Binomial Distribution and Probability Quiz

Statistical Inference & Probability

• Probabilities form the basis for statistical inference. Inferential statistics answers questions about how representative data is of a population.
• Probability starts with an observation (e.g., animal footprints) and predicts possible outcomes. Statistics analyzes observed phenomena (e.g., footprints) to infer underlying characteristics.

Frequentists vs. Bayesians

• Frequentists define probability as long-run frequency (e.g., 50% chance of heads for a fair coin).
• Frequentists require data and models.
• Advantages: objectivity and unambiguous calculations.
• Disadvantages: limited scope (can't apply to unique events) and infinite sequences don't exist in the physical world.
• Bayesians view probability as a degree of belief (subjective).
• Bayesians require prior information, data, and models
• Advantages: allows assignment of probabilities to any event.
• Disadvantages: not purely objective, can vary between observers.

Probability Distributions in R

• dbinom(x, size, prob): calculates the probability of a specific outcome (x) for binomial distributions, given specified success probability (prob) and number of trials (size).
• pbinom(q, size, prob): calculates the cumulative probability up to (including) a quantile(q) for binomial distributions
• qbinom(p, size, prob) calculate the quantile corresponding to probability (p) for binomial distributions
• rbinom(n, size, prob) simulates n random numbers from binomial distribution.
• dnorm(x, mean=0, sd=1): calculates normal probability density function (PDF) for given X values
• pnorm(q, mean=0, sd=1) : Calculates cumulative probability of a normal distribution
• qnorm(p, mean=0, sd=1): Calculates the quantile of a normal distribution
• rnorm(n, mean=0, sd=1): generates random normal data.

Normal Distribution

• The area under the normal curve is equal to 1.
• The curve is symmetric around the mean.
• Mean, mode, and median are all equal.
• Exactly half the values lie on either side of the mean.
• Standard deviation controls the spread of the distribution. Smaller SD = taller, narrower curve; larger SD = flatter, wider curve.

Correlation

• Pearson's correlation, Spearman's correlation
• Kendall's tau
• Cor() Calculates Pearson's correlation
• Cor.test() perform a hypothesis test (null hypothesis correlation=0) for Pearson's Correlation
• Rcorr() Calculate correlation coefficients using Kendall's tau

Sample Statistics and Population Parameters

• Sample statistics are calculated from a sample.
• Population parameters describe the entire population.

Hypothesis Testing

• Null hypothesis: claims no effect or no difference.
• Alternative hypothesis: claims there is an effect or a difference.
• p-value indicates the probability of observing the results if the null hypothesis was true

Linear Regression Testing

• Estimating relationships by predicting one variable from another.
• Model is linear (straight line)
• Coefficient estimates (b's) obtained through techniques.
• b0 = intercept, b1 = slope for predictors
• Sum of squares for total, model, and residual are crucial in evaluating fit of the model

Sampling Methods

• Random sampling: every member has an equal chance of selection.
• Stratified sampling: divides the population into subgroups and selects samples from each.
• Volunteer sampling: participants self-select to be in a study.
• Convenience sampling: selects participants who are readily available.
• Snowball sampling: participants provide referrals of others to participate.

Confidence Intervals

• Quantifies uncertainty in estimates of population parameters.
• Typically expressed as a range of values, within which the true population parameter likely lies (e.g., 95% CI).

Central Limit Theorem

• As sample size increases, the distribution of sample means approaches a normal distribution, regardless of the shape of the underlying population distribution.
• The mean of the sampling distribution equals the population mean.
• The standard error of the mean, a measure of variability, decreases as the sample size increases.

Type I and Type II Errors

• Type I error: rejecting a true null hypothesis (false positive).
• Type II error: failing to reject a false null hypothesis (false negative).

Regression and Test statistics

• Regression coefficients represent changes predicted by one or more independent variables.
• Different kinds of regressions exist to model different kinds of relationships (e.g. linear, non-linear).

Types of correlations (and their properties)

• Positive correlation: both variables change in the same direction.
• Negative correlation: variables change in opposite directions.
• No correlation: no relationship between variables.

Variance, Covariance, and Correlation

• Variance is the measure of data spread around the mean.
• Covariance shows the direction and strength of the relationship (joint variation) between two variables.
• Correlation is a standardized version of covariance and isn't affected by the units used to measure data. It is between -1 and + 1, a value closer to 1 or -1 identifies a strong relationship.

Partial and Semi-partial correlations

• Used when we want control for the effect of a third variable (or more) on two others.
• Partial correlation: looks at the relationship between two variables after controlling for the influence of a third (or more) variables.
• Semi-partial correlation: looks at association between variables after controlling only for the influence of a third variable on only one of the other variables.

Multiple Regression and its properties

• Multiple regression models the relationship between one dependent variable and two or more independent variables.
• Coefficients represent the change expected in the dependent variable given one unit change in the independent variable while all others are constant.

Growth Curve Modeling

• Models changes of dependent variables over time, sometimes for different groups or conditions.
• Models patterns that appear in the form of a curve of slopes.

Assumptions for Regression Analysis

• Independent errors
• Normality of residuals
• Linearity
• Homoscedasticity (constant variance)
• No multicollinearity

Polynomial Regression

• Models a curvilinear relationship between one or more independent and one dependent variables
• Useful when data shows a systematic trend or curve

Categorical Variables - Coding of Categorical Variables

• Dummy coding: one group used as reference.
• Unweighted effects coding: weights assigned for each category
• Contrast coding: weights set up according to a priori hypothesis, useful to test or prove specific patterns.
• Categorical variables can be coded using dummy codes or polynomials in multiple regression analysis to reflect the effect of the independent variables on the dependent variables.

Test your knowledge of binomial distribution, elementary events, and probability concepts with this quiz. Each question addresses fundamental principles and applications related to statistical analysis and random variables. Perfect for students wanting to solidify their understanding of probability theory.