Hard GCSE Sampling Questions

Write down the population and the sample

The population is all $1000$ students at Ashfield School. The sample actually used contains $4$ of them.

Ask who could possibly have been chosen

Every one of the $1000$ students could have been picked, so the sampling frame is the whole population.

Ask who decided whether a member took part

The researcher chose the members at random, so no one selected themselves into the sample.

Ask whether the sample is big enough

A sample of just $4$ out of $1000$ is a tiny fraction of the population. Even a perfectly random sample that small would swing wildly from one sample to the next, so no reliable estimate can come from it.

Test the sample against the first thing that can go wrong: bias in the frame

Everyone had a chance of being chosen, so there is no bias in the frame.

Test the sample against the second thing that can go wrong: self-selection

Nobody volunteered themselves, so volunteer bias is not the problem here.

Test the sample against the third thing that can go wrong: size

A sample of $4$ is far too small to say anything reliable about $1000$ students.

Rule out the claim that the sample is bigger than the population

A sample is always part of the population, and $4$ is indeed smaller than $1000$, so there is nothing wrong on that score.

Rule out the claim that this is a census

A census asks every member of the population. Only $4$ of the $1000$ students were asked, so this is a sample, not a census.

Say what a good sample would look like

A good sample is chosen at random from a list of all $1000$ students, is not made of volunteers, and is large enough that the proportion in it is a useful guide to the population.

Say what a biased sample does to an estimate

Scaling a biased sample up to the population does not fix the bias — it multiplies it. The estimate would be systematically wrong, and taking a bigger biased sample would not help either.

Say what even a perfect sample cannot do

Even a fair random sample of $4$ only gives an estimate: a different sample of $4$ would give a slightly different answer. Only asking all $1000$ students would give the exact figure.

Compare the five statements offered

Each statement makes a different claim about this sample. Checking them against the population, the frame, the way members were chosen and the sample size leaves exactly one that survives.

Pick the statement that survives

The sample is far too small: only $4$ of the $1000$ students were asked, so the estimate would not be reliable.

State the answer

The correct statement is: The sample is far too small: only $4$ of the $1000$ students were asked, so the estimate would not be reliable.

Write down what is already right about the sample

The $20$ households were chosen at random from all $1500$, so the sample is not biased. Whatever is improved must not spoil that.

Say what is still wrong with the sample

Only $20$ of the $1500$ households were asked, so the estimate could easily be a little way out. The weakness is the size of the sample, not the way it was chosen.

Say what a larger random sample does

A random sample of $250$ varies less from one sample to the next than a sample of $20$ does, so its proportion is more likely to be close to the true proportion in the population.

Rule out taking a smaller sample

A sample of $10$ is smaller still, so its estimate would swing about even more. Smaller is worse, not better.

Rule out asking the same people again

Asking the same $20$ households a second time adds no new households at all, so it gives no new information about the ones who were never asked.

Rule out the two biased methods

Asking $250$ households at the village shop, or using online volunteers, would make the sample bigger but no longer random: some members could not be chosen at all. A bigger biased sample is worse than a smaller fair one, because it gives a confident answer that is systematically wrong.

Say what the improvement does not do

Even $250$ households chosen at random still leaves $1250$ households unasked, so the answer stays an estimate. A larger sample makes the estimate more reliable; it does not make it exact.

Say what would make it exact

Only asking every one of the $1500$ households — a census — gives the exact number who recycle glass, and that is usually far too slow and expensive to do.

Compare the five suggestions

The best change is the one that keeps the sample random and covering the whole population, and increases the number of members asked. Only one of the five does both.

Check the winning option keeps the sample random

The larger sample is still chosen at random from all $1500$ households, so it is still unbiased — just more reliable.

Check the winning option really is larger

$250$ is bigger than $20$, so this really is a larger sample.

Say why bias cannot be cured by size

A small random sample is noisy but fair; a large biased sample is precise but wrong. Increasing the size of a biased sample only makes you more confident in the wrong answer, which is why the biased options are not improvements.

Say how the larger sample should actually be taken

Number all $1500$ households, then use random numbers to pick $250$ of them. That is what "chosen at random from all" means in practice.

Summarise the rule

To improve an estimate from a sample: keep it random, keep it covering the whole population, and make it bigger.

State the answer

The change that would most improve the estimate is: Take a larger random sample, for example $250$ households chosen at random from all $1500$ households.

Write down what is already right about the sample

The $40$ members were chosen at random from all $2000$, so the sample is not biased. Whatever is improved must not spoil that.

Say what is still wrong with the sample

Only $40$ of the $2000$ members were asked, so the estimate could easily be a little way out. The weakness is the size of the sample, not the way it was chosen.

Say what a larger random sample does

A random sample of $400$ varies less from one sample to the next than a sample of $40$ does, so its proportion is more likely to be close to the true proportion in the population.

Rule out taking a smaller sample

A sample of $15$ is smaller still, so its estimate would swing about even more. Smaller is worse, not better.

Rule out asking the same people again

Asking the same $40$ members a second time adds no new members at all, so it gives no new information about the ones who were never asked.

