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Inference from Sample Statistics and Margin of Error
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Percent first, then multiply
- Start:
- The sample count and the sample size, then the population size.
- Stop:
- After you get the number for the population.
- Skip:
- Details of how the survey was done if it says random.
- Why:
- Questions 1 and 4 are the same move: divide the sample, multiply by the population. A calculator or Desmos does it in seconds.
Used in questions 1, 4
2. Plus and minus the margin
- Start:
- The sample result and the margin of error.
- Stop:
- After you write both ends of the interval.
- Skip:
- The confidence level, unless the question compares levels.
- Why:
- Questions 2, 3 and 8 all need both ends. Write low and high on your paper, then test each option.
Used in questions 2, 3, 8
3. Test options against the interval
- Start:
- The options list, with your interval ready.
- Stop:
- The first option that fits entirely inside.
- Skip:
- Options that are just outside by a small amount; they are built to tempt.
- Why:
- In question 3, 15.6 looks close but is outside 15.3. Compare numbers exactly.
Used in questions 3, 8
4. Square root for sample size
- Start:
- The sample sizes and the old margin.
- Stop:
- After you find the square root of the ratio.
- Skip:
- Straight-line shortcuts, which are always the tempting wrong answers.
- Why:
- Questions 5 and 7 are both traps for the straight-line answer. Margin ÷ k needs k² times the sample.
Used in questions 5, 7
5. Stay on the mean, not the individual
- Start:
- The conclusion options.
- Stop:
- The first option that talks about the population mean or percent with the word 'plausible'.
- Skip:
- Options with 'every', 'exactly', 'all' or 'certainly'.
- Why:
- A margin of error never describes single items (question 6). It also cannot give an exact value.
Used in question 6
6. Check the 50% line for 'majority'
- Start:
- The interval ends.
- Stop:
- When you see whether 50 is inside.
- Skip:
- The sample percent alone.
- Why:
- A sample of 54% is not enough for a majority when the margin is 5 (question 8). The whole interval must be above 50.
Used in question 8
7. Compare intervals by overlap
- Start:
- The two intervals, written as low and high numbers.
- Stop:
- When you know if they share any number.
- Skip:
- The sample sizes and the size of the dots.
- Why:
- If two intervals share a number, then equal values for the two populations are plausible. If they share nothing, the difference is real. A touching edge still counts as shared (questions 13 and 16).
Used in questions 13, 16
8. Higher confidence means a wider margin
- Start:
- The confidence levels in the table.
- Stop:
- When you have placed the missing value in order from smallest to largest.
- Skip:
- The formulas for margin of error.
- Why:
- More certainty needs a wider net: 90% < 95% < 99% for the margin. Any option smaller than the 95% margin cannot be the 99% margin (question 12).
Used in question 12
9. Work back from the margin you want
- Start:
- The old margin, the new margin and the old sample size.
- Stop:
- After you square the ratio and multiply.
- Skip:
- Straight-line shortcuts such as 400 × 5/4 = 500.
- Why:
- Margin is proportional to 1/√n, so n is multiplied by (old margin ÷ new margin)². In question 15 that is 400 × (5/4)² = 625. In question 14 the table shows the same pattern: 4 times the people, half the margin.
Used in questions 14, 15