Match the strength of a conclusion to the population, measurements and uncertainty behind it.
GMAT35 min study guide8 indexed sections
BY THE END, YOU CAN
Explain population versus sample
Explain sampling limitations
Explain expected versus guaranteed outcomes
Explain uncertainty and error intervals
Explain measurement versus inference
Explain misleading scales
Explain aggregated versus subgroup data
Explain absolute versus relative change
Explain relevance and source consistency
BEFORE YOU START
Percentages, means and weighted rates
Basic distinction between evidence and assumption
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GMAT study through an editorial skill map. It does not reproduce official questions, predict exam content or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
Original GMAT-style worked examples and practice
All passages, examples and exercises here are original. No official GMAC questions, choices or explanations are reproduced. This independent preparation resource is not affiliated with or endorsed by GMAC. GMAT is a trademark of the Graduate Management Admission Council.
01 / UNDERSTAND & APPLY
Identify the population before generalizing
A population is the complete group a claim concerns; a sample is the observed subset. A statistic computed exactly for a sample is not thereby exact for the population. Specify who could enter the sample, who actually did and whether the question asks about respondents or everyone.
All scenarios in this lesson are fictional teaching constructions, not reports of research findings. We use them to test logical support, not to assert how real organizations behave.
ORIGINAL WORKED EXAMPLE
Exact sample fact, uncertain population claim
A company has 1,000 employees. A voluntary survey receives 100 replies, of which 70 favor a new schedule. Select Supported or Not supported for each of two claims: (i) 70% of respondents favor it; (ii) exactly 70% of all employees favor it.
Respondent proportion is exactly 70/100=70%, so (i) follows.
The preferences of 900 nonrespondents are unknown, and volunteering may relate to preference.
The observed proportion does not identify the exact employee-wide proportion.
Answer (i) Supported; (ii) Not supported
ORIGINAL PRACTICE · 1
In the voluntary-survey scenario, how many of all 1,000 employees are known to favor the schedule, and what is the largest possible favorable total consistent with the replies? Give both counts.
Reveal answer & reasoning
Known: 70; largest possible: 970
The 70 favorable respondents are known. If all 900 nonrespondents also favor it, favorable total=70+900=970; the 30 opposed respondents cannot be counted as favorable.
02 / UNDERSTAND & APPLY
Sampling limitations are specific, not generic doubts
Coverage error excludes parts of the target population; nonresponse leaves selected people unobserved; self-selection can make participation related to the measured attribute. Larger samples reduce some random variability under suitable designs but do not automatically remove these biases.
Random selection supports inference under the sampling design, but it does not guarantee a sample proportion equals the population proportion. Separate uncertainty due to sampling from a concrete flaw in coverage or measurement.
ORIGINAL WORKED EXAMPLE
A larger sample can retain the same bias
A transit survey interviews 500 weekday morning riders to estimate the preferences of all riders, including evenings and weekends. Would doubling only the weekday morning sample necessarily resolve the coverage problem?
The target includes evening and weekend riders.
Increasing interviews within the same morning-only frame still omits those groups.
More observations may improve precision for morning riders without fixing representation of the full target.
Answer No
ORIGINAL PRACTICE · 1
A simple random sample of 80 from 800 records has 10 errors. Does random sampling guarantee exactly 100 errors in all 800? Select Yes or No.
Reveal answer & reasoning
No
10/80=12.5% suggests an estimate of 100 under a suitable inference framework, but random sampling does not force exact proportionality. Unsampled records remain unknown.
03 / UNDERSTAND & APPLY
Expected counts are not promised outcomes
An expected value is a probability-weighted average across possible outcomes. If n indicators each have success probability p, expected total successes=np by linearity of expectation. Independence is not needed for that expectation, although it matters for many distribution calculations.
A realized count is an integer and can differ from its expectation. A noninteger expected count is meaningful because it describes an average over repetitions, not a fractional observed event.
ORIGINAL WORKED EXAMPLE
Expectation versus guarantee
Twenty invitations each have a 0.30 probability of response. What is the expected response count, and must the actual count equal it?
Represent each response by an indicator with expected value 0.30.
Sum the 20 expectations:20×0.30=6.
The actual response count may differ; no guarantee of exactly 6 follows.
Answer Expected: 6; guaranteed exactly 6: No
ORIGINAL PRACTICE · 1
Nine items each have a 0.20 chance of a specified outcome. Can the expected number be 1.8 even though actual counts are whole numbers?
Reveal answer & reasoning
Yes
Expected count=9×0.20=1.8. An expectation is an average over possible outcomes; every realized count is still an integer from 0 to 9.
04 / UNDERSTAND & APPLY
Read what an uncertainty interval actually means
An interval may describe a hard error bound, a range of possible inputs, a confidence procedure or a prediction interval. These have different interpretations. Do not treat every “±” display as a guaranteed containment statement or as a probability distribution.
For the next exercises, bounds are explicitly hard bounds: the true measurement lies within the stated closed interval. Under that assumption, nonoverlapping ordered intervals prove an ordering. Overlap may leave the order unresolved.
