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One-sample t-test: compute t, df and p-value and decide.

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t=(x̄−μ₀)/(s/√n), df=n−1

📖 Tutorial | t-Test

1. Definition
One-sample t-test uses sample s when σ is unknown: t=(x̄−μ₀)/(s/√n), df=n−1. Larger t and smaller p reject H₀.
2. Symbols
SymbolMeaning
ssample standard deviation
ttest statistic
dfdegrees of freedom n−1
ptwo-sided p-value
3. How it works
  • t=(x̄−μ₀)/(s/√n);
  • df=n−1;
  • Get the two-sided p from the t distribution;
  • Reject H₀ if p<α.
4. Steps
  1. Enter μ₀, x̄, s, n, α;
  2. Click Calculate;
  3. Read t, df and p;
  4. Decide by p<α.
5. Example
Example: H₀: μ=50, x̄=52, s=4, n=16. t=2, df=15, p≈0.064>0.05, do not reject H₀.
6. Pitfalls
Use t only when σ is unknown;
Small samples need approximate normality;
df=n−1, not n.

❓ FAQ | t-Test

t vs Z test?
Z if σ known; t if σ unknown (using sample s).
Why df=n−1?
Estimating the mean costs one degree of freedom.
What does p≈0.064 mean?
Not significant at 0.05; do not reject H₀, but near the boundary.
Do I need a large sample?
t is suited to small samples, but the population should be approx. normal.
How to read the two-sided p?
Under H₀, the two-sided probability that |t|≥observed.