• Statistics 2. Lecture notes
  • 1 Introduction
  • 2 Three worlds
    • 2.1 Questions
  • 3 Population and samples
    • 3.1 Population and sample: examples
    • 3.2 Sampling with and without replacement
    • 3.3 Questions
  • Probability calculus
  • 4 Probability
    • 4.1 Probability axioms
    • 4.2 Probability interpretations:
    • 4.3 Events – basic concepts
      • 4.3.1 Elementary event
      • 4.3.2 Compound event
      • 4.3.3 Sample space (set of elementary events)
      • 4.3.4 Union of events
      • 4.3.5 Intersection of events
      • 4.3.6 Complementary event
      • 4.3.7 Conditional probability
      • 4.3.8 Mutually exclusive (disjoint) events
      • 4.3.9 Independent events
    • 4.4 Probability – basic rules
      • 4.4.1 Complement rule
      • 4.4.2 Addition Rule
      • 4.4.3 Multiplication rule
    • 4.5 Exercises
  • 5 Bayes’ formula
    • 5.1 Law of total probability
    • 5.2 Bayes’ theorem
    • 5.3 Examples
    • 5.4 Templates
    • 5.5 Links
    • 5.6 Zadania
  • 6 Combinatorics
    • 6.1 Variations
    • 6.2 Combinations
    • 6.3 Questions
    • 6.4 Exercises
  • 7 Random variables
    • 7.1 Probability distribution
    • 7.2 Expected value
    • 7.3 Variance and standard Deviation
    • 7.4 Cumulative distribution function (CDF)
    • 7.5 Transormations of random variables
      • 7.5.1 Adding a constant to a random variable
      • 7.5.2 Multiplying a random variable by a constant
      • 7.5.3 Adding random variables
    • 7.6 Templates
    • 7.7 Exercises
  • 8 Discrete distributions
    • 8.1 Bernoulli distribution
    • 8.2 Binomial distribution
    • 8.3 Poisson distribution
    • 8.4 Templates
    • 8.5 Questions
    • 8.6 Exercises
  • 9 Continuous distributions
    • 9.1 Density function
    • 9.2 Uniform distribution
    • 9.3 Gaussian distribution
      • 9.3.1 Standardized normal distribution
      • 9.3.2 Sum and difference of variables with normal distribution
      • 9.3.3 Approximation of binomial distribution by normal distribution
    • 9.4 Templates
    • 9.5 Exercises
  • Inferencial statistics
  • 10 Sampling distributions
    • 10.1 Statistics and parameters
    • 10.2 Sampling distribution of the sample mean
    • 10.3 Central limit theorem (CLT)
    • 10.4 Point estimators – biased and unbiased
    • 10.5 Statistical inference
    • 10.6 Links
    • 10.7 Questions
    • 10.8 Exercises
  • 11 Confidence intervals for a mean
    • 11.1 Confidence interval and confidence level
    • 11.2 Interval Estimation of the Mean
    • 11.3 Additional Notes
    • 11.4 Links
    • 11.5 Templates
    • 11.6 Questions
    • 11.7 Exercises
  • 12 Confidence intervals for a proportion
    • 12.1 Formulas
    • 12.2 Links
    • 12.3 Templates
    • 12.4 Zadania
  • 13 Sample size
    • 13.1 Determination of sample size when estimating the mean
    • 13.2 Determination of sample size when estimating a proportion
    • 13.3 Rounding
    • 13.4 Templates
    • 13.5 Exercises
  • 14 Hypothesis testing — one mean
    • 14.1 Hypotheses and statistical tests
    • 14.2 Test statistic and rejection region
    • 14.3 Errors
    • 14.4 Do we accept the null hypothesis?
    • 14.5 Elements of a typical statistical test
    • 14.6 Types of alternatives
    • 14.7 Notation of null hypotheses with non-strict Inequalities
    • 14.8 One-sample mean test based on the \(z\) statistic
    • 14.9 One-sample \(t\) test
    • 14.10 Practical use of \(t\) and \(z\) tests
    • 14.11 Links
    • 14.12 Templates
    • 14.13 Exercises
  • 15 Hypothesis testing — one proportion
    • 15.1 Single proportion test based on the \(z\) statistic
    • 15.2 p-value
