A filling process targets 10.0 mL, while the mean of 12 independent vials is 10.3 mL. Is 0.3 mL a real process shift, or ordinary fluctuation from a small sample?
This unit asks one question.
What supports a claim that one population mean differs from a pre-specified target?
The target comes from outside the sample, before you inspect it
For a one-sample comparison, the reference μ₀ is not the current sample mean or the number that looks most favorable. It is fixed in advance by the question: a process target, a validated historical reference, or a physical zero. Moving it after seeing data changes the question the test answers.
Write the null and two-sided alternative as:
H0: μ=μ₀H1: μ≠μ₀
The estimated effect is x̄−μ₀. If the unit is mL, the effect is also in mL. Define whether that difference is practically large before looking at a p-value.
t scales a difference by its standard error
When population SD is unknown, estimate the standard error of the mean with sample SD s.
SE(x̄)=s/√n
t=(x̄−μ₀)/(s/√n)
The numerator is the observed effect and the denominator is the expected sampling fluctuation. The same 0.3 mL produces a larger |t| if s is smaller or independent n is larger. Because s is estimated from the same sample, the reference distribution has heavier tails than a normal distribution: the t distribution, with n−1 degrees of freedom.
The 95% CI x̄ ± t*×SE is an interval for the mean, not an effect. Checking whether it contains μ₀ shares the same decision boundary as a two-sided α=.05 t-test. To express an effect CI, subtract μ₀ from both endpoints.
p-value and CI are two expressions of the same evidence
Under H0 and the model assumptions, the p-value is the tail probability of observing |t| at least this large. A CI shows the range of means or effects compatible with the data. The p-value is compact for a threshold decision; the CI reveals direction, magnitude, and uncertainty.
For example, if the effect is 0.30 mL and its 95% CI is −0.08–0.68 mL, zero remains compatible with the data, so H0 is not rejected. It does not prove that the process equals the target. Shifts from −0.08 to +0.68 remain compatible, and whether +0.68 is practically acceptable needs a separate criterion.
Whether the mean differs significantly from the target and whether individual products meet specification are different questions. A one-sample mean CI concerns the population mean; it does not guarantee individual observations or the range containing most of the population.
Assumptions are not one line saying “normality p-value passed”
For a one-sample t procedure, independence of observations and meaningful mean and SD at the experimental-unit level are especially important. Twenty instrument reads from the same batch are not batch n=20. Counting technical replicates as independent n makes SE unjustifiably small.
A normality test has low power in small samples and can flag trivial departures in large ones. Examine the histogram, dot plot, outliers, and data-generating process together. Mean inference by t can be reasonably robust with symmetric data lacking extremes, but strong skew, censoring, mixtures, or small n call for transformations, robust methods, or an explicit distributional model.
In-Silico Lab: repeat the distance between target and effect
The Lab uses a simple z model with μ₀=10 and known σ=3 to make the relationships visible. Remember that an actual one-sample t test estimates σ with s.
- With true difference 0, change the seed and see whether the p-value changes each time.
- With effect 1.0, compare CI widths for n=8 and n=80.
- Find settings with similar p-values but different effects and CIs.
- Explain in units why “significant” does not mean “important.”
Connect reference value, effect, and n to one axis.
See how the effect and independence n change the CI and p-value in a transparent training model with μ₀=10 and σ=3.
If this is your first time: What should I press?
- 1. Read the question firstIn the Lab title, check the one thing you will compare this time.
- 2. Change just one conditionInitially, change only one of the inputs: n, effect, or spread.
- 3. new composite specimen pressureNew synthetic data is created. The same conditions may vary depending on the sample.
- 4. Pictures and calculation results CompareWrite in one sentence what moves and what stays the same before and after the change.
If it gets stuckresetGo back to see the default results and change just one condition. This Lab is not a correct answer tester but a pattern observation tool.
Synthetic observations of the same settings
calculation result
We first interpret how large the effects and CIs are for each study unit, rather than whether to reject them.
educational synthetic modelbjs-comparison-sequence-v1. Actual research judgments must separately reflect experimental units, missingness, distribution, multiplicity, pre-planning, and domain criteria.
Do not extract only one line from JMP output
Connect the reference value, sample mean, t ratio, and p value for each direction.
The effective range that is compatible with the reference value is read first.
Check for shape problems outside of independence using histograms and box plots.
Test Mean can show the hypothesized value, actual estimate, t ratio, and two- and one-sided p-values. Select the direction appropriate to the research question in advance and report the exact p-value. Choosing the smallest of three displayed directions after the fact is post hoc selection.
A result statement contains six elements
The mean fill volume of 12 independent vials was 10.30 mL (SD 0.62). The mean difference from the pre-specified target of 10.00 mL was 0.30 mL, with a two-sided 95% CI of −0.09 to 0.69 mL, one-sample t(11)=1.68, p=.12. These data did not detect a mean difference from target; they did not establish equivalence to target or specification compliance of individual vials.
Report independent n, mean and SD, target, an effect with units, CI, test direction, statistic, degrees of freedom, and p-value.
Takeaways
- Set the target outside the current sample and before analysis.
- The effect is
x̄−μ₀; SE describes sampling variation of the mean. - t is the effect divided by estimated SE; degrees of freedom reflect uncertainty in estimating SD.
- CI and two-sided p-value express uncertainty from the same model differently.
- Non-significance is not proof of equality to the target or equivalence.
- Do not count technical replicates as independent n.
The next unit compares two groups rather than one target. The first distinction there is between independent and paired structures.
Official supplementary resources
The values, figures, and Lab in this article are educational synthetic material. Apply site-specific design and domain criteria to real decisions.