Command of Evidence (Quantitative) Flashcards

A researcher presents the following data: In 2010, City A had 45,000 residents and a crime rate of 12 per 1,000 residents. By 2020, the population grew to 90,000 and the crime rate dropped to 7 per 1,000 residents. A student claims: 'The total number of crimes in City A increased between 2010 and 2020.' Which of the following correctly evaluates this claim using the data?

Answer: The claim is correct because 90,000 × 0.007 = 630 crimes, compared to 45,000 × 0.012 = 540 crimes in 2010.

To evaluate a claim about total crimes, you must multiply the rate by the population. In 2010: 45,000 × (12/1,000) = 540 crimes. In 2020: 90,000 × (7/1,000) = 630 crimes. The total number of crimes did increase (540 → 630), even though the rate per 1,000 decreased. The student's claim is therefore correct, and option B performs the necessary calculation accurately.

The table below shows survey results from 400 college students asked about their weekly study hours and GPA range: | Study Hours | GPA 20 hrs | 20 | 110 | A classmate concludes: 'Students who study more than 20 hours per week are more than twice as likely to have a GPA ≥ 3.0 compared to students who study fewer than 10 hours.' Which best evaluates this conclusion?

Answer: The conclusion is supported: within the >20 hrs group, 110/(110+20) ≈ 84.6%, and within the <10 hrs group, 40/(40+80) ≈ 33.3%, and 84.6% is more than twice 33.3%.

A likelihood comparison requires within-group proportions, not raw counts. Among students studying >20 hrs: 110/130 ≈ 84.6% have GPA ≥ 3.0. Among students studying 20 hrs group is indeed more than twice as likely to have GPA ≥ 3.0. Option A incorrectly compares raw counts across groups with different sizes. Option B makes a calculation error calling it a 4.1:1 ratio.

A graph shows that a city's average monthly temperature (°F) and monthly ice cream sales (in thousands of dollars) are strongly positively correlated (r = 0.91) over 36 months. A journalist writes: 'This data proves that hot weather causes people to buy more ice cream.' Which response most precisely identifies the flaw in using this quantitative evidence to support the claim?

Answer: The flaw is that a strong correlation coefficient, regardless of its value, cannot by itself establish a causal direction — a third variable or reverse causation could explain the pattern.

Correlation, no matter how strong (even r = 0.99), does not establish causation. The journalist's error is conflating correlation with causation. A third variable (e.g., summer season driving both temperature and outdoor activity) or even reverse causation could theoretically produce the observed pattern. Option A is wrong because r = 0.91 is actually very strong. Option B is wrong because 36 months is a reasonable sample. Option D is wrong because correlating continuous variables is statistically valid.

A study reports: 'Among 1,200 participants, those who took Supplement X showed a 25% reduction in reported fatigue.' The study also notes in a footnote that it was funded by the manufacturer of Supplement X, used a self-reported fatigue scale, and had no placebo control group. A student wants to use this data to support the claim that Supplement X is effective. Which quantitative limitation most directly undermines the 25% figure as evidence for effectiveness?

Answer: Without a control group, there is no baseline to determine whether the 25% reduction exceeds what would occur naturally or from a placebo effect, making the figure uninterpretable as evidence of effectiveness.

The most direct quantitative flaw is the absence of a control group. Without knowing how much fatigue naturally decreases over the same period (or due to placebo effect), the 25% reduction has no comparative baseline — it cannot be attributed to the supplement. Option A assumes fraud without evidence. Option C is incorrect; 1,200 participants is a substantial sample. Option D is wrong; self-reported scale scores can be compared as percentages of change.

The following two data points are presented in a passage: • Statistic 1: 'Country A spends $12,000 per student annually on education, while Country B spends $4,000.' • Statistic 2: 'Country B's students outperform Country A's on international math assessments by an average of 18 points.' An author concludes: 'Spending more money per student does not improve math outcomes.' Which statement best describes how the quantitative evidence relates to this conclusion?

Answer: The evidence is consistent with the conclusion but does not fully support it, since two countries cannot establish a general trend and unmeasured variables (class size, curriculum, teacher quality) could explain the difference.

Two data points (n=2 countries) can be consistent with a conclusion but cannot establish a general rule about spending and outcomes. Countless confounding variables — curriculum differences, teacher preparation, cultural attitudes toward education, socioeconomic conditions — could explain why Country B outperforms despite lower spending. The evidence doesn't contradict spending effectiveness in general; it's simply insufficient to generalize. Option D contains a valid point but frames it as a condition for support rather than an inherent limitation of the evidence.

A table in a passage shows median household income by education level for three years: 2000, 2010, and 2020. In each year, households with a bachelor's degree earn roughly $25,000–$30,000 more than those with only a high school diploma. However, the passage also notes that the percentage of adults with bachelor's degrees rose from 26% in 2000 to 38% in 2020. A student argues: 'The income premium for a bachelor's degree has remained stable, so education continues to provide the same financial benefit over time.' What is the most sophisticated quantitative objection to this argument?

Answer: As more people earn bachelor's degrees, the credential may be subject to credential inflation — the stable nominal premium could mask a declining relative advantage if overall wages rose faster than the premium, or if the marginal degree-earner has lower earning potential than earlier cohorts.

This is the most nuanced objection. When bachelor's degree attainment rises from 26% to 38%, the population earning those degrees changes — less-selective institutions and weaker labor-market matches become more common. A nominally stable premium ($25–30K) could mask several issues: (1) if it isn't adjusted for inflation (option C raises this, but it's a secondary concern, not the most sophisticated), (2) if the composition of degree-earners has shifted such that the 'average' graduate today has lower earning potential. More importantly, the 'premium' being stable in dollar terms while the comparison group (high school diploma holders) also shifts means the metric may not capture the true relative value. Option C is a valid but more mechanical objection; the credential-inflation argument in B is the deeper, more sophisticated quantitative challenge to interpreting the trend.