Worked examples
- Worked examples
False-rejection counts and family error risk
Read two different summaries of a fictional ten-test experiment.
- Worked examples
Reconciling a mean interval with a two-sided test
Use the same model and uncertainty inputs in both calculations.
- Worked examples
A mean test with known population standard deviation
Reconstruct the standard error, standardized statistic and two-sided decision.
- Worked examples
A one-sided test with an opposite-sign statistic
Use the planned tail when the estimate points away from the alternative.
- Worked examples
Sample size and mean-estimation precision
Calculate the sample-size change needed to halve standard error under a fixed-variance model.
- Worked examples
A unit inclusion probability in two-stage sampling
Multiply the selected-group chance by the conditional within-group chance.
- Worked examples
Estimating after disproportionate stratum sampling
Compare population-weighted and unweighted estimates when small groups are oversampled.
- Worked examples
Enumerating a finite sampling distribution
List every possible sample and calculate the distribution of its mean.
- Worked examples
Systematic sampling and a periodic list
Compare exact uncertainty under a periodic systematic design and a simple random design.
- Worked examples
A rare-event sample mean at thirty observations
Inspect an exact distribution that a blanket normality rule would miss.
- Worked examples
Pricing a three-year annual-coupon bond payment by payment
Work through every discounted coupon and principal payment.
- Worked examples
Decomposing an uneven stream into an annuity and an extra payment
Check cash-flow additivity using two representations of the same dated stream.
- Worked examples
Calculating a collaborative engagement group’s aggregate stake
Add holdings with a matching denominator and test a stated participation condition.
- Worked examples
Building and reconciling a common-size balance sheet
Divide each selected amount by the same asset base without counting subtotals twice.
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Updating a control-weakness probability after an exception flag
Calculate both flag branches before interpreting the posterior.
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Converting covariance into correlation with consistent units
Normalize a covariance of −18 using standard deviations of 6% and 5%.
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Why paying current debt can raise the current ratio and lower the cash ratio
Trace both numerator and denominator changes in a stated cash repayment.
- Worked examples
Calculating a cash cycle with separate flow bases
Reconstruct each days measure before combining the result.
- Worked examples
How the debt definition changes a leverage ratio
Compare interest-bearing debt with total liabilities using the same equity base.
- Worked examples
Funding the discounted invoice amount
Compare cash costs over the actual gap between payment dates under a stated simple-rate model.
- Worked examples
Reconciling all five DuPont factors
Use the full precision of each factor and check the product against direct ROE.
- Worked examples
Estimating a price change with duration and convexity
Follow the signs, decimal yield change and squared curvature term.
- Worked examples
Expected credit loss from a two-outcome table
Derive PD × LGD × EAD from explicit default and non-default outcomes.
- Worked examples
An engagement free-rider incentive: a simple payoff example
Compare an investor’s share of a common benefit with the cost it bears alone.
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Calculating an engagement outcome rate with a clear denominator
Calculate resolved, escalated and ongoing shares without conflating them.
- Worked examples
Checking nearby order quantities around EOQ
Compare the objective value rather than assuming a rounded quantity is automatically optimal.
- Worked examples
A finite annuity compared with the perpetuity limit
Compare a bounded payment stream with an infinite stream under the same positive rate.
- Worked examples
A level-payment amortization model
Derive a payment from an annuity factor and separate interest from principal.
- Worked examples
Comparing current, quick and cash ratio components
Use one liability denominator and three explicitly defined numerators.
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Interpreting debt ratios when equity is negative
Identify why a negative debt-to-equity quotient is not a low-leverage signal.
- Worked examples
Expected revenue from a two-stage scenario model
Calculate conditional expectations, then apply the outer scenario probabilities.
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NPV of unequal annual cash-flow amounts
Discount an initial outflow and three unequal annual receipts.
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Converting odds against default into a default probability
Work through the event label, quotation direction and normalization.
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Comparing ordinary and due annuity values
Shift each payment one period earlier while preserving the valuation date.
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Counting an investable universe with overlapping exclusion lists
Avoid subtracting the same issuer more than once when screening a universe.
- Worked examples
Purchases versus COGS in payable days
Show why an inventory movement changes the flow used for a payables comparison.
- Worked examples
Quarterly and annual discounting of the same payment
Check that an effective annual conversion preserves the value of a dated single sum.
- Worked examples
Calculating the probability of at least one event
Subtract the overlap once when adding two event probabilities.
- Worked examples
Future value of end-quarter deposits
Use a quarterly rate, quarterly payment count and a defined final date.
- Worked examples
Reconciling assets, liabilities and retained earnings
Use the accounting equation and explicit equity-component assumptions.