    • 15.3 Power of a test
    • 15.4 Links
    • 15.5 Templates
    • 15.6 Exercises
  • 16 Hypothesis testing — two means
    • 16.1 Testing two means – sampling distribution
    • 16.2 Test dwóch średnich oparty na statystyce \(z\)
    • 16.3 Two-sample \(t\)-test
    • 16.4 Paired samples
    • 16.5 Formulas
      • 16.5.1 Confidence intervals for differences in parameters (formulas)
      • 16.5.2 Tests for differences in means (formulas)
      • 16.5.3 Difference in means – paired samples
      • 16.5.4 Confidence intervals – paired samples
    • 16.6 Confidence intervals versus means
    • 16.7 Effect size
    • 16.8 Links
    • 16.9 Templates
    • 16.10 Pytania
    • 16.11 Zadania
  • 17 Hypothesis testing — two proportions
    • 17.1 Two proportions test for large samples
    • 17.2 Formulas
    • 17.3 Effect size
    • 17.4 Templates
    • 17.5 Exercises
  • 18 Chi-square tests
    • 18.1 Applications of the chi-square tests
    • 18.2 Formula
    • 18.3 Hipotheses
    • 18.4 Expected frequencies
    • 18.5 Conditions for test application
    • 18.6 Degrees of freedom
    • 18.7 Rejection region
    • 18.8 Chi-square tests versus proportion tests
    • 18.9 Effect size in independence tests
    • 18.10 Links
    • 18.11 Templates
    • 18.12 Exercises
  • 19 Analysis of variance (ANOVA)
    • 19.1 One-way analysis of variance – test
    • 19.2 F statistic – formula
      • 19.2.1 Between-group variability
      • 19.2.2 Within-group variability
      • 19.2.3 F Statistic
    • 19.3 ANOVA results table
    • 19.4 Effect size
    • 19.5 Post hoc procedure
    • 19.6 Levene's test and Brown-Forsythe test
    • 19.7 Two-way analysis of variance
    • 19.8 Links
    • 19.9 Templates
    • 19.10 Exercises
  • 20 Nonparametric tests
    • 20.1 Runs test
    • 20.2 Mann-Whitney test
    • 20.3 Wilcoxon signed-rank test
    • 20.4 Kruskal–Wallis test
    • 20.5 Templates
    • 20.6 Exercises
  • 21 Linear regression
    • 21.1 Linear regression model – assumptions
    • 21.2 Linear regression model – estimation
      • 21.2.1 Confidence intervals for model coefficients
      • 21.2.2 Confidence intervals for the expected value of the dependent variable
    • 21.3 Prediction intervals
    • 21.4 Linear regression model – t-test and F-test
      • 21.4.1 Significance tests for coefficients
      • 21.4.2 F-Test
    • 21.5 Diagnostic tests for the regression model
      • 21.5.1 Testing linearity
      • 21.5.2 Testing homoskedasticity
      • 21.5.3 Testing normality of the error term
    • 21.6 Templates
  • 22 Other tests
    • 22.1 Checking conformity with the Gaussian distribution
      • 22.1.1 Kolmogrov-Smirnov test
      • 22.1.2 Shapiro-Wilk-Royston test
      • 22.1.3 Jarque-Bera test
      • 22.1.4 Anderson-Darling test
    • 22.2 Test and confidence interval for Pearson's correlation coefficient
    • 22.3 Templates
    • 22.4 Zadania
  • 23 Simulations, bootstrapping, permutations tests
    • 23.1 Simulations in proportion estimation
    • 23.2 Bootstrapping
    • 23.3 Permutation tests
    • 23.4 Exercises
  • Appendices
  • A Formulas
    • A.1 Basic probability formulas
      • Union (“A or B”) — addition rule
      • Intersection (“A and B”) — product (or chain) rule
      • Complementary event
      • Conditional probabilities (A | B – “A given B”)
      • Law of total probability
      • Bayes’ formula
    • A.2 Counting
      • A.2.1 Variations of \(n\) taken \(r\) at a time (order matters)
      • A.2.2 Combinations of \(n\) taken \(r\) at a time (order does not matter)
    • A.3 Random variables
      • A.3.1 General formulas for a discrete random variable
