From a Test Result to a Statistical Report
A one-sample t test gives a numerical result, but a reader needs more than a number to understand what the test says. A useful report connects the sample estimate to the hypothesized population mean, identifies how unusual the result would be under the null hypothesis, and states what the evidence implies about the question being investigated.
A concise report normally includes the sample mean, the test statistic, its degrees of freedom, the p-value, and a conclusion stated in context. The conclusion should match the alternative hypothesis: evidence for a mean greater than the target is not the same as evidence that the mean differs from the target.
The estimate and the test statistic answer different questions. The sample mean, \(\bar{x}\), describes the estimated population level in the original measurement units. The t statistic describes how far that estimate is from the null value in standard-error units. Its sign indicates the direction of the difference: a positive value means \(\bar{x}>\mu_0\), while a negative value means \(\bar{x}<\mu_0\).
Before using a test result, its assumptions and conditions must be reasonable. The observations should come from a random or otherwise appropriately selected sample, and should be independent. For a small sample, the population distribution should be approximately normal, with no severe outliers; for a larger sample, the t procedure is generally more robust to departures from normality. A report does not need to repeat a full diagnostic analysis, but the analyst should check these conditions and disclose important limitations.
A Reliable Reporting Sequence
The following sequence keeps the numerical result and the scientific conclusion aligned. It is a writing structure, not a substitute for checking the design and assumptions.
State what population mean is under investigation, its units, and whether the alternative claims a greater value, a smaller value, or any difference from the null value.
Report \(\bar{x}\), then the t statistic, degrees of freedom, and p-value. Keep the estimate in the original units; the t statistic and p-value are unitless.
Relate the p-value to the null model and say whether the results provide evidence for the stated alternative. If using a significance level, say whether the result is statistically significant at that level.
Do not claim that a test proves the null or alternative, and do not claim causation from a study that only observes an association.
The t statistic makes the relationship between the estimate and the null value explicit. Rearranging its definition gives
Thus, the observed difference from the null value equals the test statistic multiplied by the estimated standard error. A report should not replace the estimate with \(t\): “the mean was 2.4 minutes above the target” communicates a difference in real units, while “\(t=2.40\)” communicates a standardized difference. Both can be useful, but they do not mean the same thing.
Worked Reporting Examples
Worked Example: A Two-Sided Claim About Fill Volume
Hypothetical situation. A plant’s target is a mean fill volume of 50 milliliters. An independent random sample of 16 containers has \(\bar{x}=52.4\) mL and \(s=4.0\) mL. The sample measurements show no severe outliers and are reasonably symmetric. Test the claim that the population mean differs from 50 mL, using \(\alpha=0.05\).
The parameter is the population mean fill volume, \(\mu\). The hypotheses are \(H_0:\mu=50\) mL and \(H_a:\mu\ne50\) mL. The random sampling and independence conditions are reasonable by the stated design; the small-sample distribution check is also reasonable because the observations are reasonably symmetric and have no severe outliers.
The standard error is
The test statistic and degrees of freedom are
For a two-sided alternative, the p-value includes outcomes at least as far from zero as \(2.40\) in either direction. The t distribution with 15 degrees of freedom gives \(p\approx0.030\). Since \(0.030<0.05\), the result is statistically significant at the 5% level.
Report: The sample mean fill volume was 52.4 mL. A one-sample t test found evidence that the population mean differs from the 50 mL target, \(t(15)=2.40,\ p=0.030\). The observed mean was 2.4 mL above the target.
The conclusion is about the population mean, not every container. It also does not establish that the difference is important for manufacturing decisions; that requires considering the size and consequences of the difference in context.
Worked Example: A One-Sided Claim With Weak Evidence
Hypothetical situation. A battery maker claims that mean operating life exceeds 500 hours. In an independent random sample of 25 batteries, the mean is 498 hours and the sample standard deviation is 10 hours. The measurements have no severe outliers or pronounced skew. Assess the maker’s greater-than claim at \(\alpha=0.05\).
Let \(\mu\) be the population mean battery life in hours. The hypotheses are \(H_0:\mu=500\) hours and \(H_a:\mu>500\) hours. Independent random selection supports the sampling conditions, and the distribution check is acceptable for this sample. The standard error is
Then
Because the alternative is greater than 500, the p-value is the upper-tail probability \(P(T_{24}\ge-1.00)\), not the probability in the lower tail. It is approximately \(0.836\). This large p-value means that the observed statistic does not provide evidence for the claim that mean life exceeds 500 hours. The sample mean is actually below 500 hours, but that alone does not establish that the population mean is below 500.
