Statistics and Probability Reviewer

Statistics and Probability Reviewer

• Hypothesis: An educated guess explaining a set of facts or natural phenomena.
• Hypothesis Testing: A decision-making process evaluating population claims based on sample characteristics.
• Null Hypothesis (H₀): No difference between a parameter and a specific value, or between two parameters.
• Alternative Hypothesis (H₁): A difference exists between parameters or a parameter and a specific value.
• Level of Significance (α): A measure of evidence needed to reject the null hypothesis. A stronger effect needs more evidence (higher alpha).
• Type I Error: Rejecting a true null hypothesis.
• Type II Error: Accepting a false null hypothesis.
• Rejection Region: The values of the test statistic where the null hypothesis is rejected. Also known as the critical region.
  • Directional (One-Tailed):
    • Left Tail: Probability in the left tail of the distribution.
    • Right Tail: Probability in the right tail of the distribution.
  • Non-directional (Two-Tailed): Probability in both tails of the distribution.
• Z-test: Used when the sample size (n) is greater than 30.
• t-test: Used when the sample size (n) is less than 30.
• Population: The totality of objects, individuals, or characteristics of interest.
• Sample: A carefully selected group from a population.
• Parameter: A numerical value describing a population characteristic.
• Statistic: A numerical value describing a sample characteristic.
• Test Statistic: A statistical method to test a hypothesis (e.g., Z-test, t-test)
• P-Value Method: Another approach to hypothesis testing.

Other Elements of Hypothesis Testing

• Population-Proportion:
  • p = x/n (sample proportion)
  • q₀ = 1 - p₀
  • H₀ : p ≤ 0.5 (not a majority)
  • H₁ : p > 0.5 (a majority) Example to test for whether some value is above or below 0.5 i.e., some value constitutes a majority.
• Bivariate Data: Data dealing with two variables. Useful to understand the relationship between them.
• Scatter Plot: A graph displaying bivariate data.
• Correlation Analysis: Finds the linear relationship between variables.
• Correlation Coefficient (r): Measures the strength and direction of a linear relationship between bivariate data.
• Regression Analysis: Identifying the relationship between a dependent variable and one or more independent variables.
• Regression Line: A line that best fits the observed data; useful for prediction.
  • Independent Variable (X): The hypothesized variable affecting the dependent variable.
  • Dependent Variable (Y): The variable whose value relies on the independent variable.
• Slope: Indicates the steepness of the regression line.
• Y-intercept: The point on the y-axis where the regression line intersects.