Statistics and Variables

Overview and Descriptive Statistics

Populations, Samples, and Processes

• A Population is the entire group to be studied; N typically represents the size of a finite population.
• An Observation is a single individual entity or measurement.
• A Variable is a characteristic of interest about an individual or object within a population.
• A Census involves measuring or surveying every unit in the population.
• A Sample is a subset of the population, selected from the entire group.

Random Sample vs. Convenience Sample

• A Random Sample is chosen so every individual or object has an equal chance of selection.
• A Convenience Sample is selected based on ease of access.

Probability vs. Statistics

• In a Probability Problem, population characteristics are known, and the focus is on sample-related questions.
• In an Inferential Statistics Problem, limited knowledge about the population exists, so sample data is used to make generalizations about the population and answer related questions.

Types of Data

• A Categorical Variable has non-numerical values that can be grouped into categories.
  • An Ordinal Variable is a categorical variable with ordered categories.
  • A Nominal Variable is a categorical variable without any specific order.
• A Numerical Variable has numerical values.
  • A Discrete Variable is numerical with a finite or countably infinite set of possible values.
  • A Continuous Variable has possible values within an interval of numbers.

Measures of Location and Variability

Summation Notation

• Summation notation is a concise way to express the sum of a series of terms
• Denoted by the symbol Σ, it represents the addition of a sequence of values

Sample Mean

• A set of n observations is denoted as x1, x2, ..., xn.
• The sample mean is calculated as the sum of all observations divided by the number of observations: x̄ = (1/n) * Σxi.
• The population mean is denoted as μ, and is a fixed constant, while the sample mean varies from sample to sample.

Measures of Variability

• Measures of variability, along with measures of center, are essential to describe a dataset fully.

Sample Variance and Sample Standard Deviation

• The ith deviation about the mean is xi - x̄.
• Sample variance, denoted as s², measures the average squared deviation from the mean.
• Sample standard deviation, denoted as s, is the square root of the sample variance and has the same units those of the original data.
• Population variance is denoted by σ² = (1/N) * Σ(xi - μ)², and the population standard deviation is denoted by σ = √(σ²).
• n-1 is used as the divisor for sample variance, because xi's tend to be closer to their average x̄ than to μ, and this compensates for it.

Probability

Sample Space and Event

• An Experiment is an activity with at least two possible outcomes, where the result cannot be predicted with certainty.
• A Sample Space (S) is a listing of all possible outcomes of an experiment, expressed using set notation.
• An Event is any collection or set of outcomes from an experiment.

Set Operations

• Complement (A'): All outcomes in the sample space S that are not in A (not A).
• Union (A∪B): All outcomes in A or B or both (A or B).
• Intersection (A∩B): All outcomes in both A and B (A and B).
• Disjoint or Mutually Exclusive: A ∩ B = {} (empty set); no elements in common. P(Ø) = 0.
• Exhaustive: A ∪ B = S; includes all outcomes of the sample space.
• Mutually Exclusive and Exhaustive: A ∩ B = {} and A ∪ B = S; no elements in common and includes all outcomes of the sample space.

Axioms, Interpretations, and Properties of Probability

• Relative Frequency of Occurrence: The number of times an event occurs divided by the total number of experiment repetitions.
• Probability of an Event A, P(A): The limiting relative frequency, or the proportion of time the event A occurs in the long run.
• Axioms of Probability:
  • P(A) ≥ 0
  • P(S) = 1, where S is the sample space.
  • For infinite disjoint events, P(A1 ∪ A2 ∪ ...) = Σ P(Ai).
• **P(A) ≤ 1
• If A and B are disjoint events, then P(A ∩ B) = 0.
• For any events A and B, P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
• If A and B are disjoint events, then P(A ∪ B) = P(A) + P(B).
• Equally Likely Outcome Experiment: When all outcomes have an equal chance, the probability of event A is the number of outcomes in A divided by the total outcomes in the sample space S.

Conditional Probability

• The conditional probability of A given B is P(A|B) = P(A ∩ B) / P(B), where P(B) > 0.

Independence

• A and B are independent if and only if P(A|B) = P(A).
• If A and B are independent, then A' and B; A and B'; A' and *B' *are also independent
• If A, B are independent, P(A ∩ B) = P(A) * P(B).
• A1, A2, ..., An are mutually independent if for every k = 2,3, ..., n and every subset of indices i1, i2, ..., ik, P(Ai₁ ∩ Aiz ก... ∩ Aik) = P(Ai₁) * P(Ai₂) * ... * P(Aik)

Discrete Random Variables and Probability Distributions

Random Variables

• A Random Variable assigns a unique numerical value to each outcome in a sample space.
• A Discrete Random Variable has values from a finite set or a countably infinite sequence.
• Continuous Random Variable has an interval, or disjoint union of intervals possible values.
• Random variables are usually denoted by a capital letter.

Probability Distributions for Discrete Random Variables

• Probability distribution or probability mass function (pmf) of a discrete random variable is defined for every number x by p(x) = P(X = x) = P(all w ∈ S : X (w) = x)
• The cumulative distribution function (cdf) F(x) of a discrete random variable X with pmf p(x) is defined for every number x by:
  • F(x) = P(X ≤ x) = Σ p(y) for all y ≤ x.
  • For any number x, F(x) represents the probability that the observed value of X will be at most x.
  • For any two numbers a, b, and a - P(a ≤ X ≤ b) = F(b) – F(a – 1)

Expected Value and Variance

• For a discrete random variable X with possible values D and pmf p(x), the expected or mean value (E(X) or μX) : E(X) = μX = Σx * p(x), where the sum is over all x ∈ D.
• The expected value of any function h(X) is denoted by E[h(X)] or μh(X): E[h(X)] = Σ h(x) * p(x), where the sum is over all *x ∈ D Variance of X: *V(X) = Σ(x – E(X))² * p(x) = Σ x² * p(x) – E(X)² = E(X²) – [E(X)]², where the sum is over all x ∈ D Standard deviation (SD) of X: σX = √σ².

