Sampling, Simulation & Inference — Practice
A well-chosen thousand can speak for millions, provided you admit how far off you might be.
What these questions 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
Which study can show that a treatment causes an effect?
- A randomised experiment
- A survey
- An observational study
- A case study
Worked answer
A randomised experiment
Random assignment rules out other explanations
Problem 2
A poll is taken only outside a gym. What is the problem?
- The sample is biased
- The sample is too large
- The margin of error is too small
- Nothing
Worked answer
The sample is biased
Gym-goers are not typical of everyone
The tilt is systematic, not random
Problem 3
A poll reports 48% with a margin of error of 4%. Give the lower end of the interval.
Worked answer
44
48-4=44
Problem 4
Same poll. Give the upper end.
Worked answer
52
48+4=52
Problem 5
Why is a random sample preferred?
- It has no systematic tilt
- It is always larger
- It is cheaper
- It avoids the need for a margin of error
Worked answer
It has no systematic tilt
Random selection gives every member an equal chance
So errors are noise, not bias
Problem 6
Estimate the margin of error for a random sample of 400, as a percent.
Worked answer
5
1√400=120=0.05
Problem 7
Estimate the margin of error for a sample of 2500, as a percent.
Worked answer
2
150=0.02
Problem 8
About what sample size gives a margin of error of 5%?
Worked answer
400
√n = 20
n=400
Problem 9
A poll gives 52% with margin 3%. Can a lead be claimed?
- No — the interval includes 50%
- Yes — 52 is above 50
- Yes — the margin is small
- Not enough information
Worked answer
No — the interval includes 50%
The range is 49% to 55%
It contains 50%, so the data has not decided
Problem 10
To halve a margin of error, by what factor must the sample size grow?
Worked answer
4
Halving 1√n needs √n to double
So n must quadruple
Problem 11
A study finds coffee drinkers live longer. What can be concluded?
- There is an association
- Coffee causes longer life
- Longer life causes coffee drinking
- Nothing at all
Worked answer
There is an association
This is observational
Association can be reported; cause cannot
Problem 12
A sample of 100 gives 60%. Give the lower end of the 95% interval, as a percent.
Worked answer
50
Margin =10%
60-10=50
Problem 13
A simulation of fair coins gives a result as extreme as the observed one in 0.8% of trials. What follows?
- Good evidence the coin is not fair
- The coin is definitely fair
- Nothing can be said
- The simulation was wrong
Worked answer
Good evidence the coin is not fair
0.8% is well under 5%
Chance is a poor explanation for what was seen
Problem 14
What does 95% confidence mean?
- Repeating the process, 95% of such intervals would capture the truth
- There is a 95% chance the truth is in this interval
- 95% of people agree
- The sample was 95% accurate
Worked answer
Repeating the process, 95% of such intervals would capture the truth
Confidence is a property of the procedure over many repetitions
Problem 15
One candidate is 49 to 55%, the other 45 to 51%. Do the intervals overlap? Answer yes or no.
Worked answer
yes
49 to 51 is in both
So the data has not separated them
Problem 16
A 1936 poll asked two million people and still called the election wrong. Why?
- The sample was biased
- The sample was too small
- The margin of error was miscalculated
- The votes were miscounted
Worked answer
The sample was biased
It surveyed car and telephone owners, who were wealthier than average
Size cannot rescue a biased sample