ACT Math study guide · Chapter 36
Statistical Inference
A sample statistic, like a poll's reported percentage, estimates a true population value but is almost never exactly equal to it. A margin of error expresses how far that estimate might reasonably be from the truth. The plausible interval for the population value is the sample estimate plus or minus the margin of error: estimate − margin of error to estimate + margin of error.
Use the margin of error and confidence conditions stated or implied by the problem; the interval is being interpreted under that survey or sampling procedure.
Checking whether a claimed value is plausible means checking whether it falls inside the computed interval, nothing more. A value just outside the interval isn't automatically ruled out in a strict mathematical sense, but for this kind of question, “plausible” means falling within the stated interval; a value outside it is treated as not supported by the survey.
Margin of error shrinks as sample size grows: a larger random sample reduces sampling variability, producing a more precise estimate and a narrower interval, all else being equal. Sample size and representativeness are not the same thing. A bigger sample does not automatically fix bias or make a sample representative; representativeness depends on how the sample was selected, and a very large but poorly selected sample can still give a confidently wrong picture of the population. Different random samples from the same population naturally produce different sample statistics, purely from which individuals happened to be selected; this is sampling variability, not a mistake or a flaw in the method. A simulation that repeatedly draws random samples from a population with a known true proportion will show sample proportions clustering around that true value, with some natural spread, illustrating sampling variability directly.
Where this appears as a model: a poll result reported with a margin of error. A poll of 500 voters showing 52% support with a margin of error of ±4% means the true population support plausibly falls between 48% and 56%, not that 52% is the exact value.
▸ Building a plausible interval
Worked Example A poll of 600 people finds that 41% support a ballot measure, with a margin of error of ±4%. Find the interval of plausible values, and determine whether 44% support is plausible.
Compute the interval, then check whether the claimed value falls inside it.
41% − 4% = 37%; 41% + 4% = 45%
The interval is 37% to 45%
Since 37% ≤ 44% ≤ 45%, 44% support is plausible
▸ Sample size and margin of error
When margin of error is described as inversely proportional to the square root of sample size, multiplying the sample size by a given factor divides the margin of error by the square root of that same factor. Quadrupling a sample size cuts the margin of error in half; increasing a sample size ninefold cuts the margin of error to a third of its original size.
▸ Testing a population claim with simulation
A simulation can also test whether a population claim is consistent with observed data, not just estimate a range of plausible values. Start by assuming the claim is true, then simulate many random samples under that assumption and record how often a result at least as extreme as the observed result occurs.
If results at least as extreme as the observed result occur only rarely under the claim, the observed result is evidence against the claim, and it can be rejected under the decision rule given in the problem. If the observed result is not unusual under the claimed value, the correct conclusion is to fail to reject the claim. Failing to reject does not prove the claim is true; it means the sample did not provide strong enough evidence against it.
Worked Example A claim states that 40% of a population favors a proposal. In 1,000 simulated samples built from that claim, a result at least as extreme as the observed sample result occurred only 12 times. What conclusion is best supported?
A result this rare under the claim is treated as evidence against it, not as confirmation.
12 out of 1,000 is a rare outcome under the claim
The claim should be rejected under a reasonable decision rule; this does not prove the claim false, only that the observed data is not well explained by it
Worked Example A poll of 250 people finds that 46% support a measure, with a margin of error of ±5%. Is 53% support plausible?
Recognize. This is a plausibility check. Compute the interval from the estimate and margin of error, then see whether the claimed value falls inside it.
Set up.
46% − 5% = 41%; 46% + 5% = 51%
Solve.
The plausible interval is 41% to 51%
Check. 53% is greater than the upper bound of 51%, so it falls outside the interval.
Best route. Direct arithmetic. No formula lookup is needed since the margin of error is already given.
Common error. Concluding 53% is “close enough” since it's only two points past the boundary, instead of applying the stated interval as a hard cutoff.
