates need not be distinct) that can be formed from the 𝑁 elements of the population have the same selection chances.
• If the population consists of N=5 children, a=Janine, b=Josiel, c=Jan, d=Eryl, and e=Eariel, a sample of size n=2 will be selected using SRSWOR while specifying the sample.
• A sample of size 2 is represented by the {x1, x2} set notation as a description of of 𝑥₁ and 𝑋₂ are the two distinct elements included in the sample.
• Ω = {{a,b},{a,c},{a,d},{a,e},{b,c},{b,d},{b,e},{c,d},{c,e},{d,e}}
• By definition of SRSWOR, all 10 sample points/samples are given equal selection chances.
• If the sample of size 𝑛 is selected from a population of size 𝑁 using SRSWOR, then the sample space will contain sets containing 𝑛 elements that are given equal chances of selection.
• 𝑁(𝑁−1)(𝑁−2)...(𝑁−𝑛+1) / 𝑛(𝑛−1)(𝑛−2)...(2)(1)
• When if the population consists of 𝑁=5 children, and sample of size 𝑛=2 can be selected using SRSWR.
• It is possible describe the sample space using a notation of a sample of size 2 by an ordered pair, (x1,x2) to which 𝑥₁ is the element selected on the first draw while x₂ is the element selected on the second
• Ω ={(a,a),(a,b),(a,c),(a,d),(a,e),(b,a),(b,b),(b,c),(b,d),(b,e),(c,a), (c,b), (c,c),(c,d), (c,e), (d,a), (d,b), (d,c), (d,d), (d,e),(e,a), (e,b),(e,c),(e,d), (e,e)}
• By SRSWR definition, all 25 sample points (samples) are given chances of selection.
• Generally, sample of size 𝑛 selected using SRSWR is denoted by an ordered n-tuple (x1,x2,...,xn) where xi is the element selected on the ith draw.
• If a population size 𝑁 using SRSWR, then 𝑁 ordered n-tuples and by definition, all of them are given equal selection chances.

Definition of Events

• An event is a sample space subset whose probability is defined.
• If the outcome of an experiment is one of the sample points belonging in event, then events occurred.
• Capital Latin letters (A,B,C,...) are a typical specification for any event of interest.

Illustration of Rolling a die

• To compute the probability of rolling a die, specify the following events:
• The sample space can be given to Ω = {1, 2, 3, 4, 5, 6}
• Event of observing an add = {1,3,5}
• Event of not observing any add = {2,4,6}
• If the outcome is less than 3 = {1,2}
• Event 𝐴 = event that Janine is included in the sample
• Event B is where event that Janine and Jan are both included in the sample

Impossible and Sure Events.

• Impossible event is regarded as an empty set Ø.
• Sure event is defined as the sample space Ω.
• Always define the probabilities for empty subset and the sample space.
• Outcome of Occurring Event:
• An even that occurs as sample point belonging in set Ø does not contain any sample points and thus making it impossible for event that happen.
• Ω contains all possible outcomes of the experiment and hence making the sure event that will always occur.

Additional Events

• Subsets aside from the impossible and sure event of sample space
• If the collection of sample points in spaces do not belongs in A then they are A complement set.
• Denoted as A’ or Ac
• “the complement of A”
• If the collection of sample points belonging in at least one of A and B then they are a A union B set.
• Denoted as AUB “A union B”
• If the collection of sample points belonging in both A and B, then they are a intersection B.
• Denoted as A∩B

Other Types of Events

• Union events containing “n events”: A₁UA₂U…UAn is the sample point collection that belongs in at least one of A1, A2, ..., An. Event occurs is at one from n events.
• Intersection events of “n events”: A₁∩A₂∩…∩An is the "sample points collection that belong in each one of A1, A2, ..., An. Event occurs if all of the n events occur.

Example Tossing Two dice

Ω = {(x,y) | x ε {1,2,3,4,5,6} and y ε {1,2,3,4,5,6}}.

• Total number of sample points is 36

Mutually Exclusive Events

• Two events are mutually exclusive if, A∩B = Ø, that is, A and B have no elements in common.
• If one event in the collection occurs, then any one of the other events in the collection cannot occur.

Set Theory and Probability Theory

• Probability theory: sure event corresponds to set theory: universal set, and is written Ω If The event A will occur corresponds to the set of the element A occurring by the formula A and it relates to A U Ω which contains all elements of A.
• Noncompliance by A to the event: Ac Ω
• Probability of theory of the intersection/union of sets.
• Theory for all related elements of events.
• Example exercises include determining probability that each of 3 cleaning companies will work, finding events in rolling a fair die, and determining the probability of family size.

Probability Definition

• Probability of an event A denoted by P(A): must satisfy the following properties.
  • Nonnegativity: 0 ≤ P(A) ≤ 1
  • Norming Axiom: P(Ω) = 1
  • Finite Additivity: A = A₁ U A₂ U ... U An, then P(A) = P(A₁) + P(A2) + ... + P(An).
• A function assigning a measure of chance that event A will occur.

Probability Interpretation

• A probability measure that is close to 1 is a very large chance of occurrence.

