Exam Notes for PSY3204C - Key Concepts and Formulas

Errors in Hypothesis Testing

• Decision Errors: Incorrect conclusions drawn from hypothesis testing regarding an unknown or undetected reality.

  • Not merely procedural mistakes; correct methods may yield incorrect conclusions.

  • Errors stem from limitations in using sample statistics to estimate population parameters.

• Probability of Errors: Impossible to entirely eliminate errors in hypothesis testing.

Types of Errors

• Type II Error (𝛽):

  • Failing to reject the null hypothesis when it is false.

  • Researchers can try to reduce the chance of this error, but cannot set its level in advance.

  • Example: Not detecting an existing effect.

• Type I Error (𝛼):

  • Rejecting the null hypothesis when it is actually true.

  • This type of error can be set in advance (e.g., 5% significance level).

  • Example: Finding effects that do not actually exist.

Statistical Power

• Definition: The probability of obtaining a statistically significant result when the research hypothesis (H1) is true.

  • Directly related to the power to detect significant results when they exist.

  • Power = 1 - 𝛽 (Type II error probability).

  • Higher Type I error leads to a lower Type II error, affecting power.

Effect Size

• Definition: Measures the degree of difference between populations, indicating the presence of a significant effect.

  • Key to assessing how large an effect is rather than merely whether it exists.

• Cohen's d:

  • Formula: d = \frac{\mu1 - \mu0}{\sigma} where:

  • \mu_0 = mean under the null hypothesis (H0)

  • \mu_1 = mean under the alternative hypothesis (H1)

  • \sigma = common standard deviation of distributions.

  • Effect sizes categorized as small (d = .2), medium (d = .5), and large (d = .8).

ANOVA (Analysis of Variance)

• Purpose: Assess the effect of one or more factors on a dependent variable by comparing group means.

• Sum of Squares (SS): Total variability calculated by summing squared differences from the mean.

  • SS{total} = SS{between} + SS_{within}

• Mean Squares (MS): Average variability calculated from SS divided by degrees of freedom (df).

F-Ratio in ANOVA

• Test Statistics: Ratio comparing between-group variance to within-group variance.

  • F = \frac{MS{between}}{MS{within}}

  • Cutoff F-ratio used to determine significance (e.g., F(df{between}, df{within})).

Two-Way ANOVA

• Definition: Examines the effects of two categorical independent variables on a dependent variable.

  • Main effect: Individual effect of each factor.

  • Interaction effect: The combined effect of factors on the dependent variable, which can differ depending on the levels of other factors.

Correlation

• Scatter Plot: Visual representation of the relationship between two variables.

• Correlation Coefficient (r): Measures the degree of linear correlation, ranging from -1 (perfect negative) to +1 (perfect positive).

  • Formula: r = \frac{\Sigma ZX ZY}{N}

• Coefficient of Determination (r^2): Proportion of variance accounted for in one variable by the other.

Regression Analysis

• Purpose: To examine the prediction capabilities between independent and dependent variables.

• Regression Equation: \hat{Y} = a + bX where:

  • a = intercept

  • b = slope (change in Y for a unit change in X).

• Least Squares Criterion: Minimize the sum of squared differences between observed and predicted values.

• Coefficient of Determination (R²): The percentage of variation in the criterion variable explained by the predictor.

Chi-Square Tests

• Purpose: Assess relationships between categorical variables.

  • Observed vs. Expected Frequencies: Used to determine if distributions differ significantly from expectations.

General Exam Tips

• Understand definitions and formulas.

• Familiarize yourself with different types of errors in hypothesis testing and their implications.

• Practice ANOVA calculations and understand the significance of F-ratios.

• Review correlation coefficients and regression to familiarize yourself with predictive relationships.