Statistics Sample: Hypothesis Test Solved in Handwriting

A complete two-sample t-test showing every step a stats student writes by hand — setup, computation, table lookup, conclusion.

The problem

A typical intro-stats problem: a researcher collects test scores from two groups (control and treatment) and wants to know if the treatment has a statistically significant effect at the α = 0.05 level.

Control group: n₁ = 20, mean = 72.4, s₁ = 8.1. Treatment group: n₂ = 22, mean = 78.9, s₂ = 7.6. Test at α = 0.05.

What the handwritten solution shows

State hypotheses. H₀: μ₁ = μ₂ (no effect). H_a: μ₁ ≠ μ₂ (two-sided test).
Compute the test statistic. Scrawl writes out the pooled standard error formula, substitutes the values, simplifies the fraction, and arrives at t ≈ -2.68.
Degrees of freedom. df = n₁ + n₂ - 2 = 40.
Look up critical value. For a two-sided test at α = 0.05 with df = 40, the critical t is ±2.021.
Decision and conclusion.|t| = 2.68 > 2.021, so reject H₀. In plain English: there is sufficient evidence at the 5% level that the treatment changes mean scores.

Notation Scrawl renders cleanly

Subscripts, square roots, Greek letters (μ, σ, α), and the pooled-variance fraction are all rendered in proper handwriting style — not as text approximations. This is the main pain point for students using general-purpose chatbots for stats homework.

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