Statistical Hypothesis Testing and Interpretation

Posted on Jul 21, 2026 in Statistics

Statistical Test Selection Matrix

  • T-Test (1-Sample Mean): Testing a single group’s average (μ) against a single historical benchmark value.
  • 1-Prop Z-Test (1-Sample Proportion): Testing a single group’s percentage or proportion (p) against a benchmark value.
  • 2-Prop Z-Test (2 Independent Proportions): Comparing percentages or categorical success rates between two completely separate groups.
  • 2-Sample T-Test (2 Independent Means): Comparing the average scores or numerical measurements between two separate groups.
  • Paired T-Test (Dependent Means): Comparing means from matched or connected pairs (e.g., the same subjects measured “Before” vs. “After” on a scale).

Blueprint for Writing Interpretations

If You Reject the Null Hypothesis (H0)

Sentence 1 (Rarity statement): “It is rare to get a sample [average / difference of averages / percentage] of [insert sample value] if the real [average / difference / percentage] is [insert H0 value].”

Sentence 2 (Conclusion): “Therefore, we have enough evidence to believe that [state the alternative hypothesis H1 in plain words].”

If You Do Not Reject the Null Hypothesis (H0)

Sentence 1 (Rarity statement): “It is very common [or not rare] to get a sample [average / difference of averages / percentage] of [insert sample value] if the real [average / difference / percentage] is [insert H0 value].”

Sentence 2 (Conclusion): “Therefore, we do not have enough evidence to believe that [state the alternative hypothesis H1 in plain words].”

Confidence Interval (CI) Rules

Standard Interpretation Template

“We are [X]% confident that the true [percentage / average] of [describe context/population] is between [Lower Bound] and [Upper Bound].”

Critical Conceptual Nuances

Does the interval cover the actual true value? Always select “Cannot tell”. The true population parameter is a fixed, unknown constant; an individual sample interval either covers it or doesn’t, but we can never know without testing the whole population.

Connecting CI to a Two-Tailed Hypothesis Test

  • Significance Level (α): α = 100% – Confidence Level% (e.g., a 95% CL corresponds to an α = 5% two-tailed test).
  • If the H0 value falls OUTSIDE the interval: Choose Reject H0 (sufficient evidence to claim a change).

Handling Errors, Sizes, and Condition Failures

Condition Failures (The “Stop” Rule)

If a mandatory check fails (e.g., np0(1-p0) < 10 for a proportion test), stop immediately.

Write exactly: “We can see that this value is NOT greater than 10! Therefore, we cannot continue with this question because the conditions are not met!” Do not write out a curve, a test statistic, or a p-value.

Sample Size Calculations (n)

  • Rounding: Always round UP to the nearest next whole integer, even if the decimal is tiny (e.g., 270.6025 → 271).
  • Interpretation phrasing: “A sample of at least [n] [units/customers] must be taken to be able to estimate [p or μ] to within [E] with [X]% confidence.”

Type I vs. Type II Errors in Context

  • Type I Error (False Alarm): Rejecting a true H0. Context template: Concluding that the true parameter has changed/deviated from H0 when it actually has not.
  • Type II Error (Missed Detection): Failing to reject a false H0. Context template: Believing/concluding the parameter equals the baseline H0 value, when it is actually true that it has changed to the H1 condition.