▸ Practice
Question 1
- AA. 48% to 54%
- BB. 50% to 58%
- CC. 51% to 57%
- DD. 57% to 63%
Question 2
- FF. No, because 60% is less than 62%
- GG. Yes, 60% is within the interval
- HH. It is impossible to say without more data
- JJ. No, 60% is outside the interval
Question 3
- AA. It is impossible to say without more data
- BB. Yes, because 44% is close to 50%
- CC. No, the interval does not reach 50%
- DD. Yes, the interval extends past 50%
Question 4
- FF. It decreases
- GG. It stays the same
- HH. It increases
- JJ. It becomes zero
Question 5
- AA. Survey A
- BB. They are equally precise
- CC. Cannot be determined
- DD. Survey B
Question 6
- FF. 46% to 50%
- GG. 42% to 54%
- HH. 45% to 51%
- JJ. 48% to 51%
Question 7
- AA. No, 50% is outside the interval
- BB. Yes, 50% is within the interval
- CC. It is impossible to say without more data
- DD. No, because 50% is greater than 48%
Question 8
- FF. Measurement error in the survey instrument
- GG. Sampling variability from which individuals happen to be selected
- HH. Dishonest responses from participants
- JJ. A flaw in the researcher's calculations
Question 9
- AA. Proportions spread evenly from 0% to 100%
- BB. Proportions clustering around 50%, with some variability
- CC. Every sample showing exactly 40%
- DD. Proportions clustering around 40%, with some variability
Question 10
- FF. No, because the population itself changes between polls
- GG. Yes, repeating the same method guarantees the same result
- HH. No, though it should be close, due to sampling variability
- JJ. Yes, because the margin of error eliminates variability
Question 11
- AA. 68% to 72%
- BB. 65% to 75%
- CC. 60% to 80%
- DD. 67% to 73%
Question 12
- FF. It is cut in half
- GG. It stays the same
- HH. It is cut to a quarter
- JJ. It doubles
Question 13
- AA. Yes, because 32% is close to 28%
- BB. No, 32% is outside the interval
- CC. Yes, 32% is within the interval
- DD. It is impossible to say without more data
Question 14
- FF. Decreasing the sample size
- GG. Increasing the sample size
- HH. Asking more opinion-based questions
- JJ. Surveying the same people twice
Question 15
- AA. The survey of 2,000 people
- BB. Neither can be close without a census
- CC. The survey of 20 people
- DD. Both are equally likely to be close
Question 16
- FF. 57% to 59%
- GG. 56% to 60%
- HH. 52% to 64%
- JJ. 55% to 61%
Question 17
- AA. No, 50% is outside the interval
- BB. Yes, because 50% is a common benchmark
- CC. It is impossible to say without more data
- DD. Yes, 50% is within the interval
Question 18
- FF. Sampling variability due to random chance in selection
- GG. Bias in the underlying population
- HH. An error in the stated true proportion
- JJ. A flaw in how the simulation was designed
Question 19
- AA. No, 68% is outside the interval
- BB. Yes, because 68% is close to 62%
- CC. It is impossible to say without more data
- DD. Yes, 68% is within the interval
Question 20
- FF. 50
- GG. 200
- HH. 500
- JJ. 2,000
▸ Answers and Explanations
Show the answers and explanations
1: C. The plausible interval is the sample estimate plus or minus the margin of error: 54% − 3% = 51%, and 54% + 3% = 57%. Choice A subtracts the margin of error twice, computing the interval as if the estimate were 51% instead of 54%.
2: G. The plausible interval is 62% − 4% = 58% to 62% + 4% = 66%. Since 58% ≤ 60% ≤ 66%, 60% is a plausible value. Choice H assumes any value below the sample estimate is automatically implausible, without checking it against the actual interval.
