Conditions for a One-Sample T Test

Why Conditions Matter

A one-sample t test uses a sample mean to assess a claim about a population mean. Its t statistic can be calculated from a sample mean, a sample standard deviation, and a hypothesized mean. But the calculation alone does not show that the reference t distribution is appropriate. That distribution depends on how the observations were obtained and on the population model.

Checking conditions is therefore part of the statistical argument, not a formality after the arithmetic. In particular, distinguish an exact t reference distribution from an approximate one. Sampling without replacement from a finite population can create dependence; the usual 10% guideline makes that dependence small enough to treat as negligible in many applications, but it does not make the observations exactly independent.

What Random Selection or Assignment Supports

A probability-based method of selecting individuals from a population supports inference from the sample to that population. A simple random sample, for example, gives each possible sample of the specified size an equal chance of selection. The method should be described, not merely assumed from the fact that data were collected.

Random assignment answers a different question. In an experiment, randomly assigning units to a condition helps make the assigned groups comparable on average, supporting conclusions about the effect of the assigned condition under the study design. Random assignment is not the same as randomly sampling from a wider population: it does not by itself establish that the study participants represent that population.

For a one-sample mean analysis, identify the population mean the inference is intended to address and explain how the observations relate to it. If the data come from a convenience sample, for instance, the formula for a t statistic remains computable, but the usual probability-based justification for generalizing to a population is missing. Random selection also does not guarantee every other condition: observations can still be dependent, and the population need not be normal.

Independence and the 10% Guideline

The standard one-sample model treats the observations as independent and drawn from the same population. Independence means that knowing one observation does not change the probability distribution of another. It can be doubtful when the same person contributes several measurements, when people are sampled in related groups, or when one observation can influence another. A large sample size does not automatically resolve dependence caused by the design.

Sampling without replacement from a finite population creates dependence: once one unit has been selected, the composition of the remaining pool has changed. When the sample is a small fraction of a much larger population, that effect is often sufficiently small to use the independence model as an approximation. The common 10% guideline says that, for sampling without replacement, the sample size should be no more than 10% of the population size:

Here \(n\) is the sample size and \(N\) is the finite population size. Satisfying this guideline does not turn dependent observations into exactly independent ones. It supports treating the dependence as negligible for an approximate analysis. When observations are sampled independently, rather than without replacement from a finite population, this particular correction is not needed; other sources of dependence still need consideration.

Population Shape and the Exact t Result

For an exact one-sample t test, the model is that \(X_1,\ldots,X_n\) are independent observations from the same normal population, \(N(\mu,\sigma^2)\), with \(\sigma^2\) the population variance. The sample must have at least two observations so that \(s\) is defined. Under the null hypothesis, the population mean is \(\mu_0\). The test statistic is

Under that model and the null hypothesis, \(T\) has a t distribution with \(n-1\) degrees of freedom. The normal-population condition is what permits the exact result for samples of any size; a small sample does not make the condition less important.

The short justification shows why the assumptions matter. If the observations are independent and normal, the standardized sample mean \(Z=\sqrt{n}(\bar{x}-\mu_0)/\sigma\) has a standard normal distribution under the null. The sample variance also has a known distribution under this model: \((n-1)s^2/\sigma^2\) has a chi-square distribution with \(n-1\) degrees of freedom, and it is independent of \(Z\). Substituting these quantities into the statistic gives

A standard normal variable divided by the square root of an independent chi-square variable divided by its degrees of freedom has a t distribution. Thus the exact t result follows from the independent normal model; it is not a consequence of the statistic’s algebra alone.

In some settings, a t procedure is used as an approximation when the observations are independent or nearly independent and the sample mean is approximately normal. This is not the exact result above. The quality of that approximation depends on the data-generating distribution and sample size, so there is no universal sample-size cutoff that guarantees it. In particular, a small sample with substantial skewness or an extreme observation may not be well described by the exact normal model, and one should not claim exactness just because software reports a t statistic.

Putting the Conditions Together

A conditions check should connect each claim to the study design or the population model. It is useful to separate whether the data support the intended inference from whether the t reference distribution is exact.

Why the Distinction Matters

Conditions determine what a test result can reasonably mean. A small p-value from an exact t reference distribution is meaningful only if that reference distribution is justified by the model. If the data are dependent, nonrandomly selected, or inconsistent with the normal model, the calculation may not provide the claimed inference.

Keep the language of the conclusion aligned with the condition check. A random sample can support generalization to its sampling population, but not automatically to other populations. Random assignment can support a causal comparison under its design, but does not establish population representativeness. An approximately independent sample supports an approximate analysis, not an exact one.

Common Mistake

Wrong reasoning: “The sample is less than 10% of the population, so the observations are independent and the t distribution is exact.”

Correction: For sampling without replacement, the 10% guideline says the dependence is often small enough to ignore for an approximate analysis. It does not remove the dependence. An exact t reference requires genuinely independent observations from a normal population. Describe the design accurately and label the inference exact or approximate accordingly.

Likewise, finding a numerical t statistic does not demonstrate that the conditions hold. Check the sampling or assignment mechanism, independence, and population shape before interpreting the result.