Try
Twenty independent measurements from a stipulated normal process have sample mean 14.2 minutes and sample standard deviation 3.6 minutes. The question is whether the population mean μ exceeds 12 minutes. Before entering summaries, state H₀:μ=12 and Hₐ:μ>12, and choose α=0.05. A second question asks for a 95% interval for μ.
The standard deviation describes variation among measurements. The standard error of their mean is s/√n. It is smaller because averaging independent measurements reduces variability. Neither number proves the measurements were independent or the process was normal; those are supplied assumptions in this worked example.
Predict
Predict whether the reference should be standard normal or Student t when the population standard deviation is unknown and the sample SD is used. Then decide whether two measurements from the same person should be entered as independent groups. The sampling unit determines the method before arithmetic begins.
A paired analysis is a one-sample analysis of within-pair differences. It needs the SD of those differences, not just the two marginal SDs. A Welch analysis instead compares two independent groups and estimates their variance contributions separately.
Experiment
Choose one-sample t, with n=20, mean 14.2, sample SD 3.6, null mean 12, greater alternative and confidence 0.95. Compute both outputs and compare them with this independently calculated reference record.
| Quantity | Calculation | Reference |
|---|
| Standard error | 3.6/√20 | about 0.80498447 minutes |
| t statistic | (14.2−12)/SE | 2.73297197 |
| Degrees of freedom | n−1 | 19 |
| Greater-tail probability | P(T₁₉≥2.73297197) | 0.00660675 |
| 95% critical t | Lower t probability 0.975 | 2.09302405 |
| Mean interval | 14.2 ± critical t·SE | [12.515148,15.884852] minutes |
At α=0.05 the test rejects μ=12 in favor of a larger mean, under the model. The interval estimates the mean duration, not the durations of 95% of individual cases. A p-value does not state the chance that μ=12 is true. The shared confidence setting describes a two-sided interval; the alternative controls the test tail separately.
Understand
For a paired worked case, 12 independent people each perform a first and second task. Define d=first−second, with mean difference −2.4 minutes and difference SD 3 minutes. Assume a normal distribution of differences. Select paired t and enter these difference summaries, not the marginal summaries. Testing mean difference zero against less gives t≈−2.771281, df=11 and p≈0.00909264. The 95% interval is approximately [−4.306109,−0.493891] minutes. The sign says the first task has a lower estimated mean duration. Swapping the definition changes the sign and the appropriate directional hypothesis.
Normality for this paired calculation concerns the differences. Raw measurements in each column need not individually be normal. Missing or mismatched pairs can change the target population and cannot be silently repaired by treating the remaining columns as independent.
Now use independent groups: A has n=18, mean 52 and SD 8; B has n=22, mean 46 and SD 10. Assume independent random observations from normal populations. Welch’s estimate is 6. Its SE is √(8²/18+10²/22), with approximate degrees of freedom
df=(8²/18+10²/22)² / [(8²/18)²/17+(10²/22)²/21]≈37.988988.
Keep the noninteger degrees of freedom in the numerical calculation. For a zero difference null and greater alternative, t≈2.108054 and p≈0.02083966. The 95% critical t is about 2.024413, giving interval [0.238059,11.761941]. Welch’s reference is approximate; it does not require assuming equal population variances. A null of equal means does not itself imply equal variances.
For small samples, severe skew or outliers can invalidate a t interpretation; summary statistics alone cannot reveal these features. Random sampling fractions and dependence also need inspection. A sample size exceeding 30 is not a universal guarantee. If sample SD is zero, the usual t statistic can become undefined; a software refusal should prompt measurement/model investigation, not an invented automatic p-value.
Apply
Return to the paired example and reverse the difference to second−first. Enter mean +2.4 with the same SD and n, choose greater, and predict the interval and p-value before calculating. Explain why this is a relabeling rather than new evidence.
Check the orientation repair
The statistic becomes +2.771281 and the greater-tail p-value remains about 0.00909264 because the alternative was reversed along with the difference. The interval is [0.493891,4.306109]. If the mean sign changed but the less alternative stayed fixed, the p-value would instead be about 0.99090736. Always state what a positive difference means. Repeated calculations of the same data are not independent replications.
For another design check, suppose only separate before and after means and SDs survived. The mean difference is recoverable for complete pairs, but its variance depends on covariance. Ask for the paired differences or their SD; no calculator setting can reconstruct information absent from the summaries.
Continue
Review paired mean inference and Welch inference for deeper derivations. In practice, identify the independent unit, define the oriented parameter, check shape/design, compute the reference probability and critical value, and finish with a conclusion limited to the actual study.