ORIGINAL WORKED EXAMPLE
Use the worst endpoints
Device A reads 10 with a hard absolute error bound 0.5; B reads 12 with the same bound. Is B’s true value certainly greater, and what is its minimum possible advantage?
A true value lies in [9.5,10.5]; B in [11.5,12.5].
Smallest B minus largest A=11.5−10.5=1.0.
Every permitted pair has B>A by at least 1.0 unit.
Answer Yes; minimum advantage 1.0 unit
ORIGINAL PRACTICE · 1
A has hard interval [9,11] and B [10,12]. Is B certainly greater than A?
Reveal answer & reasoning
No
A=11,B=10 is legal and makes B smaller; A=9,B=12 makes B larger. Overlap alone does not fix the ordering.
ORIGINAL PRACTICE · 2
An estimate is 40±3 under an explicitly guaranteed absolute error bound. Is a true value of 44 possible? Select Yes or No.
Reveal answer & reasoning
No
The allowed closed interval is [37,43]. A true value of 44 violates the stated hard bound. This reasoning would need revisiting if ±3 represented a non-guaranteed statistical interval instead.
05 / UNDERSTAND & APPLY
Separate what was observed from what was inferred
A measurement can directly record a count, duration or reading. An inference may classify, predict or explain that observation using a model. A measured association does not automatically establish a causal effect, because assignment, confounders and reverse causation may remain unresolved.
Measurement definitions also matter: completed orders differ from placed orders, gross sales from net sales, and average service time from the longest wait. A source can be numerically accurate yet irrelevant to the target if it measures a different concept.
ORIGINAL PRACTICE · 1
A table shows teams using tool Z had higher output than other teams. Does it establish that tool Z caused the difference?
Reveal answer & reasoning
No
The table establishes an association under its measurements. Teams might differ in staffing, workload or selection into tool use; no causal design or sufficient controls are specified.
06 / UNDERSTAND & APPLY
A scale can exaggerate appearance without changing values
The original scale comparison displays values 98 and 100 twice. In the full-scale panel bars start at zero; in the truncated panel they start at 96. The second drawing makes the visible segments appear in a 2:1 ratio, but the values are 100/98≈1.0204, not 2.
Read numerical values and axis origins before interpreting visual ratios. A truncated line-chart axis can reveal small changes legitimately when labeled; the problem is treating the truncated visual length as a ratio of complete values.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
Original values. Always read the axis origin before comparing the lengths of bars.
ORIGINAL WORKED EXAMPLE
Absolute and relative change
Scale figure: What is the change from 98 to 100 in absolute units and as a percentage of the initial value?
Absolute change=100−98=2 units.
Relative change=2/98×100%≈2.04%.
The truncated panel’s doubled visible segment does not mean the measured value doubled.
Answer 2 units; approximately 2.04%
ORIGINAL PRACTICE · 1
A rate rises from 4% to 5%. State the percentage-point increase and the relative percent increase.
Reveal answer & reasoning
1 percentage point; 25% relative increase
Subtract rates:5%−4%=1 percentage point. Relative increase divides by the original rate:1/4=25%.
07 / UNDERSTAND & APPLY
Aggregation can reverse a subgroup comparison
Combining groups changes weights. A category with stronger performance within every subgroup can still have a weaker overall rate if it has more observations in a difficult subgroup. Compute both the numerators and denominators rather than averaging subgroup percentages.
The original success table has easy tasks: A 9/10=90%, B 72/90=80%; hard tasks: A 36/90=40%, B 3/10=30%. These figures are constructed to illustrate a weighting reversal, not a study of real methods.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
Constructed fictional data. A is higher within each group; B has more easy tasks.
ORIGINAL WORKED EXAMPLE
A reversal produced by group composition
Subgroup figure: Which method has a higher success rate in each difficulty group, and which has the higher combined rate?
Easy:90%>80% and hard:40%>30%, so A is higher in each group.
A total success=(9+36)/(10+90)=45/100=45%.
B total success=(72+3)/(90+10)=75/100=75%. B has more easy tasks, so its overall rate is higher.
Answer Within both groups: A; combined: B
ORIGINAL PRACTICE · 1
Subgroup figure: If each method were summarized using equal weights on its easy and hard rates, what rates would result for A and B?
Reveal answer & reasoning
A: 65%; B: 55%
A=(90%+40%)/2=65%; B=(80%+30%)/2=55%. Equal weighting answers a standardized comparison, not the actual aggregate rate in the table.
08 / UNDERSTAND & APPLY
Check source relevance before reconciling claims
A report of “100 sales” and one of “98 fulfilled orders” need not conflict if cancellations explain a difference in definitions, but do not invent those cancellations as established facts. First request or locate the measurement definitions.
A disciplined conclusion names its scope: in this sample, under this model, within these hard bounds, or for this subgroup. Such qualifiers are part of an accurate answer rather than signs of indecision.
KEEP THE BIG IDEAS
Quick revision
Sample facts do not automatically generalize exactly to a population.
Expected is not guaranteed; a range is not a probability distribution.
Inspect definitions and axis origins before comparing.
Aggregate and subgroup rates use different weights and can reverse.
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