Mathematics Advanced case studies

Mathematics Advanced · Put your knowledge to work in context.

This independent study page provides practice for Mathematics Advanced. Its editorial learning areas include Algebra, equations and functions; Differential and integral calculus; Analytic geometry and trigonometry; Vectors and linear structure; Sequences and series; Probability and statistics; Mathematical modeling; Proofs and reasoning. The questions develop subject knowledge and application through worked reasoning, distinctions between competing answers, and practical scenarios. Study the relevant guide before attempting a mixed set, then explain the evidence or calculation that supports each answer. Topics on this exam: Algebra, equations and functions, Differential and integral calculus, Analytic geometry and trigonometry, Vectors and linear structure, Sequences and series, Probability and statistics, Mathematical modeling, and Proofs and reasoning. The Mathematics Advanced case studies are 4 cases and about 2,196 words.

Strain-Gauge Calibration: Least Squares After a Suspect Reading

A calibration technician records a strain gauge's output y (mV) at five known loads x (kN): (1, 2.10), (2, 3.95), (3, 6.10), (4, 7.85), (5, 10.20). The intended model is y = a + bx. During the run, the technician notices that the fifth reading was taken while a connector was loose and suspects a constant +1.00 mV offset on that reading only. The lab's quality procedure says a slope change of more than 8% after correcting a suspect point must trigger a full recalibration, not just a note in the log.

621 words, about 3 min 2

Screening Test: PPV, Repeat Testing, and a Limited Confirmation Budget

A regional clinic screens 10,000 adults, of whom 2,000 are in a high-risk subgroup with family history. The general-population prevalence is 0.80%; in the high-risk subgroup it is 6.0%. The screening test has sensitivity 92% and specificity 96%. The clinic can run a second, independent confirmatory test that costs 60 currency units per patient; treatment started on a false positive causes 250 units of avoidable harm, while a true case missed because treatment is delayed costs 400 units.

661 words, about 3 min 3

Auditing a Raffle's Expected Payout and Fairness Claim

A student council runs a raffle to fund a trip. Each ticket costs 5. There are 400 tickets sold. The prizes are: one grand prize of 300, two second prizes of 100 each, and ten third prizes of 20 each.

448 words, about 2 min 4

Interpreting a Confidence Interval from a Small Sample

Assume independent packet masses from an approximately normal population. A student measures the mass of 9 randomly selected packets of seeds from a production line and obtains a mean of 50.2 g with a sample standard deviation of 1.5 g. The student constructs a 95% confidence interval for the population mean using the normal critical value 1.96 and reports 50.2 ± 0.98, i.e., (49.22, 51.18). A classmate says the interval is wrong because the sample is small and the population standard deviation is unknown.

466 words, about 2 min