Bernoulli Random Variable

• A Bernoulli experiment: an experiment with only two possible random outcomes: success and failure.
• Bernoulli random variable: whose only possible values are 0 and 1.

Binomial Probability Distribution

• Binomial experiment:
  • Consists of n trials.
  • Each trial has only two outcomes: success, failure.
  • Outcomes are independent between trials.
  • Success probability, p, remains constant between trials.
• A binomial random variable, X: the number of successes in n trials. Denoted: X ~ B(n, p).

Continuous Random Variables and Probability Distributions

Probability Density Functions

• A continuous probability distribution describes the random variable completely and is used to compute probabilities associated with a random variable.
• Probability density function (pdf), f(x):
  • A function defined for all real numbers (i.e., x ∈ (-∞, +∞)).
  • A smooth curve which describes the probability distribution for a continuous random variable X through area under the curve. Let a < b, P(a ≤ X ≤ b) = ∫ f(x) dx from a to b.
• The cumulative distribution function (cdf) F(x) for a continuous rv X is defined, for every number x, by F(x) = P(X < x) = ∫ f(y) dy from -∞ to X.
• F(x)* is the area under the density curve to the left of x.
  • P(X > a) = 1-2
  • P(a ≤ X ≤ b) = F(b)-F(a)
• Expected value and variance, of a continuous rv X:
  • μX = E(X)
  • μh(x) = E[h(X)]
  • σχ2=V(X)
  • σX

Uniform Distribution

• The random variable X has a uniform distribution on the interval [a, b].
• Probability density function (pdf):
  • f(x; a, b) = 1/(b-a), when a ≤ X ≤ b
  • f(x; a, b) = 0, otherwise

The Normal Distribution

• The Normal Distribution has two parameters: μ,σ (or μ, σ²), −∞ < μ +∞ and σ > 0. We write the random variable X ~ Ν(μ, σ²). The pdf is:
• f(x) = 1 / (σ√(2π)) * e^((- (x-μ)^2) / (2σ²))

Joint Probability Distributions and Random Samples

Statistic and the Distribution

• A Population Parameter is a numeric measure of a population.
• A Sample Statistic is a numerical characteristic of the sample.
• A Point Estimate is a single value of the selected point estimator, computed from a given sample.

The Distribution of the Sample Mean

• When taking a random sample X1, X2, ..., Xn from a distribution with mean μ and standard deviation σ, the sample mean x̄ has these properties:
  • E(x̄) = μ
  • V(x̄) = σ²/n and σχ=σ/√n**.
• Random samples from a normal distribution
  • Given the rv's X1, X2, ..., Xn from a normal distribution with mean μ and standard deviation σ, then for any n.
    • E(x̄) = μ
    • V(x̄) = σ²/n.
    • x̄ ~ N(μ, σ²/n)
    • T(0) ~N(nμ, nσ²)
• Central Limit Theorem (CLT):
  • The rv's X1, X2, ..., Xn should be a random sample from a distribution with mean value μ and standard deviation σ. If n is sufficiently large (n > 30), then,
    • E(x̄) = μ, and V(x̄) = σ²/n
    • x̄ ~ N(μ, σ²/n)
    • T(0) ~N(nμ, nσ²)

Confidence Intervals with a Single Sample

Why do we need a Confidence Interval?

• A point estimate alone is highly likely to be wrong.
• A Confidence Interval is an interval that is used to estimate the population true mean, is more reliable than a single estimate.
• Confidence Level: Represents the degree of reliability, is expressed as 100(1 – α)%.
• The random interval is defined as x̄ ± 1.96 (σ/√n).

Derive the Confidence Interval

• For confidence interval derivation, assume the population is normal or n > 30 (large) and σ² is known.
• Questions to consider:
  • What is the difference b/w Random Interval and Confidence Interval?
  • How to interpret a Confidence Interval?
  • Is a wider CI or a narrower CI better?
  • How to make a Confidence Interval smaller when the confidence level keeps the same value?
  • How to calculate a 99% CI for μ? Given the same sample, which is wider: a 95% CI or a 99% CI?

Two-Sided CI with Known σ: Z-CI

• Assumptions include, the sample size is normal or large (n ≥ 30) and that the standard deviation σ is known.
• A two-sided 100(1 – α)*% Confidence Interval (CI):
• x ± zα/2*(σ/√n).

The Width of Z-CI and Sample Size

• Given a population with σ fixed, the factors that may effect CI width. Given a 100(1 – α)*% *Confidence interval, is: x ± zα/2 (σ/√n). The margin of error, m = zα/2 (σ/√n)

Confidence Interval when σ is unknown: t - CI

• In situation's where σ is unknown, a t-CI is used, which involves the t distribution. The t distribution has a degree of freedom parameter, v,= n − 1.

Two-Sided t - CI with Unknown

• The underlying distribution must be either normal or the sample size must be large (n ≥ 30)
• σ² must also be unknown

This quiz covers fundamental concepts in statistics, including variables, hypothesis testing, and measures of variability. It explores variable types like nominal and discrete, and delves into hypothesis testing components such as null and alternative hypotheses. It further tests understanding of statistical measures like sample mean and standard deviation.