GCSE Sampling Practice Questions

Write down the two numbers you need

$12$ of the students in the sample walk to school, out of $30$ students altogether.

Write the proportion as a fraction

The proportion is the number in the group you want over the total size of the sample, so it is $\frac{12}{30}$.

Simplify the fraction

The highest common factor of $12$ and $30$ is $6$, so divide top and bottom by $6$ to get $\frac{2}{5}$ in its simplest form.

Say what a good sample has to do

A sample is only fair if every one of the $600$ students at Hillside School has the same chance of being chosen. Then the sample is likely to look like the population.

Test each method against that rule

Asking whoever the researcher happens to meet, asking at the cycling club, using online volunteers, or asking only the students in Year 9 all leave some students with no chance of being picked, so each of those samples is biased before the survey even starts.

State the answer

The best method is: Choose $50$ students at random from a list of all $600$ students. That way the sample of $50$ is a fair picture of all $600$ students, so what it says about cycle lanes can be trusted.

Write down the two sample sizes

Ali asked $25$ students; Beth asked $120$ students. Both samples were chosen at random from the same $1500$ students at Fairview College.

Note that neither sample is biased

Both samples are random samples of the whole population, so neither estimate is systematically too high or too low. The only difference between them is size.

Say what sample size does

A larger random sample varies less from one sample to the next. Beth's $120$ students are more likely to contain the same proportion who travel by bus as the whole population does than Ali's $25$ are.

Rule out the claim that they are equally reliable

The two samples are not the same size — $25$ is not $120$ — so there is no reason to expect them to be equally reliable.

Rule out the claim about more than half the population

A sample of $25$ is nowhere near half of $1500$: even doubled it is only $50$. And no sample, however large, makes an estimate CERTAIN — only a census does that.

State the answer

The correct statement is: Beth's estimate is likely to be more reliable, because a sample of $120$ is larger than a sample of $25$, and a larger random sample is more likely to represent the whole population.

Write the sample result as a fraction

$13$ of the $20$ members in the sample borrow e-books, which is $\frac{13}{20}$ of the sample.

Scale the proportion up to the population

Assuming the sample is representative, about $585$ of the $900$ members borrow e-books.

Check the scaling

$585$ out of $900$ is the same proportion as $13$ out of $20$, so the scaling is right.

Ask whether that number is exact

Only $20$ of the $900$ members were actually asked. Nothing at all is known for certain about the other $880$.

Show why it cannot be exact

Take a second random sample of $20$ members. It would almost certainly contain a different number who borrow e-books, and so would give a different estimate. Two honest samples cannot both give THE exact figure, so no single sample does.

Rule out the wrong scalings

The $13$ is only the number in the sample, not in the population, and subtracting from $900$ answers a different question. The population estimate must keep the sample proportion $\frac{13}{20}$, and only $585$ does that.

Say what the sample DOES tell you

The sample is still worth having: $585$ is the best estimate available, and a random sample has no reason to be systematically too high or too low.

Say how the estimate could be tightened

A larger random sample would vary less, so its estimate would be closer to the truth more of the time.

Say what would make it exact

Only a census — asking every one of the $900$ members — gives the exact number.

State the answer

The correct statement is: It is about $585$, but this is only an estimate: a different random sample of $20$ members could give a different answer.

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.