📘 When and How to Use a One-Sample t-Test

This app helps students understand when and why to use a t-test instead of a z-test, including equations, step-by-step computations, and examples — no graphs.

🧠 When to Use a t-Test

Population is normal or approximately normal
Population variance σ² is unknown
Use sample standard deviation s and degrees of freedom df = n − 1

📐 Test Statistic Formula

t = (x̄ − μ₀) / (s / √n)
where:

x̄ = sample mean
μ₀ = hypothesized mean
s = sample standard deviation
n = sample size

📑 Summary Table: One-Sample t-Test (Variance Unknown)

Alternative Hypothesis	P-value	Reject H₀ if...
H₁: μ ≠ μ₀	2P(Tₙ₋₁ > \|t₀\|)	t₀ > tₐ⁄₂,ₙ₋₁ or t₀ < −tₐ⁄₂,ₙ₋₁
H₁: μ > μ₀	P(Tₙ₋₁ > t₀)	t₀ > tₐ,ₙ₋₁
H₁: μ < μ₀	P(Tₙ₋₁ < t₀)	t₀ < −tₐ,ₙ₋₁

🧪 Example: Golf Club Restitution

Given:
Sample mean x̄ = 0.83725, s = 0.02456, μ₀ = 0.82, n = 15
Degrees of freedom: df = 14

Step-by-Step:

Step 1: Compute Standard Error:
SE = s / √n = 0.02456 / √15 = 0.00634

Step 2: Compute t-statistic:
t = (0.83725 − 0.82) / 0.00634 = 2.681

Step 3: Find critical value at α = 0.05 (right-tailed):
t₀.05,14 = 2.145

Step 4: Compare and conclude:
2.681 > 2.145 ⇒ Reject H₀

Step 5: P-value (from software):
P ≈ 0.0179

✅ Final Summary

Test Statistic (T): 2.681
Degrees of Freedom: 14
P-value: 0.0179
Critical Value (α = 0.05): 2.145
Decision: Reject H₀

📘 T-test vs Z-test Comparison

Aspect	t-test	z-test
Population Variance Known?	No (use s)	Yes (use σ)
Distribution Assumption	Normal or approx. normal	Normal
Degrees of Freedom	n − 1	N/A (standard normal)
Test Statistic	t = (x̄ − μ₀) / (s / √n)	z = (x̄ − μ₀) / (σ / √n)
Used When	σ unknown, small n	σ known, large n
Table/Dist Used	Student’s t	Standard Normal