Question 1

What is a one-sample t-test?

Accepted Answer

A one-sample t-test is a parametric hypothesis test that determines whether the mean of a single sample is significantly different from a specified value (μ₀). Examples: "Is our product weight significantly different from the labelled 500g?" or "Is the average website load time significantly above the 2-second target?" It uses the t-distribution to compute a p-value that accounts for sample size and variability.

Question 2

What is the null hypothesis in a one-sample t-test?

Accepted Answer

H₀: μ = μ₀ (the population mean equals the hypothesised value). The alternative depends on your tail choice: H₁: μ ≠ μ₀ (two-tailed), H₁: μ > μ₀ (upper/right one-tailed), or H₁: μ < μ₀ (lower/left one-tailed). Two-tailed is the default choice unless you have a strong directional hypothesis specified before data collection.

Question 3

What is Cohen's d and how do I interpret it?

Accepted Answer

Cohen's d = (x̄ − μ₀) / s measures effect size — how large the difference is relative to the standard deviation. Guidelines: d = 0.2 small, d = 0.5 medium, d = 0.8 large. A significant p-value (p < 0.05) with d = 0.1 means the effect is real but tiny. Always report effect size alongside p-values to distinguish statistical from practical significance.

Question 4

What are the assumptions of a one-sample t-test?

Accepted Answer

1) Independence: each observation is independent. 2) Normality: the population distribution is approximately normal. For n ≥ 30, the Central Limit Theorem makes this less critical. 3) Random sampling. The t-test is fairly robust to mild normality violations when n ≥ 15. For seriously non-normal data (heavy skew, outliers), consider a Wilcoxon signed-rank test instead.

Question 5

How do I interpret the 95% confidence interval for μ?

Accepted Answer

The 95% CI is x̄ ± t*(s/√n), where t* is the critical t-value for df degrees of freedom at α = 0.05. If μ₀ falls outside this interval, you will reject H₀ at α = 0.05 (two-tailed). Interpretation: "We are 95% confident the true population mean lies between the lower and upper bounds."

Question 6

One-sample t-test vs z-test: which should I use?

Accepted Answer

Use a t-test when the population standard deviation σ is unknown (almost always). Use a z-test only when σ is known from prior research or theory. For n ≥ 30, both give virtually identical p-values. For small samples, the t-test is essential — it has heavier tails that account for the uncertainty in estimating σ from data.

p-value	Interpretation	Decision (α = 0.05)
p < 0.001	Very strong evidence against H₀	Reject H₀
0.001 ≤ p < 0.01	Strong evidence against H₀	Reject H₀
0.01 ≤ p < 0.05	Moderate evidence against H₀	Reject H₀
0.05 ≤ p < 0.10	Weak evidence against H₀	Fail to reject H₀
p ≥ 0.10	Little to no evidence against H₀	Fail to reject H₀

d value	Effect size	Practical meaning
< 0.2	Negligible	Difference too small to be practically meaningful
0.2–0.5	Small	Noticeable but modest effect
0.5–0.8	Medium	Visible effect in most contexts
> 0.8	Large	Substantial, clearly perceptible difference

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Effect Size (Cohen's d) Reference

One-Sample t-Test Step by Step

Common Mistake: Confusing Statistical and Practical Significance

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