3: C. The plausible interval is 44% − 5% = 39% to 44% + 5% = 49%. Since the interval's upper bound (49%) does not reach 50%, a majority is not plausible. Choice D miscalculates the upper bound of the interval, overestimating how far it extends.
4: F. A larger sample size gives a more precise estimate, which shrinks the margin of error for the same confidence level. Choice H reverses the relationship between sample size and margin of error.
5: D. A smaller margin of error indicates a more precise estimate. Survey B's margin of error (±2.5%) is smaller than Survey A's (±5%), so Survey B is more precise. Choice A reports the survey with the larger margin of error as more precise, reversing the relationship between margin of error and precision.
6: H. The plausible interval is 48% − 3% = 45% to 48% + 3% = 51%. Choice J only adjusts the interval on one side, using the sample estimate itself as the lower bound instead of subtracting the margin of error.
7: B. The plausible interval from the previous question is 45% to 51%. Since 45% ≤ 50% ≤ 51%, 50% is a plausible value. Choice D incorrectly concludes that a value greater than the sample estimate must fall outside the interval, without checking it against the actual bounds.
8: G. Each random sample includes a different, randomly chosen subset of individuals, so the sample proportion naturally shifts from one sample to the next; this is sampling variability, not a flaw in the survey. Choice F attributes the variation to a flaw in the measurement process instead of the expected variability from random sampling.
9: D. Since the population's true approval rate is 40%, repeated random samples should produce sample proportions that cluster around 40%, with some natural variability from sample to sample. Choice A describes a uniform spread across all possible values, which does not reflect how random samples behave around a fixed true proportion.
10: H. Because each poll draws a new random sample, sampling variability means the second poll's result will likely be close to 35% but not necessarily exactly 35%. Choice F incorrectly attributes the expected difference to the population changing, instead of to the randomness of which individuals are sampled.
11: B. The plausible interval is 70% − 5% = 65% to 70% + 5% = 75%. Choice A narrows the interval instead of using the full stated margin of error of ±5%.
12: F. If the margin of error is inversely proportional to the square root of the sample size, and the sample size quadruples, the margin of error is divided by √4 = 2, so it is cut in half. Choice J assumes the margin of error scales the same direction as the sample size instead of being inversely related to its square root.
13: B. The plausible interval is 28% − 3.5% = 24.5% to 28% + 3.5% = 31.5%. Since 32% is greater than the upper bound of 31.5%, it is not a plausible value. Choice C assumes any value near the sample estimate must be plausible, without checking it against the actual computed interval.
14: G. A larger sample size reduces sampling variability, which directly narrows the margin of error. Choice F reverses the relationship, describing a smaller sample size as the way to reduce the margin of error.
15: A. A larger sample size produces a smaller margin of error and a more precise estimate, so the sample proportion from the 2,000-person survey is more likely to be close to the true population proportion. Choice C reverses the relationship, favoring the much smaller and less precise sample.
16: J. The plausible interval is 58% − 3% = 55% to 58% + 3% = 61%. Choice F narrows the interval on both sides instead of using the full stated margin of error of ±3%.
17: A. The interval from the previous question is 55% to 61%. Since 50% is less than the lower bound of 55%, it is not a plausible value. Choice D assumes 50% is automatically plausible without checking it against the computed interval.
18: F. The spread of sample proportions around the true value of 0.3 reflects the natural variability introduced by which individuals happen to be included in each random sample. Choice J assumes the observed spread indicates a design flaw instead of expected sampling variability.
19: A. The plausible interval is 62% − 4.9% = 57.1% to 62% + 4.9% = 66.9%. Since 68% is greater than the upper bound of 66.9%, the claim is not consistent with the survey. Choice D assumes a value close to the sample estimate must be within the interval, without checking the actual computed bounds.
20: J. A larger sample size produces a smaller margin of error, so the largest listed sample size, 2,000, would most likely produce the smallest margin of error. Choice F selects the smallest sample size instead of the largest, reversing the relationship between sample size and margin of error.