Statistics and Probability Quiz

Statistics II - Exam Study Guide

• Probabilities form the basis for statistical inference, used to answer questions about how representative data are of a population.
• Probability involves starting from a situation (e.g., an animal) and determining possible outcomes (e.g., footprints). Statistics focuses on analyzing existing data (e.g., footprints) to infer characteristics of the population (e.g., the animal).
• Frequentists define probability as long-run frequency. For example, if a coin is fair (50% heads), half the experiments are expected to land on heads.
• Frequentists require data and a model. They're objective, but their scope is limited to sequences that don't exist in the physical world.
• Bayesians' view of probability is subjective; it's the degree of belief that an intelligent agent assigns to an event's truth. Probabilities are based on thought processes and assumptions, not the world.
• Bayesians require prior information, data, and a model. They aren't purely objective, and their approach can be broader.

Probability Distributions

• Binomial Distribution: "Either something is or isn't" (e.g., success, failure). A single observation has a 0 or 1 outcome.
• Binomial Distribution in R: dbinom(x, size, prob) calculates a single probability; pbinom() calculates the cumulative probability; rbinom() generates random numbers; qbinom() computes the quantile.
• Normal Distribution (Gaussian): Described by two parameters: the mean (µ) and standard deviation (σ). The shape is symmetrical around the mean, and there's a predictable distribution of the data within a certain number of standard deviations from the mean.
• Normal Distribution in R: dnorm(), pnorm(), rnorm(), qnorm() functions in R are used to calculate or simulate normal distributions.

Relationships Between Models and Data

• Regression and Relationships: Statistical methods for establishing and measuring relationships. Data = Model + Error.

Correlation

• Types of Correlation:
  • Positive: variables change in the same direction
  • Negative: variables change in opposite directions
  • No correlation: there is no relationship between the variables.
• Pearson Correlation: measures the linear relationship between two variables.
• Spearman Correlation: measures monotonic relationship between two variables, ranking data first.
• Kendall's Tau: Another non-parametric correlation measure.

Sample Statistics and Population Parameters

• Statistics summarize properties of a sample (e.g., mean, standard deviation).
• Parameters describe characteristics of a whole population (e.g., population mean, population standard deviation). Crucial for generalizing findings.

Running and Interpreting R Output for Simple Linear Regression

• Output shows estimates, standard errors, t-values, p-values, and other statistics for the intercept and predictor.
• Significant p-values suggest a statistically significant relationship between variables.
• R-squared indicates proportion of variance explained by the model.

Hypothesis Testing

• Null Hypothesis (H0): A statement that there is no relationship or significance (typically that a population parameter is zero).
• Alternative Hypothesis (Ha): A statement that there is a relationship or significance.
• P-value assesses the probability of observing the data if the null hypothesis is true.
• Reject the Null: A low p-value indicates the null hypothesis is likely false.

Regression and Test Statistics

• Regression: a method to predict the value of one variable using one or more other variables.
• Equation of a straight line: Represents the linear relationship. Y = b₀ + b₁X + ε
• Regression Coefficients: Gradient (b₁) and Y-intercept (b₀).
• Ordinary Least Squares (OLS): Minimizes the sum of squared differences between observed values and predicted values. The best linear approximation.

Sampling Methods

• Simple Random Sampling: Every member has an equal chance of selection, good but can be time-consuming.
• Stratified Sampling: Dividing the population into meaningful sub-groups and selecting samples proportionally, creating a representative sample.
• Volunteer Sampling: Individuals choose to participate; highly prone to bias.
• Convenience Sampling: Selecting participants that are easily accessible, which can be very unrepresentative.
• Snowball Sampling: Used for hard-to-reach populations. Early participants recruit others.

Confidence Intervals

• Provide a range of plausible values for a population parameter. A 95% confidence interval implies there's a 95% chance the true value falls within that window.

Central Limit Theorem

• The distribution of the sample mean approaches a normal distribution as the sample size increases. This is crucial for using sample data to make inferences about the population mean.

Type I and Type II Errors

• Type I Error: Rejecting the null hypothesis when it's actually true (false positive).
• Type II Error: Failing to reject the null hypothesis when it's actually false (false negative).

Effect Sizes (e.g., Cohen's d)

• Quantify the practical significance of an effect. A significant finding might have little real- world importance, whereas a small effect can have significant implications if the effect size is important in the context.

Multiple Regression

• Predicting a dependent variable from two or more independent variables.
• Coefficients reflect the relationship of each independent variable to the dependent variable, holding the others constant.

Assumptions of Regression

• Independence: Observations are unrelated.
• Normality: errors are normally distributed.
• Homoscedasticity: variance of the dependent variable is equal across all levels of the predictors.
• Linearity: relationship between variables is linear.
• Multicollinearity: Predictors are not too highly correlated.

Outliers and Influential Points

• Outliers: Extreme values that deviate greatly from the rest of the data (potentially problematic)
• Influential Points: Points that heavily impact the regression line (can distort the results).

Polynomial Regression

• Models non-linear relationships; often represented as polynomial (increasing powers of x) equations.
• Useful for fitting curves, particularly when a curvilinear relationship is suspected.
• Interpreting: focus on overall fit (R²) and significance of the polynomial terms.

Growth Curve Models

• Examine how a variable changes over time.
• Includes both fixed and random effects.
• Usually used in longitudinal analyses.

Coding Categorical Variables

• Dummy Coding: One category serves as a reference point, with coefficients representing the relative difference between other categories and this reference.
• Unweighted Coding: Uses a different set of values for each group.
• Weighted Coding: Each group is assigned weights reflecting its importance in the analysis.

Interpretations of Results

• Examine the significance of effects (p-values) and also the effect sizes (e.g. R², Cohen's d) for determining the importance of the results.
• Consider all context and the validity of the data in relation to drawing relevant conclusions.