AP Statistics Cumulative AP Exam Study Guide Notes

Statistics

  • Science of collecting, analyzing, and drawing conclusions from data.
  • Descriptive: organizing and summarizing statistics.
  • Inferential: making generalizations.
  • Population: entire collection.
  • Sample: subset.
  • Variable: changes in value.
  • Data: observations.

Types of Variables

  • Categorical (Qualitative): characteristics.
  • Numerical (Quantitative): numerical data.
    • Discrete: listable sets (counts).
    • Continuous: any value (measurements).
    • Univariate: one variable.
    • Bivariate: two variables.
    • Multivariate: many variables.

Distributions

  • Symmetrical: same shape (Bell Curve).
  • Uniform: equal frequency (rectangle).
  • Skewed: one side is longer.
  • Bimodal: two large frequencies.

Describing Numerical Graphs (S.O.C.S.)

  • Shape: symmetrical, skewed, uniform, or bimodal.
  • Outliers: gaps, clusters.
  • Center: middle (mean, median, mode).
  • Spread: variability (range, standard deviation, IQR).
  • Context: in context.
  • Comparative Language: when comparing.
  • Parameter: population value.
  • Statistic: sample value.

Measures of Center

  • Median: middle point (50th percentile).
  • Mean: \mu (population), \bar{x} (sample).
  • Mode: most frequent.

Measures of Spread (Variability)

  • Range: (Max - Min)
  • IQR: (Q3 - Q1)
  • Standard deviation: \sigma (population), s (sample).
  • Variance: standard deviation squared.

Resistant Measures

  • Resistant: not affected by outliers.
    • Median
    • IQR
  • Non-Resistant: Affected by outliers
    • Mean
    • Range
    • Variance
    • Standard Deviation
    • Correlation Coefficient (r)
    • Least Squares Regression Line (LSRL)
    • Coefficient of Determination
      ^2

Comparison of Mean & Median

  • Symmetrical: mean = median.
  • Skewed Right: mean > median.
  • Skewed Left: mean < median.
  • Trimmed Mean: eliminate outliers.

Linear Transformations

  • \\mu

Combination of Variables

  • \\mu

Z-Score

  • z = \frac{x - \mu}{\sigma}

Normal Curve

  • Bell-shaped and symmetrical.
  • Empirical Rule (68-95-99.7).

5-Number Summary

  • Minimum, Q1, Median, Q3, Maximum

Probability Rules

  • Sample Space: all outcomes.
  • Event: any outcomes.
  • Complement: not in the event.
  • Union: A or B (A \cup B).
  • Intersection: A and B (A \cap B).
  • Mutually Exclusive: no intersection.
  • Independent: one event doesn't change another.

Correlation Coefficient (r)

  • Strength and direction of a linear relationship.
  • Values: [-1, 1]
  • Least Squares Regression Line (LSRL): \hat{y} = a + bx
  • Residuals (error): y - \hat{y}. Residual Plot: no pattern = linear.

Coefficient of Determination \bf{r^2}

  • Proportion of variation in y explained by (x, y).

Interpretations for LSRL

  • Slope (b): unit increase in x, y increases/decreases by slope.
  • Correlation coefficient (r): (strength), (direction), linear association.
  • Coefficient of determination (r^2): r^2% of variation in y explained by x and y.
  • Influential Points: change LSRL if removed.
  • Outliers: large residuals.

Sampling

  • Census: complete count.
  • Sampling Frame: list of population.
  • Sampling Design: method to choose a sample.

Types of Samples

  • SRS (Simple Random Sample): equal chance of being selected.
  • Stratified: divide into groups, then SRS each.
  • Systematic: every 50th.
  • Cluster Sample: based on location.
  • Random Digit Table.
  • Random # Generator: Calculator or computer program

Bias

  • Error favoring an outcome.
  • Sources: Voluntary Response, Convenience Sampling, Undercoverage, Non-response, Response, Wording.

Experimental Design

  • Observational Study: observe outcomes.
  • Experiment: imposes treatment.
  • Experimental Unit: receives treatment.
  • Factor: explanatory variable.
  • Level: specific value for factor.
  • Response Variable: what you measure.
  • Treatment: experimental condition.
  • Control Group: compare to factor.
  • Placebo: no active ingredients.
  • Blinding: subjects unaware.
  • Double Blinding: neither subjects nor evaluators know.

Principles of Experimental Design

  • Control: keep variables constant.
  • Replication: use many subjects.
  • Randomization: assign subjects randomly.
  • Cause and effect: well-designed, controlled experiment.

Experimental Designs

  • Completely Randomized: units allocated randomly.
  • Randomized Block: units blocked, then assigned.
  • Matched Pairs: units matched, then assigned.
  • Confounding Variables: effect cannot be separated.
  • Randomization: reduces bias.
  • Blocking: reduces variability.

Random Variables

  • Discrete: a count.
  • Continuous: a measure.
  • Discrete Probability Distributions.
  • \\muX = \sum xi p(x_i)
  • \\sigmaX^2 = \sum (xi - \muX)^2 p(xi)

Special Discrete Distributions

  • Binomial Distributions: two outcomes, fixed trials, independent, same probability.
    • \\mu_X = np
    • \\sigma_X = \sqrt{npq}
  • Geometric Distributions
  • Poisson Distributions

Continuous Random Variables

  • Normal Distributions: unimodal, bell-shaped curves.

Normal Distributions

  • Use graphs – dotplots, boxplots, histograms, or normal probability plot.

Sampling Distributions

  • Central Limit Theorem: n > 30, sampling distribution is approximately normal.

Confidence Intervals

  • Steps: Assumptions, Calculations, Conclusion.

T-Distributions

  • Compared to standard normal curve: Centered around 0, More spread out and shorter, More area under the tails

Hypothesis Tests

  • Null Hypothesis: H_0
  • Alternate Hypothesis: H_a
  • P-Value: probability of observed result.
  • Steps: Assumptions, Hypotheses, Calculations, Conclusion.

Type I and II Errors

  • Type I Error: Reject H0 when H0 is true (probability is \\alpha).
  • Type II Error: Fail to reject H0, and H0 is false (probability is \\beta).

Chi-Square Tests

  • Goodness of Fit (univariate).
  • Independence (bivariate