AP Statistics Unit 1-9 Lecture Notes

Unit 1: Introduction to Statistics

Data Types

• Quantitative Data:
  • Deals with numbers (e.g., heights, class size, population).
  • Represented numerically.
• Categorical Data:
  • Deals with names and labels (e.g., eye color, hair color).
  • Cannot be numerically represented.
  • Represented with two-way tables.

Two-Way Tables

• Used to represent categorical data.
• Shows intersections between variables.
• Marginal Relative Frequency: Percentage of data in a single row or column vs total.
• Joint Relative Frequency: Percentage of data in a single group vs total.
• Conditional Relative Frequencies: Percentage in a single category given a specific group.

Quantitative Data

• Described using the acronym "C-SOCS":
  • Context
  • Shape: Symmetrical, skewed, number of peaks.
  • Outliers: Identify any extreme data points.
  • Center: Mean or median.
  • Spread: Range, standard deviation, IQR.

Basic Statistical Terms

• Mean: Average of all values.
• Standard Deviation: Measure of variation.
• Median: 50th percentile.
• Range: Difference between max and min values.

Box Plots

• Use the five-number summary: Minimum, Q1, Median, Q3, Maximum.
• IQR (Interquartile Range): Q3 - Q1.
• Identifies low-end and high-end outliers.

Percentiles and Frequency

• Percentile: Percentage of values below a certain point.
• Cumulative Relative Frequency: Cumulative percentages from intervals.

Z-scores

• Measures the number of standard deviations a value is from the mean.

Data Transformation

• Addition/Subtraction: Shape and variability remain the same; center changes.
• Multiplication/Division: Center and variability change proportionally.

Density Curves and Normal Distribution

• Density Curve: Shows probability distribution with an area of one.
• Normal Distribution: Follows the 68-95-99.7 rule.

Calculator Commands for Normal Distribution

• normPDF: Finds probability at a specific value.
• normCDF: Probability between an interval.
• Inverse Normal: Finds value for a given percentile.

Normal Probability Plot

• Plots actual vs theoretical Z-values to check normality.

Unit 2: Correlation and Regression

Scatter Plots and Correlation

• Acronym SEED:
  • State in context.
  • Examine direction (positive/negative).
  • Outliers noted.
  • Form (linear/nonlinear).
  • Strength of correlation.

R Value (Correlation Coefficient)

• Ranges from -1 to 1.
• Closer to -1 or 1 indicates stronger correlation.

Regression Lines

• Best Fit Line (y-hat): Predicts values based on the slope and intercept.
• Residuals: Difference between actual and predicted values.

Least Squares Regression Line

• Minimizes sum of squared residuals.
• S Value: Average distance predicted values are from LSR.
• R-square Value: Coefficient of determination.

Impact of Outliers

• Affects slope and y-intercept of regression lines.

Residual Plots

• Used to determine if linear functions are best.
• Clear pattern implies linear function may not be best.

Unit 3: Sampling Methods

Common Sampling Methods

• Simple Random Sample (SRS): Equal chance for all.
• Stratified Random Sample: Divides population by shared traits.
• Cluster Sample: Divides into clusters, samples entire clusters.
• Systematic Random Sample: Regular intervals with a random start.

Poor Sampling Methods

• Convenience Sample: Chooses easy-to-reach individuals.
• Voluntary Response Sampling: Participants choose to join.

Shortcomings and Biases

• Undercoverage: Leaving out groups.
• Non-response Bias: Selected individuals don't respond.
• Response Bias: False or misleading answers.
• Wording Bias: Poorly phrased questions.

Observational Studies vs. Experiments

• Observational Study: Observes without influencing subjects.
• Experiment: Manipulates variables to measure effects.

Principles of Experimental Design

• Comparison, Random Assignment, Control, Replication.

Terminology

• Factors, Levels, Confounding.
• Placebo: Fake treatment.
• Blinding: Single and Double Blind.

Experimental Designs

• Randomized Block Design: Divides subjects into blocks and assigns treatments.
• Matched Pairs Design: Pairs subjects based on characteristics.