Report: The sample mean battery life was 498 hours. A one-sample t test did not provide evidence that the population mean exceeds 500 hours, \(t(24)=-1.00,\ p=0.836\). The result is not statistically significant at the 5% level.
This conclusion does not prove that the mean is exactly 500 hours or that the maker’s claim is false. It says that this sample does not provide support for the greater-than claim under the specified test.
Worked Example: Evidence for a Greater-Than Claim
Hypothetical situation. A laboratory checks whether the mean output of a process exceeds its 100 milliliter target. An independent random sample of 12 runs has \(\bar{x}=104\) mL and \(s=6\) mL. The observations are approximately symmetric with no severe outliers. Test \(H_0:\mu=100\) mL against \(H_a:\mu>100\) mL at \(\alpha=0.05\).
The random sampling and independence conditions are reasonable as described. Because \(n=12\) is small, the approximate symmetry and absence of severe outliers are important support for using the one-sample t procedure. The estimated standard error is
The observed statistic is
The alternative is one-sided and points to values above 100 mL, so the p-value is the upper-tail probability for \(t=2.309\) with 11 degrees of freedom: \(p\approx0.021\). Since \(0.021<0.05\), there is statistically significant evidence for a population mean above the target.
Report: The sample mean output was 104 mL. A one-sample t test provided evidence that the population mean output exceeds the 100 mL target, \(t(11)=2.31,\ p=0.021\). The estimated mean was 4 mL above the target.
The p-value is not the probability that the process mean is above 100 mL. It is calculated under the null hypothesis of a 100 mL population mean. The estimate of 104 mL gives the observed difference in the original units; the test quantifies how inconsistent that result is with the null model.
How to Word the Conclusion Carefully
A well-formed conclusion links the p-value to the alternative hypothesis and names the population quantity in context. If the p-value is small relative to a prespecified \(\alpha\), the data provide evidence against \(H_0\) and in favor of \(H_a\). If it is not small, the data do not provide sufficient evidence for \(H_a\). Neither outcome establishes that a hypothesis is certainly true.
The direction of the alternative matters. For \(H_a:\mu>\mu_0\), evidence is described as evidence that the population mean is greater than the null value. For \(H_a:\mu<\mu_0\), describe evidence for a smaller mean. For \(H_a:\mu\ne\mu_0\), the test considers deviations in either direction; the sample mean tells the reader which direction the observed estimate takes.
A p-value is computed assuming the null hypothesis and the test model. It is the probability of a test statistic at least as extreme as the observed one, using the tail or tails specified by \(H_a\). It is not the probability that \(H_0\) is true, the probability that the result occurred “by chance,” or the size of the effect. Report the p-value with enough precision to be useful; avoid reporting it as zero. If a software output gives a very small value, a form such as \(p<0.001\) is preferable to \(p=0\).
Statistical significance is also not the same as practical importance. A small difference can produce a small p-value when the estimate is precise, while a meaningful difference can remain uncertain with limited data. Reporting \(\bar{x}\), its units, and its difference from \(\mu_0\) helps readers assess the size of the observed result alongside the test evidence.
Common Reporting Mistakes
- Reporting only the p-value. A p-value without the estimate, direction, and context is incomplete. State the sample mean and what population mean was tested.
- Using the wrong tail. Choose the p-value direction from \(H_a\), not from whichever tail makes the result look more significant.
- Saying “accept the null.” A non-significant result does not demonstrate that \(H_0\) is true. Say that the data do not provide sufficient evidence for the alternative.
- Calling the p-value the probability the null is true. It is calculated assuming the null model; it is not a probability assigned to the hypothesis.
- Omitting units or overstating scope. Report the mean in its measurement units, and restrict the conclusion to the population represented by the study design.
Check Your Understanding
Use the reporting principles in this tutorial to answer the following questions.
- A one-sample test reports \(\bar{x}=18.6\) minutes, \(t(19)=2.10\), and a two-sided \(p=0.049\). What additional context is needed to write a complete conclusion?
- For \(H_a:\mu<12\), should the p-value be taken from the lower tail, upper tail, or both tails of the t distribution? Explain.
- In a report, what does the sign of the t statistic say about the sample mean relative to the null value?
- Why is “there is a 3% probability that \(H_0\) is true” not a correct interpretation of \(p=0.03\)?
- A test is not statistically significant at \(\alpha=0.05\). Give a careful contextual conclusion that does not claim the null hypothesis has been proven.