Descriptive Statistics (univariate)

- qualitative/quantitative variables
- continuous/discrete variables
- random variable
- sample mean
- sample variance (defining formula)
- sample variance (computational formula)
- histograms
- histogram area = probability
- transformations for changing histogram shape
- comparative boxplots
- quantile-quantile plots

Descriptive Statistics (bivariate)

- scatterplot (constant variance)
- histograms on scatterplots
- contingency tables

Distributions (Calculus required)

- probability density function (pdf)
- probability mass function (pmf)
- Bernoulli 
- Binomial
- Derivation of Binomial
- Poisson
- Derivation of Poisson
- Uniform
- Normal/Gaussian
- Exponential 
- Power-law
- Expected Value (E)
- Variance (V)
 
Probability

- Standardization
- Derivation of Sampling Distribution
- Derivation of E[sample mean]
- Derivation of V[sample mean]
- Central Limit Theorem
- prob( a < sample mean < b) = integral/sum of pdf/pmf
- prob( observed sample mean ) = nonsense
- prob( population parameter ) = nonsense
- Basic set/event theory, Venn diagrams
- Basic axioms of prob, conditional prob
- Independence, and Bayes' theorem.

Bivariate (Calculus required)

- correlation coefficient (r)
- invariance of r under shift/scale transformations
- sensitivity of r to clusters/outliers
- Ordinary Least-square Regression (OLS)
- meaning of regression coefficients
- Derivation of OLS estimates of regression coefficients
- Model versus Data 
- Extrapolation
- transformations
- polynomial regression
- multiple regression
- overfitting
- collinearity 
- interaction
- residual plots 
- Analysis of Variance (ANOVA) in regression
- Derivation of ANOVA in regression
- R-squared
- standard deviation of errors

Inference with Confidence Interval (CI)

- Derivation of (CI)
- 2-sided CI for 1 mean
- 2-sided CI for 2 means
- 1-sided CI (upper and lower) for 1 mean     All 1-side CIs are skipped this quarter
- 1-sided CI (upper and lower) for 2 means
- 2-sided CI for 1 proportion
- 2-sided CI for 2 proportions
- 1-sided CI (upper and lower) for 1 proportion
- 1-sided CI (upper and lower) for 2 proportions 
- CI for paired data
- determination of upper vs. lower confidence bound
- random vs. observed CI
- interpretation of CI in terms of confidence
- interpretation of CI in terms of probability
- coverage property of CIs
- determination of minimum sample size
- unknown vs. known population variance
- t-distribution
- CIs based on t-distribution

Hypothesis Testing (HT)

- HT with p-value
- HT with rejection region
- 2-sided 1-sample HT for mean
- 2-sided 2-sample HT for means
- 1-sided 1-sample HT for mean
- 1-sided 2-sample HT for means
- 2-sided 1-sample HT for proportion
- 2-sided 2-sample HT for proportions
- 1-sided 1-sample HT for proportion
- 1-sided 2-sample HT for proportions
- HT for paired data
- Determination of 2-sided vs. 1-sided
- Significance level as prob(Type I)
- chisquared HT of multiple proportions in 1 population
- chisquared HT of multiple proportions in multiple populations (i.e. test of homogeneity)
- power
- 1-way ANOVA F-test
  
Inference in Regression

- probability model for regression
- predicted response = conditional mean
- prob ( a < random y < b)
- CI and HT of regression parameters in simple regression
- CI and HT of regression parameters in polynomial regression
- CI and HT of regression parameters in multiple regression
- F-Test of model utility in multiple regression
- HT of correlation coefficient
- Confidence Interval (CI) for the true mean y(x)
- Prediction Interval (PI) for a single y, at a given x
- Derivation of CI and PI for prediction
- Classification and Regression problems (e.g., neural nets)

Statistical Software

- R