Unit 4: Probability and Random Variables

Probability Basics

• Probability ranges from 0 to 1.
• Long-term vs. short-term unpredictability.

Simulation

• Uses models to mimic real-world events.
• Four-step process: Define problem, model, perform, estimate probability.

Probability Rules

• Mutually Exclusive: Events can't occur together.
• Independence: Outcome of one doesn’t affect the other.
• Addition Rule: For mutually exclusive and non-mutually exclusive events.
• Multiplication Rule: For independent and non-independent events.
• Complement Rule: Probability of not occurring is 1 minus probability of occurring.

Visualization of Probability

• Venn diagrams, two-way tables, probability trees.

Random Variables

• Discrete: Countable values.
• Continuous: Any value within a range.
• Binomial: Fixed number of trials.
• Geometric: Trials until the first success.

Mean and Standard Deviation of Random Variables

• Use calculator for one-variable stats.
• Discrete example calculation.

Transforming Probability Distributions

• Addition/Subtraction: Shape unchanged, center changes.
• Multiplication/Division: Shape unchanged, center and variability change.

Binomial Random Variables

• Acronym BINS: Binary, Independent, Number of trials, Set probability.

Calculator Commands for Binomial Random Variables

• binomialPDF: Probability of exactly x successes.
• binomialCDF: Cumulative probability up to x.

Geometric Random Variables

• Models number of trials until first success.
• Calculator commands: geometricPDF, geometricCDF.

Unit 5: Sampling Distributions

Statistic vs. Parameter

• Statistic: From a sample.
• Parameter: From a population.

Sampling Distribution

• Probability distribution obtained through repeated sampling.
• Unbiased estimator: Statistic's distribution equals parameter.

Proportions

• Repeated samplings, plotting sample proportions.
• Conditions: Random sampling, 10% condition, large counts.

Calculating Z-scores

• Used to find probabilities of sample proportions.

Central Limit Theorem

• Sampling distribution is normal if sample size ≥ 30.

Unit 6: Inference for Proportions

Point Estimate

• Statistic estimating population parameter.

Confidence Interval vs. Confidence Level

• Interval: Range of values estimating parameter.
• Level: Probability parameter falls within a range.

Confidence Interval Interpretation

• We are X% confident the interval from Y to Z captures the true parameter in context.

Factors Affecting Confidence Interval

• Higher confidence level = wider interval.
• Larger sample size = narrower interval.
• Bias does not affect margin of error.

Panic for Confidence Interval

• Parameter, Assumptions, Name of test, Interval, Conclusion.

Confidence Interval Calculation

• Use calculator or formula.

Significance Tests

• Phantoms:
  • Parameter, Hypothesis, Assumptions, Name of test, Test statistic, Obtain p-value, Make decision, State conclusion.

Type I & II Errors

• Type I: False positive; null hypothesis is true but rejected.
• Type II: False negative; null hypothesis is false but not rejected.
• Power: Probability of correctly rejecting a false null hypothesis.

Unit 7: Inference for Means

Key Differences

• Use T instead of Z for mean.
• Degrees of freedom: n - 1.

Panic Procedure

• Same as proportions but with means.

Interpretation

• Similar to proportions but refers to means.

Significance Tests

• Similar to proportions but for means.

Unit 8: Chi-Squared Tests

Chi-Squared Tests Overview

• Tests for goodness of fit, homogeneity, and association/independence.

Goodness of Fit

• Compares observed distribution to expected.
• No parameter/sample statistic.

Homogeneity

• Compares distributions across multiple groups for a single variable.

Association/Independence

• Tests for association between two variables in one sample.

Unit 9: Inference for Slope

Confidence Intervals and Significance Tests

• For true population slope.
• Check assumptions for linearity, independence, equal variance, normality.

Degrees of Freedom

• n - 2 for slope inference.

Conclusion

• Always in context.

These notes provide a comprehensive review of the major concepts across various units in AP Statistics, serving as a helpful study guide for the course.