Slide 1Steps for Testing a HypothesisDescriptive vs Inferential StatisticsProbabilities for a Normal DistributionProbability & Alpha LevelsChi-Square (χ²)Chi-Square (χ²)χ² Goodness of FitFrequency Distribution for Null χ²Frequency Distribution for Null χ²χ² Goodness of FitChi-Square (χ²)χ² Test of Independenceχ² Test of Independenceχ² Test of Independenceχ² Test of IndependenceHYPOTHESIS TESTING & CHI-SQUARED1Steps for Testing a Hypothesis1. Assume There Is No Effect (Null Hypothesis)2. Collect Data3. Calculate the Probability of Getting Such Data, or Even More Extreme Data, If the Null Hypothesis Is True4. Decide Whether to Reject or Retain the Null Hypothesis2Descriptive vs Inferential Statistics•Descriptive•Used to describe the properties of the data•Definitive•Ex. Central tendency, variability, frequency distribution•Inferential•Uses data from a sample to make estimates about the population•Probabilistic •Ex. Chi-square, correlation, t-tests3Probabilities for a Normal Distribution4Probability & Alpha Levels•Probability (p)•The likelihood that the scores in your data would have occurred by chance alone (rather than due to a variable in your study)•Determined by the statistical test of your data•Alpha level (α)•Probability value used to define the point at which the occurrence of something is “very unlikely” to have happened by chance alone•Chosen by the researcher•Common alpha levels•α = .05 (5%)*•α = .01 (1%)•α = .001 (0.1%)•Probabilities lower than alpha mean that the data is not likely to have happened by chance•p < .05* •p < .01 •p < .001 5Chi-Square (χ²)•Nominal/Categorical (and Ordinal/Rank) data•Frequencies for each category•Compares observed proportions (sample) to the expected proportions for the null hypothesis (population)•Assumptions•Random Sampling•Data are drawn from the population at random•Independence of Observations•Each observed frequency comes froma different participant•Size of Expected Frequencies•fe >56Chi-Square (χ²)•Goodness of Fit (One-Way)•1 variable•Compares sample to the population (based on null hypothesis)•H0: No difference between the sample distribution and the population distribution•Test of Independence (Two-Way)•2 variables•Compares the frequency distribution of one variable within the categories of the second variable•H0: No difference in the distribution of one variable at different levels of the second variable7χ² Goodness of Fit•Observed Frequency (fo)•Number of participants from the sample who are classified in a particular category•Sample•Expected Frequency (fe)•Number of participants that would be predicted to be classified in a particular category if the null hypothesis is true •Hypothetical Population8XfofeX1X2eeofff22)(Frequency Distribution for Null χ²9αFrequency Distribution for Null χ²10•Degrees of Freedom (df)•The number of free choices that exist when you are determining the null hypothesis or the expected frequencies•Categories (C) 1Cdfχ² Goodness of Fit11α = .016.63α = .053.84χ²Frequency1dfChi-Square (χ²)•Goodness of Fit (One-Way)•1 variable•Compares sample to the population (based on null hypothesis)•H0: No difference between the sample distribution and the population distribution•Test of Independence (Two-Way)•2 variables•Compares the frequency distribution of one variable within the categories of the second variable•H0: No difference in the distribution of one variable at different levels of the second variable12feX1X2TotalY1fr1Y2fr2Totalfc1fc2nχ² Test of Independence•Expected Frequency (fe)•fc: frequency total for column•fr: frequency total for row•n: total of all cells (sample size)13nfffrcenffrc 11nffrc 12nffrc 21nffrc 22χ² Test of Independence14•Degrees of Freedom (df)•C: number of columns•R: number of rowfeX1X2TotalY1fr1Y2fr2Totalfc1fc2n)1)(1(  CRdfχ² Test of Independence •Effect Size•A measurement of the absolute magnitude of a treatment effect, independent of the size of the sample(s) being used•Phi (φ)•Used for 2x2 tables (2 variables with 2 categories)•Cramér’s V•Used for tables larger than 2x2 (any variable has more than 2 categories)•df*: Either (C-1) or (R -1), whichever is smaller15n2*)(2dfnVχ² Test of Independence16•Phi (φ)•φ = 0.10 small•φ = 0.30 medium•φ = 0.50 largeSmall EffectMediumEffectLarge Effectdf* = 1 0.10 0.30 0.50df* = 2 0.07 0.21 0.35df* = 3 0.06 0.17 0.29•Cramér’s