Sampling, Simulation & Inference — Challenge
A well-chosen thousand can speak for millions, provided you admit how far off you might be.
What these problems cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Repeating — Every fraction becomes a decimal that either stops or repeats forever.
- Random sample — Every member has an equal chance of being picked — the only kind you can trust.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Factor — A number that divides evenly into another.
- Range — The set of outputs that actually occur.
- Association — A pattern linking the two variables.
- Bias — A flaw in how the sample was gathered that tilts the result in one direction.
- Margin of error — How far the sample result might sit from the truth.
Problem 1
What does a 95% confidence interval mean?
- If the sampling were repeated many times, about 95% of such intervals would contain the true value
- There is a 95% chance the sample mean is correct
- 95% of the data lies inside the interval
- The sample is 95% accurate
- 95% of samples are unbiased
Worked answer
If the sampling were repeated many times, about 95% of such intervals would contain the true value
The confidence level describes the method, not any single interval that has already been calculated.
Repeating the sampling many times and building an interval each time, about 95% of those intervals would capture the true population value.
It is not a statement about where the data lies: most of the data usually falls outside a confidence interval for the mean, because the interval describes the mean rather than the spread.
Problem 2
A random sample of 900 voters is taken. Using the estimate 1√n, what is the margin of error?
- 1.1%
- 2.2%
- 3.3%
- 4.7%
- 9%
Worked answer
3.3%
√900=30. So the margin is 130 ≈ 0.033. That is about 3.3%.
Problem 3
A poll of 400 people reports 55% support with a margin of error of 5%. Which conclusion is justified?
- Support is plausibly between 50% and 60%
- Support is exactly 55%
- Support is certainly above 55%
- Support is below half
- The sample was too small to say anything
Worked answer
Support is plausibly between 50% and 60%
The interval is 55 ± 5, which runs from 50% to 60%. The estimate is the centre of that range, not a fact on its own. Saying support is exactly 55% ignores the margin, and the range does not reach below 50%, so the last two options are both wrong.
Problem 4
Two polls are taken on the same race, one of 1000 people and one of 4000. The larger poll’s margin of error is what fraction of the smaller poll’s?
- 14
- 12
- 1√2
- 2
- 4
Worked answer
12
The margin is proportional to 1√n. The ratio is 1/√40001/√1000=√10004000=√14. =12. Four times the people buys only half the margin, which is why polls rarely exceed a couple of thousand.
Problem 5
A researcher wants to show that a new fertiliser causes taller plants. Which design supports that conclusion?
- Randomly assign plots to fertiliser or no fertiliser
- Survey farmers who already use it
- Compare fields that use it with fields that do not
- Measure one treated field over several years
- Ask agricultural experts for their opinion
Worked answer
Randomly assign plots to fertiliser or no fertiliser
Only random assignment of the treatment rules out other explanations. In every other option the plots that got fertiliser may differ in soil, climate or care, and any of those could be the real cause. Those designs can show an association; they cannot establish cause.
Problem 6
Before collecting data, a researcher chooses a one-sided test for a coin favouring heads. A coin gives 32 heads in 50 flips. A large fair-coin simulation produces 32 or more heads in about 3.2% of runs. At a 5% threshold, what follows?
- There is evidence that the coin favours heads
- The coin is certainly biased
- The coin is certainly fair
- There is no evidence at the stated threshold
- The chance the coin is fair is 3.2%
Worked answer
There is evidence that the coin favours heads
The alternative was chosen in advance: more heads than a fair coin would typically produce.
The one-sided simulated tail probability is about 3.2%, below 5%.
This is evidence against fairness in favour of heads, not proof. A two-sided test uses both tails and gives a different probability.
Problem 7
A survey is conducted by calling landline telephones between 10 a.m. and 2 p.m. on weekdays. What is the main flaw?
- The sample is biased toward people at home during the day
- The sample is too small
- The questions are leading
- The margin of error is too large
- There is no flaw
Worked answer
The sample is biased toward people at home during the day
Who can be reached decides who is in the sample: retired people and those at home in the daytime are over-represented. The tilt is in a fixed direction, so calling more numbers the same way makes the wrong answer more precise, not more accurate. Nothing in the question says anything about the sample size or the wording.
Problem 8
Candidate A polls 51% ± 3% and Candidate B polls 47% ± 3%. Which statement about these individual intervals is correct?
- They do not overlap
- They overlap from 48% to 50%, but overlap alone does not test the difference
- The true support levels must be equal
- Overlap proves the difference is insignificant
- Candidate A is certain to win
Worked answer
They overlap from 48% to 50%, but overlap alone does not test the difference
A has interval [48%,54%] and B has [44%,50%].
The overlap is [48%,50%].
To assess the difference, use a test or interval for the difference itself, accounting for related estimates from the same poll.