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Lesson details
- Estimated study time
- 1 hr 30 min
Learning objectives (6)
The 31.875-day Year 1 sample mean is known. The mean for all 480 qualifying Year 1 invoices is fixed by that population but not revealed in the packet. The first is a statistic; the second is a parameter.
Watch the statistic move
The repeated-sample example uses one visible eight-value population and several four-value samples. Every sample follows the same role and arithmetic, yet the sample means differ. That variation is what an estimator faces before the sample is drawn.
Do not confuse it with a changing business process. Repeated samples from one fixed population create sampling variability. Samples from different years can also reflect real population change, definition changes, and source changes. Juniper's −2.5-day difference does not identify which mechanism dominates.
Standard deviation and standard error answer different questions
For Year 1, sample standard deviation is about 5.85235 days. That describes spread among the 16 invoice outcomes. The simple mean standard error is:
[ SE(\bar{x})=5.852349955/\sqrt{16} =1.463087489\text{ days}. ]
The standard error estimates spread among sample means under the stated design and conditions. It is smaller because a mean aggregates several observations. It is not a description of a typical invoice.
More data do not solve every uncertainty
If the sample standard deviation stayed similar, increasing the simple sample from 16 to 64 invoices would halve the formula's standard error: sample size is multiplied by four, its square root is multiplied by two, and division by that square root halves the result. It would not:
- pull missing invoices into the frame;
- correct a wrong issue date;
- change the target from settled invoices to all issued invoices;
- remove within-customer dependence; or
- prove future settlement behavior is stable.
Depth checkpoint: choose only after inspecting the design
A finite-population correction may matter because 16 invoices were selected without replacement from a finite frame; the relevant facts are the actual sampling fraction and whether the stated frame is the target. A clustered method may matter if several invoices share customers or another dependence source; the relevant facts include cluster keys, cluster sizes, selection stages, and within-cluster similarity. This module asks learners to identify those facts and stop. It does not teach either adjusted variance calculation.
Exit check
Complete the standard-error diagnostic. Then write one sentence with both 5.85235 and 1.46309, attaching each number to the object whose spread it describes.