- Basic Math and Science Data Interpretation and Probability Questions and Answers Flashcards

A disease affects 1% of a population. A diagnostic test is 95% sensitive (correctly detects 95% of diseased individuals) and 90% specific (correctly identifies 90% of healthy individuals). If a randomly selected person tests positive, what is the probability they actually have the disease?

Answer: Approximately 8.8%

This requires Bayes' Theorem. P(Disease) = 0.01, P(Healthy) = 0.99. P(Positive | Disease) = 0.95; P(Positive | Healthy) = 0.10 (since specificity is 90%, false positive rate is 10%). Total P(Positive) = (0.95)(0.01) + (0.10)(0.99) = 0.0095 + 0.099 = 0.1085. P(Disease | Positive) = 0.0095 / 0.1085 ≈ 0.0876 ≈ 8.8%. The counterintuitively low result occurs because the disease is rare — most positive tests come from the large healthy population generating false positives.

A lab technician combines three solutions: 200 mL at 15% concentration, 350 mL at 8% concentration, and 150 mL at 20% concentration. What is the concentration of the resulting mixture?

Answer: 12.6%

Concentration is a weighted average by volume, not a simple arithmetic average. Total solute = (200)(0.15) + (350)(0.08) + (150)(0.20) = 30 + 28 + 30 = 88 mL. Total volume = 200 + 350 + 150 = 700 mL. Final concentration = 88 / 700 ≈ 0.1257 = 12.6%. The common mistake is averaging the three percentages: (15 + 8 + 20)/3 = 14.3%, which ignores the differing volumes.

A committee of 3 is randomly selected from a group of 5 engineers and 4 technicians. What is the probability that the committee contains at least 2 engineers?

Answer: 25/42

Total ways to choose 3 from 9 people: C(9,3) = 84. 'At least 2 engineers' means exactly 2 or exactly 3 engineers. Exactly 2 engineers: C(5,2) × C(4,1) = 10 × 4 = 40. Exactly 3 engineers: C(5,3) × C(4,0) = 10 × 1 = 10. Favorable outcomes = 40 + 10 = 50. Probability = 50/84 = 25/42 ≈ 0.595. A common error is calculating P(exactly 2 engineers) only, which gives 40/84 = 10/21.

A production line records quarter-over-quarter changes of +20%, −15%, and +10%. A manager reports the average quarterly growth as (20 − 15 + 10) / 3 = 5%. A data analyst disputes this figure. Which statement is correct?

Answer: The analyst is correct; the true average quarterly rate is approximately 3.9%, calculated using the geometric mean

For compound percentage changes, the geometric mean is the correct average, not the arithmetic mean. Multiply the growth factors: 1.20 × 0.85 × 1.10 = 1.1220. The geometric mean rate = (1.1220)^(1/3) − 1 ≈ 1.0389 − 1 = 3.89% per quarter — not 5%. The arithmetic average overstates growth because percentage losses have a larger absolute impact on a smaller base than the equivalent percentage gain. A 20% gain then a 15% loss does NOT return to the same level.

The following 10 science quiz scores are listed in order: 8, 12, 15, 19, 23, 28, 34, 41, 47, 52. What is the interquartile range (IQR)?

Answer: 26

With 10 ordered values, split into two halves of 5. Lower half: 8, 12, 15, 19, 23 → Q1 = 15 (middle value). Upper half: 28, 34, 41, 47, 52 → Q3 = 41 (middle value). IQR = Q3 − Q1 = 41 − 15 = 26. A common error is using the overall median (average of 23 and 28 = 25.5) or miscounting which values belong to each half.

A scientific instrument has a 3% probability of producing a false reading on any single measurement. Three independent measurements are taken. What is the probability that at least one of the three readings is false?

Answer: 8.73%

Use the complement rule: P(at least one false) = 1 − P(none false). P(a single reading is accurate) = 0.97. P(all three accurate) = (0.97)³ = 0.9409 × 0.97 = 0.912673. P(at least one false) = 1 − 0.912673 ≈ 0.0873 = 8.73%. The common error is simply adding 3% + 3% + 3% = 9%, which overcounts because it assumes the false-reading events are mutually exclusive, which they are not.