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 It is fixed in advance by the question: a process target, a validated historical reference, or a physical zero.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.”
기준값·효과·n을 한 축에 연결하세요
μ₀=10, σ=3인 투명한 교육 모형에서 효과와 독립 n이 CI와 p-value를 어떻게 바꾸는지 봅니다.
처음이라면: 무엇을 눌러야 하나요?
- 1. 질문을 먼저 읽기Lab 제목에서 이번에 비교할 한 가지를 확인합니다.
- 2. 조건 하나만 바꾸기처음에는 n, 효과, 산포 같은 입력 중 하나만 바꾸십시오.
- 3. 새 합성 표본 누르기새 합성 데이터가 만들어집니다. 같은 조건도 표본에 따라 달라질 수 있습니다.
- 4. 그림과 계산 결과 비교하기바꾸기 전후 무엇이 움직이고 무엇이 그대로인지 한 문장으로 적어보십시오.
막히면 초기화로 돌아가 기본 결과를 본 뒤 조건 하나만 바꾸십시오. 이 Lab은 정답 판정기가 아니라 패턴 관찰 도구입니다.
같은 설정의 합성 관측
계산 결과
기각 여부보다 효과와 CI가 연구 단위에서 얼마나 큰지 먼저 해석합니다.
교육용 synthetic model · bjs-comparison-sequence-v1. 실제 연구 판단에는 실험단위, 결측, 분포, 다중성, 사전계획과 도메인 기준을 별도로 반영해야 합니다.
Do not extract only one line from JMP output
기준값, 표본평균, t ratio와 방향별 p값을 연결합니다.
기준값과 양립하는 효과 범위를 먼저 읽습니다.
히스토그램·box plot으로 독립성 밖의 모양 문제를 점검합니다.
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.