Contingency Tables and Distributions

Contingency Tables

Contingency tables are used to summarize the relationship between two categorical variables. They display how individuals are distributed across each variable.

• Contingency Table: A table showing the distribution of individuals along each variable.

• Marginal Distribution: The totals for each row or column in a contingency table.

• Conditional Distribution: The distribution of one variable for cases that satisfy a condition on another variable. For example, the distribution of Event B given Event A occurs.

Example: Eye Color and Gender

• Marginal Distribution of Gender: Male: 60%, Female: 40%

• Conditional Probability Example: Percentage of females with blue eyes:

Sampling and Experimental Design

Sampling Concepts

Sampling is the process of selecting a subset of individuals from a population to estimate characteristics of the whole population.

• Population: The entire group of individuals or instances about whom we hope to learn.

• Sample: A representative subset of the population.

• Sample Survey: A study that asks questions of a sample drawn from a population.

• Randomization: Each individual has a fair, random chance of selection.

• Census: A sample that consists of the entire population.

• Population Parameter: A numerically valued attribute of a model for a population (e.g., mean income).

• Sample Statistic: A value calculated for sample data (e.g., sample mean).

• Sampling Frame: The list of individuals from whom the sample is drawn.

Types of Sampling Methods

• Simple Random Sample (SRS): Each set of n elements in the population has an equal chance of selection.

• Stratified Random Sampling: Population divided into strata, random samples drawn from each stratum.

• Cluster Sampling: Entire groups (clusters) chosen at random; clusters are heterogeneous.

• Multistage Sampling: Combines several sampling methods.

• Systematic Sample: Individuals selected systematically from a sampling frame.

Types of Bias

• Voluntary Response Bias: Individuals choose whether to participate.

• Undercoverage Bias: Some population members are less represented.

• Nonresponse Bias: Large fraction of those sampled fails to respond.

• Response Bias: Survey design influences responses.

Experimental Design

• Observational Study: No manipulation of factors; can be retrospective or prospective.

• Experiment: Manipulates factor levels to create treatments, randomly assigns subjects, compares responses.

• Factor: Variable whose levels are manipulated.

• Response Variable: Variable whose values are compared across treatments.

• Levels: Specific values chosen for a factor.

• Treatment: Process or intervention applied to experimental units.

• Block: Groups of similar experimental units; randomize within blocks.

• Randomization: Assign units to treatment groups randomly.

• Control: Control aspects not being studied.

• Replicate: Use as many subjects as possible.

• Statistically Significant: Observed difference unlikely to have occurred naturally.

• Types of Experiments: Completely Randomized Design (CRD), Randomized Block Design (RBD), Matched Pair Design.

• Blinding: Individuals unaware of treatment allocation.

• Single/Double Blind: Single: one group blinded; Double: both groups blinded.

• Placebo: Treatment known to have no effect.

• Placebo Effect: Response to placebo treatment.

• Confounding: Effects of two factors cannot be separated.

• Lurking Variable: Variable associated with both explanatory and response variables.

Probability and Its Rules

Basic Probability Concepts

Probability quantifies the likelihood of events in random phenomena.

• Random Phenomenon: Outcomes are possible, but which will occur is unknown.

• Trial: Single attempt or realization.

• Outcome: Value measured or observed in a trial.

• Event: Collection of outcomes; denoted by capital letters.

• Sample Space: Collection of all possible outcomes; denoted by S or .

• Law of Large Numbers (LLN): Long-run relative frequency approaches true probability as trials increase.

• Independence: Occurrence of one event does not affect the probability of another.

• Probability: Number between 0 and 1 indicating likelihood; for event A.

• Empirical Probability: Based on observed frequencies.

• Theoretical Probability: Based on models;

• Personal Probability: Subjective degree of belief.

Rules of Probability

• Probability Assignment Rule: ,

• Complement Rule:

• Addition Rule (Disjoint Events):

• Multiplication Rule (Independent Events):

• General Addition Rule:

• Conditional Probability:

• General Multiplication Rule:

• Independence:

• Bayes Rule:

Tree Diagrams

Tree diagrams help visualize conditional probabilities and sequences of events.

• Example: Calculating using Bayes Rule with given probabilities.

Binomial Distribution

Definition and Properties

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Conditions: Two possible outcomes (success/failure), constant probability , independent trials, fixed number .

• Success/Failure Condition: Binomial model is approximately normal if and .

Binomial Model Formulas

• Probability of k successes:

• Mean:

• Standard Deviation:

• Binomial Coefficient:

Sampling Distributions and Central Limit Theorem

Sampling Distribution

The sampling distribution describes the distribution of a statistic (e.g., sample mean) over all possible samples from the same population.

• Sampling Distribution: Distribution of statistics over all possible samples.

• Sampling Distribution Model: Practical model for theoretical sampling distribution.

• Sampling Error: Variation from sample to sample.

Central Limit Theorem (CLT)

• Statement: For large , the sampling distribution of the sample mean is approximately normal, regardless of population distribution, if observations are independent.

• Sampling Distribution Model for Mean: If independence and random sampling are met, and is large,

Example: Battery Life

• Population Mean: hours

• Population SD: hours

• Sample Size:

• Mean of Sample Mean:

• SD of Sample Mean:

• Model:

• Interval for 99.7%:

Worked Examples and Applications

Probability Table Example

Note: The sum of probabilities must equal 1.

Binomial Example: Defective Reams

• Given: ,

• Probability of exactly 4 defective reams:

• Mean:

• SD:

Conditional Probability Example: Animal Shelter

• P(Male | Cat):

• P(Cat | Female):

• P(Female | Dog):

Independence Example

• Definition 1:

• Definition 2:

• Application: For the animal shelter, being male and being a dog are independent because both definitions are satisfied.

Tree Diagram Example: Drunk Driving Checkpoint

• P(Drink): 0.12

• P(Not Drink): 0.88

• P(Detain | Drink): 0.8

• P(Detain | Not Drink): 0.2

• P(Detain):

• P(Drink | Detain):

Summary Table: Sampling Methods

Summary Table: Types of Bias

Summary Table: Types of Experimental Designs

Summary Table: Probability Rules

Summary Table: Binomial Distribution

Summary Table: Sampling Distribution of the Mean

Additional info: These notes expand on brief points with academic context, definitions, formulas, and examples, and include summary tables for quick reference.