• A probability measure that is close to 0 shows a very small chance of occurrence.

• A probability measure value of 0.5 is half occurrence chances. Probability of 1 means that you can be sure that set will be always happen.

• On the hand, probability of 0 is value of impossible set of events.

• In summary, with regards to the errors in finding probabilities:

• The probabilities that a couple will have 0, 1, 2, 3, or 4 or more children are, 0.42, 0.36, 0.25, 0.12, and -0.15 respectively have errors which can be detected in a similar manner according to what probabilities must hold.

• Person tosses a biased die three times with at least one dot on each side, if the probability that the sum of the number of of dots in all three the tosses is 2 for 0.0625.

• The other problems can be answered according to the basic probabilities.

  Probability assignment approaches:

• A prior probability using classical approaches

• A posteriori assigning with relative frequencies.

• Assigning of Subjectively of assign probabilities using the indirect assessment and knowledge.

• The method of using a priori or classical approach assigns probabilities equal to number of possible experimental outcomes:

P(A) = no. Of the desirable outcomes / total set of outcomes.

• Classic equation to show and give you theoretical compute amount of probability.
  • Experiments: of throwing die while coin and card experiments with people sampling.

Assigning probabilities:

Steps to assigning probabilities

• Specify the data. Make sure that each item meets equal requirements.

• To give any specific amount of sets, show the points denoted by A and determine its probabilities.

• P(A) = n(A) over elements that are included.

• Relative Frequency: Method of assigning each experiment a equal number total number of experiment

  Using a posteriori and probability

  • Empirical Probability (A) = #A/ number of experiment repetitions of the trials.

Empirical Notes on Probability

• Approach : the method defines events being repetitive processes and continuous loops
• Advantage of A Posterior: there are no restrictions such random experiment such as A and B as separate experiment that provides sample spaces.

Subjective Probability

-Intuitions, the personal beliefs, the informal is required to provide and determine amount of information available.

Rules of Counting

• Generalized Basic Principle:
  • If Stage 1 has 𝑛 1 distinct possible outcomes, Stage 2 has 𝑛 2 distinct possible outcomes, and so on until Stage 𝑘 with 𝑛 𝑘 distinct possible outcomes,
• Total unique possible outcomes can be counted.

  • there are (𝑛₁)⋅(𝑛₂)⋅(𝑛₃)⋅⋅⋅(𝑛𝑘) possible outcomes for the experiment.

  • Examples:

• 7-place license plates with the first 3 places occupied by letters and the last 4 by numbers
  • Assume letters and numbers can repeat = 175,760,000 total arrangements
  • Assume letters and numbers cannot repeat = 78,624,000 total arrangements

Factorial Notation:

• Factorial Notation is a compact form of representation for the product of the first 𝑛 consecutive positive integers
• denoted by 𝑛! where 𝑛 is a positive integer.
• 𝑛!=𝑛⋅(𝑛−1)⋅(𝑛−2)⋅⋅⋅(2)⋅( 1 )with 0!=1
• An 𝑟-permutation of set 𝑍 is an ordered arrangement of 𝑟 distinct elements from 𝑍
  • If 𝑍 has 𝑛 distinct elements, then the number of 𝑟-permutations of 𝑍 is denoted by 𝑃(𝑛, 𝑟) or 𝑛𝑃𝑟
• An 𝑟-combination of set 𝑍 is a subset of 𝑍 that contains 𝑟 distinct elements
  • If 𝑍 has 𝑛 distinct elements, then the number of 𝑟-combinations of 𝑍 is denoted by 𝐶(𝑛, 𝑟) or (𝑛𝑟 )
• P(𝑛, r) = 𝑛∗(𝑛−1)∗(𝑛−2)∗…∗(𝑛−𝑟+1)= 𝑛! / (𝑛−𝑟)!
• C(𝑛, r) = P(𝑛,𝑟) / r! = 𝑛! / (𝑛−𝑟)!r!
• Event Definition:*
• An eventis a subset of the sample space whose probability is defined.
• An event is expressed in capital letters
• Event Equations:*
• impossible event: ∅(empty set as elements cannot be selected).
• Sure Event: 𝒮(the entire sample space because every possibility will fall within set)
• 𝐴∪𝐵 is the set of all sample points that belong to 𝐴 or 𝐵.
  • “𝐴 union 𝐵.”
    • A UNION B set is just all the elements in both separated sets combined.
  • occurred if only event𝐴 occurred, only event𝐵 occurred, or both𝐴 and 𝐵 occurred.
• 𝐴∩𝐵is the set of all sample points that belong to both 𝐴 and 𝐵.
  • “𝐴 intersection 𝐵.”
• A Intersection B is not the sets combined.
  • occured if events 𝐴 and 𝐵 occurred simultaneously.
• 𝐴°or 𝐴’ (complement) is the set of all sample points in the sample space that don’t belong to 𝐴
  • occurred if event 𝐴 did not occur.
• *Event A is included by definition, where A is a sample in set
• Properties Of Probability:*
• Probability Function: P(A) in a = -P(AUB) where b.
  • P for ( intersection) .
  • C. if amount that they must have = a 1/pa1. Properties of Probability Formula Equations.