      • A.3.2 General formulas for a continuous random variable
      • A.3.3 Binomial distribution
      • A.3.4 Poisson distribution
      • A.3.5 Hypergeometric distribution
      • A.3.6 Uniform (continuous) distribution
      • A.3.7 Exponential distribution
      • A.3.8 Gaussian (normal) distribution
      • A.3.9 Standard normal distribution
      • A.3.10 Sampling normal distribution of the sample mean (approximation for large \(n\))
    • A.4 Transformations of random variables
      • A.4.1 Adding a constant to a random variable
      • A.4.2 Multiplying a random variable by a constant
      • A.4.3 Addition of random variables
      • A.4.4 Addition of \(n\) independent and identically distributed (i.i.d.) random variables
    • A.5 Statistical inference
      • A.5.1 Confidence interval
      • A.5.2 Sample size determination
      • A.5.3 Hypothesis testing for a population parameter
      • A.5.4 Hypothesis testing for a difference of population parameters
      • A.5.5 Confidence intervals for a difference of population parameters
      • A.5.6 Hypothesis tests for the difference of parameters – paired samples
      • A.5.7 Confidence intervals for the difference of parameters – paired samples
      • A.5.8 Chi-square test
      • A.5.9 ANOVA (one-way)
    • A.6 Linear regression
      • A.6.1 The model
      • A.6.2 Point estimates
      • A.6.3 Confidence intervals for the coefficients
      • A.6.4 Confidence intervals for the expected value of the dependent variable
      • A.6.5 Prediction intervals
  • B Using normal distribution
  • C Templates
    • C.1 Bayes' formula
    • C.2 Distributions
      • C.2.1 Discrete random variable
      • C.2.2 Parametric discrete distributions
      • C.2.3 Continuous distributions
    • C.3 Confidence intervals
      • C.3.1 Sample size
    • C.4 Tests for means and proportions
      • C.4.1 Tests for 1 mean and 1 proportion
      • C.4.2 Tests and confidence intervals for 2 means
      • C.4.3 Tests and confidence intervals for 2 proportions
    • C.5 Chi-square test
    • C.6 ANOVA i test Levene'a
    • C.7 Nonparametric tests
    • C.8 Other tests
      • C.8.1 Regression
      • C.8.2 Normality check
      • C.8.3 Correlation
  • D Tables
    • D.1 Standard normal distribution (z) table
    • D.2 Student-t distribution table
    • D.3 Chi-square distribution table
    • D.4 F distibution tables
    • D.5 Other tables
      • D.5.1 Standard normal distribution table – alternative version
  • E Using R
    • E.1 R – typical problems
    • E.2 How to do it in R
      • E.2.1 Reading data
      • E.2.2 Probability distributions
      • E.2.3 Simulating using R
  • F Using spreadsheets
    • F.1 Spreadsheets – typical problems
      • F.1.1 Regional settings
      • F.1.2 Array formulas
      • F.1.3 Excel vs Google sheets
    • F.2 How to do it in spreadsheets
      • F.2.1 Probability distributions
  • G Schedule of laboratory meetings
    • G.1 Suggested schedule for economic analytics statistics class
  • Literature
  • Published with bookdown

Statistics 2. Lecture notes

D Tables

D.1 Standard normal distribution (z) table

https://drive.google.com/file/d/1Mg7Ouhg-wPn9lNdjW4J1xF7lNnmiq2hc/view

D.2 Student-t distribution table

https://drive.google.com/file/d/1MrBfJJmjokksZ4RKYwwDJ3Mt7etl7COE/view

D.3 Chi-square distribution table

https://drive.google.com/file/d/1N2Tva9N-5hSmwBq7XPQr-sM80nz1cvjP/view

D.4 F distibution tables

https://drive.google.com/file/d/1NDmvPkN12UvVWCN2gGPY9tByzLh4QlbV/view

D.5 Other tables

D.5.1 Standard normal distribution table – alternative version

https://drive.google.com/file/d/1Mpf_PyizUOpol3DfNfqMsRMx3Osk859w/view