MASH summaries all

Binomial Probability

Formula Overview

• Binomial Probability Formula:

  • For n independent trials with probability of success p.

  • Denoted as: P(X = x) = (n choose x) * p^x * (1-p)^(n-x)

Derivation Using Tree Diagrams

• Visual representation of outcomes from multiple trials.

• Example: Basketball Free Throws

  • Probability of success (getting the ball in): p = 0.7

  • Probability of failure: 1 - p = 0.3

  • For 3 attempts, the probability of 2 successes:

    • Tree diagram highlights 3 successful paths.

    • Calculation:

      • P(2 successes) = (0.7 x 0.7 x 0.3) x 3 = 0.441

Binomial Calculation Example

• Given: n = 3, p = 0.7, x = 2

• Calculate:

  • P(X = 2) = (3 choose 2) * (0.7^2) * (0.3^1)

  • Result: P(X = 2) = 0.441

Probability Distributions Summary

Types of Distributions

• Binomial Distribution:

  • Discrete outcomes, defined by parameters n and p.

  • Notation: X ~ B(n, p)

• Normal Distribution:

  • Continuous outcomes, characterized by mean (µ) and variance (σ²).

  • Notation: X ~ N(µ, σ²)

Key Characteristics

• Binomial: Only 6 discrete outcomes.

• Normal: Symmetrical and bell-shaped; uses Empirical Rule (68/95/99.7% coverage within standard deviations).

Statistical Inference – Hypothesis Testing

Overview of Hypothesis Testing Steps

1. Hypotheses:

• Null Hypothesis (H0): No difference (e.g., µ = value)

• Alternative Hypothesis (HA): There is a difference (e.g., µ ≠ value)

2. Test Statistic:

• Summary number from the data.

3. Null Distribution & P-value:

• Comparison of the test statistic with the null distribution to determine the likelihood.

4. Decision Making:

• If p is low (< 0.05), reject H0.

5. Check Assumptions:

• Ensure independence, normality, constant variance, etc.

Common Hypothesis Tests

• Z-test, t-test for means, chi-square tests, ANOVA.

Statistical Terms & Definitions

Key Symbols

• Variable: Characteristic that varies.

• Population vs. Sample: Entire group vs. subset used to infer about the population.

• Data Types: Nominal, Ordinal, Discrete, Continuous.

Describing Data Distribution

• Shape, Center (mean, median), Spread (SD, IQR).

• Skewness: Left-skewed, Right-skewed, Symmetrical.

Probability

Basic Definitions

• Marginal: P(A)

• Joint: P(A and B)

• Conditional: P(A | B)

• Independence: Events do not affect each other's occurrence.

Special Rules

• Mutually Exclusive Events: P(A and B) = 0

• Bayes' Theorem: Calculating conditional probabilities.

Correlation and Regression

Understanding Relationships

• Correlation: Measures the strength/direction between two variables.

• Regression Coefficient (β): Used to model relationship between variables and predict outcomes.

• Model Fit: Coefficient of determination (R²) measures explained variation.