Rule out the two biased methods

Asking $400$ members at the reading group, or using online volunteers, would make the sample bigger but no longer random: some members could not be chosen at all. A bigger biased sample is worse than a smaller fair one, because it gives a confident answer that is systematically wrong.

Say what the improvement does not do

Even $400$ members chosen at random still leaves $1600$ members unasked, so the answer stays an estimate. A larger sample makes the estimate more reliable; it does not make it exact.

Say what would make it exact

Only asking every one of the $2000$ members — a census — gives the exact number who borrow e-books, and that is usually far too slow and expensive to do.

Compare the five suggestions

The best change is the one that keeps the sample random and covering the whole population, and increases the number of members asked. Only one of the five does both.

Check the winning option keeps the sample random

The larger sample is still chosen at random from all $2000$ members, so it is still unbiased — just more reliable.

Check the winning option really is larger

$400$ is bigger than $40$, so this really is a larger sample.

Say why bias cannot be cured by size

A small random sample is noisy but fair; a large biased sample is precise but wrong. Increasing the size of a biased sample only makes you more confident in the wrong answer, which is why the biased options are not improvements.

Say how the larger sample should actually be taken

Number all $2000$ members, then use random numbers to pick $400$ of them. That is what "chosen at random from all" means in practice.

Summarise the rule

To improve an estimate from a sample: keep it random, keep it covering the whole population, and make it bigger.

State the answer

The change that would most improve the estimate is: Take a larger random sample, for example $400$ members chosen at random from all $2000$ members.

Write down what the sample found

Of the $20$ students in the sample, $13$ own a bicycle and $7$ do not. The two groups add up to the whole sample, which is a useful check.

Write each group as a fraction of the sample

$\frac{13}{20}$ of the sample own a bicycle and $\frac{7}{20}$ do not.

Check the two fractions add to one

Every member of the sample is in exactly one of the two groups, so the two proportions must add up to $1$.

Simplify the fraction for the group you want

Dividing top and bottom by $1$ gives $\frac{13}{20}$.

State the assumption you are making

Assume the random sample is representative of the whole population.

Estimate the number in the population who do

About $585$ of the $900$ students own a bicycle.

Estimate the number in the population who do not

About $315$ of the $900$ students do not.

Check the two estimates add to the whole population

Every one of the $900$ students is in one group or the other, and the two estimates do add to $900$ — so the scaling is consistent.

Subtract to find the difference

About $270$ more students own a bicycle than do not.

Check the difference a second way

The sample has $13 - 7 = 6$ more in one group than the other, which is $\frac{6}{20}$ of the sample. Scaling that fraction straight up to $900$ gives $270$ — the same answer from a different route.

Check the answer is sensible

The difference must be smaller than the whole population, and it is.

Say why this is only an estimate

Only $20$ of the $900$ students were actually asked. A different random sample of $20$ would almost certainly contain a different number, so the answer is an estimate, not an exact figure.

Say how the estimate could be made more reliable

A larger random sample varies less from one sample to the next, so it is more likely to represent the whole population. Asking more people (still chosen at random) would make the estimate more reliable.

Say what would make the estimate worse

If the students had not been chosen at random — for example if only one year group, or only the people who chose to reply, had been used — the sample would be biased and scaling it up would give a misleading answer.

State the answer in context

About $270$ more of the $900$ students own a bicycle than do not.

Write down what the sample found

Of the $50$ adults in the sample, $32$ read a newspaper and $18$ do not. The two groups add up to the whole sample, which is a useful check.

Write each group as a fraction of the sample

$\frac{32}{50}$ of the sample read a newspaper and $\frac{18}{50}$ do not.

Check the two fractions add to one

Every member of the sample is in exactly one of the two groups, so the two proportions must add up to $1$.

Simplify the fraction for the group you want

Dividing top and bottom by $2$ gives $\frac{16}{25}$.

State the assumption you are making

Assume the random sample is representative of the whole population.

Estimate the number in the population who do

About $768$ of the $1200$ adults read a newspaper.

Estimate the number in the population who do not

About $432$ of the $1200$ adults do not.

Check the two estimates add to the whole population

Every one of the $1200$ adults is in one group or the other, and the two estimates do add to $1200$ — so the scaling is consistent.

Subtract to find the difference

About $336$ more adults read a newspaper than do not.

Check the difference a second way

The sample has $32 - 18 = 14$ more in one group than the other, which is $\frac{14}{50}$ of the sample. Scaling that fraction straight up to $1200$ gives $336$ — the same answer from a different route.

Check the answer is sensible

The difference must be smaller than the whole population, and it is.

Say why this is only an estimate

Only $50$ of the $1200$ adults were actually asked. A different random sample of $50$ would almost certainly contain a different number, so the answer is an estimate, not an exact figure.

Say how the estimate could be made more reliable

A larger random sample varies less from one sample to the next, so it is more likely to represent the whole population. Asking more people (still chosen at random) would make the estimate more reliable.

Say what would make the estimate worse

If the adults had not been chosen at random — for example if only one year group, or only the people who chose to reply, had been used — the sample would be biased and scaling it up would give a misleading answer.

State the answer in context

About $336$ more of the $1200$ adults read a newspaper than do not.