Probability and Contingency Tables

Analyzing Categorical Data with Contingency Tables

Contingency tables are used to summarize the relationship between two categorical variables. They display the frequency distribution of variables and allow for calculation of probabilities and assessment of independence.

• Marginal Probability: The probability of a single event occurring, found by summing across rows or columns.

• Joint Probability: The probability of two events occurring together, found at the intersection of row and column.

• Conditional Probability: The probability of one event given that another has occurred.

• Independence: Two events are independent if .

Example Table:

Use the table to calculate probabilities such as and test for independence.

Randomness and Probability

Basic Probability Rules

Probability quantifies the likelihood of events. Fundamental rules include:

• Addition Rule: For mutually exclusive events, .

• Multiplication Rule: For independent events, .

• Complement Rule: .

Example: If 25% of students have a free logo t-shirt, the probability that two randomly selected students both have one is .

Sampling Distributions and Central Limit Theorem

Sampling Distributions

A sampling distribution is the probability distribution of a statistic (such as the mean) from repeated samples of the same size from a population.

• Mean of Sampling Distribution: Equal to the population mean .

• Standard Deviation (Standard Error):

Central Limit Theorem (CLT): For large sample sizes (), the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

Example: If the average family spends $822, and the sample size is $1120.

Normal Model and Standard Deviation as a Ruler

Normal Distribution

The normal distribution is a symmetric, bell-shaped curve characterized by its mean and standard deviation .

• Standardization (Z-score):

• Empirical Rule: Approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3.

Example: To find the probability that a value is more than 2 standard deviations above the mean, use the normal table for .

Linear Regression

Regression Analysis

Linear regression models the relationship between a dependent variable and an independent variable .

• Regression Equation:

• Least Squares Method: Minimizes the sum of squared residuals to find the best-fitting line.

• Interpretation: is the slope (change in per unit change in ), is the intercept.

Example: If the regression equation is , then for GPA = 3.0, .

Testing Hypotheses

Hypothesis Testing

Hypothesis testing is a statistical method for making decisions about population parameters based on sample data.

• Null Hypothesis (): The default assumption (e.g., no effect).

• Alternative Hypothesis (): The competing claim.

• Test Statistic: Calculated from sample data to assess evidence against .

• P-value: Probability of observing data as extreme as the sample, assuming is true.

Example: To test if the mean is greater than a certain value, calculate the z-score and compare to critical values.

Comparing Groups and Counts

Comparing Means and Proportions

Statistical tests can compare means (t-tests) or proportions (z-tests) between groups.

• Two-sample t-test: Compares means from two independent groups.

• Chi-square test: Compares observed counts to expected counts for categorical data.

Example: To compare the effectiveness of three types of pain relievers, use ANOVA or chi-square tests depending on the data type.

Applications: Blood Drive, Repair Service, Decision Trees

Binomial and Normal Approximations

When dealing with counts of events (e.g., number of Type B blood donors), use the binomial distribution. For large , the normal approximation applies:

• Binomial Probability:

• Normal Approximation: for large

Example: Probability that at least 2 of the first 20 donors have Type B blood: .

Expected Value and Variance in Service Contracts

Expected value quantifies the average outcome over many trials. Variance measures the spread.

• Expected Value:

• Variance:

Example: For a repair contract, calculate the expected number of repairs and the standard deviation using the probabilities of each outcome.

Decision Trees for Probability

Decision trees visually represent possible outcomes and their probabilities, aiding in complex probability calculations.

• Branches: Each branch represents a possible event.

• Leaf Nodes: Show the final outcome and its probability.

Example: Calculating the probability of success or failure under different weather conditions using a tree diagram.

Summary Table: Key Probability Distributions

Additional info: Some explanations and formulas have been expanded for clarity and completeness. Decision tree and repair service examples have been generalized for academic context.