Which procedure is appropriate here?
Two questions, in this order. First: does the test match the situation — is the variable quantitative or binary, is the claim about a mean or a proportion, and is σ genuinely known or being estimated from these very data? Second: do its conditions hold? A procedure is appropriate only if it survives both.
Make a choiceNothing is graded until you commit to one. The simulation below is already running the test a careless analyst would reach for in this situation, so you can see what it delivers before you decide.
Say it right
Seven things people say about conditions and assumptions. Three of them are correct. Tap each one.
The habit worth buildingBefore you compute anything, say out loud where the numbers came from and what had to be true for the formula to mean what you want it to mean. A condition is a statement about the data’s origin, not about the data’s appearance — and no amount of staring at the numbers on the page will tell you how they were chosen.
Three assumptions, one repair
Sample size is not a general‑purpose fix. It repairs exactly one of the three things on this list. Shape is the forgiving one: the Central Limit Theorem says the sampling distribution of x̄ drifts toward normal as n grows no matter what the population looks like, so a skewed population is a problem that expires. How long it takes depends on how skewed — one rough guide is that you want n above roughly 25γ², which asks for about 25 observations at a skewness of 1 and about a hundred at a skewness of 2. Treat that as a hint, not a law; the curve above is the actual answer for the actual population. Dependence is unforgiving: if observations come in correlated groups, every extra observation adds less information than the formula credits it with, so the error stays put as n climbs. A non‑random sample is worse than unforgiving — it is anti‑forgiving. The bias does not shrink with n, but the standard error does, so a large biased sample converges confidently onto the wrong number.
The normality condition is about the population, not about your histogram. This trips up nearly everyone. The condition the t‑test needs is that the population be roughly normal, or that n be large enough for the CLT to take over. Your sample is only evidence about that population — and it is thin evidence when n is small, which is precisely when the condition matters most. A sample of 8 from a perfectly normal population routinely looks lumpy and lopsided; a sample of 8 from a badly skewed population routinely looks fine. So the sample histogram is worth a glance for outliers and obvious multiple peaks, but the stronger evidence is usually what you know about the variable itself. Charges, incomes, durations, and counts of rare events are right‑skewed as a matter of subject knowledge, whether or not this particular sample shows it.
The independence condition has a quiet version and a loud version. The loud version is repeated measurement — two eyes, two knees, four visits from the same patient, five students from the same classroom — and it inflates the false‑alarm rate exactly as the ρ slider shows. The quiet version is sampling without replacement: once you have drawn someone, they cannot be drawn again, so the draws are not strictly independent. That dependence is negligible as long as the sample is a small slice of the population, which is where the familiar 10% condition comes from. Note which way it errs: sampling without replacement makes x̄ less variable than the formula thinks, so ignoring it makes the test conservative rather than reckless. It is the only violation on this page that errs in your favour, and it is still worth knowing about.
When a condition fails, the answer is a different procedure — not a softer sentence. A failing condition is not a reason to write “results should be interpreted with caution” under a p‑value you no longer believe. If np₀ is too small, the count really does follow a binomial distribution, and you can add binomial probabilities directly: that is the exact binomial test, and it carries no large‑count condition at all. If the population is badly skewed and n is small, there are procedures that never assumed normality in the first place. If the observations come in clusters, collapse each cluster to one number and test those. In every case you are trading some power for an answer that means what it says — which is the only kind worth reporting.
One closing distinction, because the simulation makes it easy to blur. Everything on this page is about the false‑alarm rate: how often a test rejects a claim that is true. That is the promise α makes, and it is the promise a broken assumption breaks. It is a statement about the long‑run behaviour of the procedure, not about your one study — a single p‑value of 0.03 is not “a 3% chance the claim is true,” and it never was, in a well‑behaved world or a broken one. What the meter above tells you is narrower and more useful than that: whether the machine you are about to trust is calibrated.