An original synthetic sample of 25 independently measured outputs has mean 102 units and sample SD 5 units. The reference target is μ0=100. Is a mean difference of 2 units large relative to the estimated sampling variation? Define the population mean μ and choose the alternative before calculating a statistic.
Predict
Predict the sign of the statistic for an above-target sample mean. Then decide whether a two-sided question should use only the upper tail because the observed sign is positive. It should not: a prespecified two-sided procedure treats sufficiently large deviations in either direction as evidence.
Experiment
The estimated SE is 5/√25=1. The t statistic is (102−100)/1=2, with df=24. For a two-sided test, use the supplied reference p≈0.05694. At α=0.05, fail to reject the null. The estimate remains 2 units above target; the evidence threshold does not change the observed difference.
Now distinguish a lower-tailed, upper-tailed and two-sided question using the same statistic. Their p-values differ because their rejection regions differ. Use the reference information supplied for each, rather than inventing more precise values from a rough sketch.
Understand
The usual one-sample statistic is t=(x̄−μ0)/(s/√n). The numerator compares the sample estimate with the null mean. The denominator estimates the sample mean’s standard deviation using sample spread. Under independent normal sampling with positive variance, this statistic follows a t distribution with n−1 degrees of freedom when the null is true.
For a two-sided alternative, use the probability of a t magnitude at least as large as observed. For an upper alternative, use the upper tail beyond the signed statistic; for a lower alternative, use the lower tail. A large statistic in the opposite direction does not support the intended one-sided alternative, though it may reveal an important unexpected finding that should be reported transparently.
Check independence, a credible connection to the target population or process, and distributional suitability. Small-sample t inference is sensitive to serious skew and outliers. Large n can improve the approximation for many distributions but cannot guarantee validity under arbitrary tails or dependence. Repeated copies of one measurement do not provide independent evidence.
If s=0, the usual formula divides by zero. Do not force a finite t value or declare certainty from a collapsed sample. Investigate measurement resolution, data duplication and the model. Classical continuous t calibration does not automatically cover degenerate cases.
A p-value measures discrepancy under a null model, not the chance the null is true. Failure to reject does not establish exact equality. Rejection does not imply that the difference is useful or causal. A mean interval and contextual effect threshold help communicate magnitude and remaining uncertainty.
Under matching conditions, a conventional two-sided level-α t test and the corresponding 100(1−α)% t interval agree on whether μ0 is excluded, apart from rounding conventions at a boundary. This relationship depends on using the same model, data and two-sided procedure; it should not be generalized to unrelated approximate or one-sided calculations.
Apply
Write a full conclusion for the worked test, including the alternative and target units. Then audit a small dataset containing one extreme duration. Explain what should be investigated before reporting a t result and why deleting an inconvenient but valid value is not a statistical correction.
Continue
Next preserve pairing when measurements come from matched units. The same t machinery can apply to differences, but the correct data and sample size change.
Apply · explain · revise
Practice and reasoning
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For n=9, x̄=13, s=3 and null μ0=10, find t.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A symmetric t reference gives upper-tail area 0.012 for |t|. Find the usual two-sided p-value.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
The usual one-mean t statistic is:
If all observations are identical and s=0, the usual t formula:
A suitable sample has n=16, x̄=52, s=8. Test μ=50 versus μ>50 using supplied df=15 upper-tail P(T≥1)≈0.1666 and α=0.05.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Nine durations include one extremely large observation. Why should a small-sample t result not be reported without examining it?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
