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Hypothesis Testing

One-sample Z test: compute Z, p-value and decide whether to reject H₀.

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Z=(x̄−μ₀)/(σ/√n)
e.g. 0.05

📖 Tutorial | Hypothesis Testing

1. Definition
Hypothesis test: set H₀: μ=μ₀, compute Z=(x̄−μ₀)/(σ/√n); the p-value is the chance of such an extreme result under H₀. Reject H₀ if p<α.
2. Symbols
SymbolMeaning
H₀null hypothesis μ=μ₀
Ztest statistic
p-valueextreme probability under H₀
αsignificance level (e.g. 0.05)
3. How it works
  • Z=(x̄−μ₀)/(σ/√n);
  • Two-sided p=2(1−Φ(|Z|));
  • Compare p with α;
  • Reject H₀ if p<α, else do not reject.
4. Steps
  1. Enter μ₀, x̄, σ, n, α;
  2. Click Calculate;
  3. Read Z and the p-value;
  4. Decide by p<α.
5. Example
Example: H₀: μ=100, x̄=103, σ=15, n=36. Z=1.2, p≈0.2301>0.05, do not reject H₀.
6. Pitfalls
“Do not reject H₀” ≠ “accept H₀”;
A small p-value does not mean a large effect;
If σ is unknown, use a t-test.

❓ FAQ | Hypothesis Testing

What is a p-value?
Under H₀, the probability of observing the current or a more extreme result.
Why reject when p<α?
A rare event unlikely in one trial casts doubt on H₀.
Does rejecting H₀ prove it?
No; it is statistical evidence; a Type-I error remains possible.
Two-sided vs one-sided?
Two-sided allows deviation either way; here we use two-sided.
How to choose α?
Commonly 0.05 or 0.01; smaller α makes rejection harder.