Cours 3

• Statistical Inference


  Process of drawing conclusions about a population based on sample data
• Statistical Inference


  • Involves making inferences about unknown population parameters based on sample statistics
  • Allows us to make generalizations about populations from samples
  • Helps us to draw conclusions and make decisions based on data
  • Provides a way to test hypotheses and theories in a systematic manner
  • Can help to identify patterns and relationships in data
• Parameter estimation


  Trying to learn the true values of the parameters (e.g. mean, standard deviation) of a population distribution
• Data prediction


  Using sample statistics to predict the probability of future observations from the population
• Model comparison


  Comparing different statistical models to determine which one best fits the data
• Betrand Russel: '"Probability is the most important concept in modern science, especially as nobody has the slightest notion of what it means."'
• Probability


  A measure of the likelihood of an event occurring, a number between 0 and 1
• Interpretations of probability


  • Classical view
  • Frequentist view
  • Bayesian view
• Frequentist view of probability

  Probability is the proportion of times an event occurs in an infinite number of repetitions of an experiment
• Bayesian view of probability


  Probability expresses belief or uncertainty about a parameter, and can be updated with new data
• Addition rule


  Used to calculate the probability of the union of two events: P(A or B) = P(A) + P(B) - P(A and B)
• Multiplication rule


  Used to calculate the probability of the joint occurrence of two events: P(A and B) = P(A) * P(B|A)
• Conditional probability


  The probability of an event given that another event has occurred, denoted as P(A|B)
• Probability distribution


  A mathematical function that describes the likelihood of different outcomes in a random process
• Probability distributions


  • Uniform
  • Binomial
  • Normal
  • Poisson
• There are many more probability distributions besides the ones presented, but these are the most widely used
• Hypothesis testing


  The process of using sample data to determine whether to reject or fail to reject a null hypothesis about a population parameter
• Purpose of hypothesis testing

  • Prove something: Assess evidence in favor or against a claim
  • Make decisions based on data
  • Generalize findings from a sample to a population
• Hypothesis tests can only determine with a probability of error whether H0 is rejected (never accepted)
• Key components in hypothesis testing

  • Null hypothesis (H0)
  • Alternative hypothesis (H1)
  • Test statistic
  • Significance level (α)
  • P-value
  • Decision rule
• Types of hypothesis testing

  • Parametric tests (e.g. t-test, Z-test, ANOVA)
  • Non-parametric tests (e.g. Wilcoxon signed-rank test, Mann-Whitney U test)
  • Regression analysis
  • Chi-square test
• Null hypothesis (H0)


  A statement that there is no effect, relationship, or difference; represents the status quo or a baseline condition
• Alternative hypothesis (H1)


  A statement that contradicts the null hypothesis; represents an effect, relationship, or difference; the claim being tested
• One-tailed vs two-tailed tests

  One-tailed test: Directional hypothesis (e.g. H1: μ1 > μ2)

  Two-tailed test: Non-directional hypothesis (e.g. H1: μ1 ≠ μ2)
• Steps in hypothesis testing

  1. State the Null and Alternative Hypotheses

  2. Choose the Appropriate Test Statistic

  3. Calculate the Test Statistic

  4. Determine the P-value

  5. Make a Decision
• Hypothesis testing


  The process of testing a claim, relationship, or difference
• Null hypothesis (H0)


  The claim being tested
• George Gunnesch-Luca
• Statistics and Quantitative Research Methods in Psychology and Cognitive Sciences
• 25 / 48

  
• One-tailed test


  Directional hypothesis (e.g., H1: µ1 > µ2)
• Two-tailed test


  Non-directional hypothesis (e.g., H1: µ1 =! µ2)
• Formulating Hypotheses


  1. State the Null and Alternative Hypotheses

  2. Choose the Appropriate Test Statistic

  3. Determine the Significance Level (α)

  4. Calculate the Test Statistic and P-Value

  5. Compare the P-Value to the Significance Level and Make a Decision
• Research question: Is there a difference in height between two groups?
• Null hypothesis (H0)


  There is no difference in height between group A and group B
• Alternative hypothesis (H1)


  There is a difference in height between group A and group B
• Type I error


  Occurs when we reject the null hypothesis when it is true
• Type II error


  Occurs when we fail to reject the null hypothesis when it is false
• Power of a test

  The probability of correctly rejecting the null hypothesis when it is false
• Factors affecting power


  • Sample size
  • Effect size
  • Significance level (α)
  • Variability in the data