Hypothesis Testing Notes

Confidence Level and Alpha
  • Confidence Level: A measure of how confident we are that our estimate includes the population parameter. The commonly used level is 95%.
  • Alpha (α): The significance level calculated as:
  • Formula: 1 - Confidence Level (in decimal)
  • Example: For a 95% confidence level, α = 1 - 0.95 = 0.05
  • When dealing in percentages, convert: 100% - Confidence Level\n - Example: 100 - 95 = 5% → 0.05 (as a decimal)
Degrees of Freedom
  • Degrees of Freedom (df): Calculated as sample size (n) minus one.
  • Example: If n = 45, df = 45 - 1 = 44.
Test Statistic and P-value Computation
  • Test Statistic: For a two-sample test using t-distribution.
  • T-value and P-value: Utilize calculators or statistical software to compute.
  • Example: If test statistic (t) = 2.3, P-value for two-tailed test might result in approximately 0.02625.
Decision Making in Hypothesis Testing
  1. Null Hypothesis (H₀): Typically posits no effect or no difference.
  2. Rejecting or Not Rejecting:
  • If the P-value < α, reject H₀.
  • If P-value ≥ α, do not reject H₀.
  1. Critical Values
  • Determine critical values based on t-distribution (for t-tests) or z-distribution (for z-tests).
  • Example critical values for α = 0.05 in a two-tailed test might be ±2.015 (for df = 44).
Test Scenarios
  • Scenario A (T-test with df = 44):
  • Test statistic (t) is outside critical values, rejection of H₀ occurs (example: t = 2.3).
  • Scenario B (Z-test with 99% confidence):
  • Given a P-value of 0.035 when α is 0.01, conclusion is to reject H₀ as P-value < α.
Working with Critical Values
  • Comparison of Test Statistic with Critical Values:
  • If a test statistic lies between critical values, do not reject H₀.
  • If it lies beyond, do reject H₀.
Hypothesis Testing Flow
  1. Establish Hypotheses:
  • Null (H₀): mu = hypothesized value.
  • Alternative (H₁): mu does not equal hypothesized value.
  1. Determine Confidence Level and Calculate α.
  2. Compute Test Statistic and P-value.
  3. Compare P-value with α:
  • Draw conclusions based on comparison: reject or not reject.
  1. Confidence Intervals:
  • Calculate using mean ± (critical value * standard error).
Final Thoughts
  • Conclusively articulate your findings based on the statistical analysis and the context of the hypothesis test.
  • Use structured statements to convey whether H₀ was rejected or not and what that indicates about the population mean.