Sampling and scatter graphs – Year 11 GCSE maths revision notes to download as PDF

Worked example 1: stratified sampling

A school of 800 pupils has 200 in Year 7, 180 in Year 8, 220 in Year 9 and 200 in Year 10. A stratified sample of 80 is needed.

Year 7: \(\dfrac{200}{800}\times80=20\). Year 8: \(\dfrac{180}{800}\times80=18\). Year 9: \(\dfrac{220}{800}\times80=22\). Year 10: \(\dfrac{200}{800}\times80=20\). Check that \(20+18+22+20=80\).

Worked example 2: estimating from a sample

In a sample of 50 from 2000 pupils, 12 walk to school. Estimate how many in the school walk.

If a sample is representative, the proportion in the sample estimates the proportion in the population: \(\dfrac{12}{50}\times2000=480\).

Worked example 3 (Higher): fish in a lake

40 fish are tagged and released. Later 50 are caught and 8 have tags. Estimate the population.

\(N=\dfrac{40\times50}{8}=250\). Assumptions: the population stays the same size (no births, deaths or migration), the tags do not fall off, and the tagged animals have mixed fully with the others.

Worked example 4: equation of the line

The line passes through \((2,10)\) and \((8,34)\). Find its equation and estimate \(y\) when \(x=5\).

Gradient \(=\dfrac{34-10}{8-2}=4\). Using \(y=4x+c\) and the point \((2,10)\): \(10=8+c\), so \(c=2\). The equation is \(y=4x+2\). In context: each extra hour of revision adds about 4 marks. For \(x=5\) the estimate is \(y=22\).

Worked example 5: Why a larger sample helps

A random sample of 10 pupils gives a mean height of 158 cm and a random sample of 200 pupils gives a mean of 162 cm. Which is better?

The sample of 200 is better. A small sample can by chance contain mostly short or tall pupils, while a large sample averages out these differences. If the two means differ a lot, trust the larger sample. A larger sample does not remove bias, though: a biased sample of 10 000 is still biased.

Worked example 6: Stratified sampling with uneven groups

A club has 45 juniors, 30 seniors and 25 veterans. Choose a stratified sample of 20 members.

The club has 100 members, so the fraction is \(\dfrac{20}{100}=\dfrac15\). Juniors \(45\div5=9\), seniors \(30\div5=6\), veterans \(25\div5=5\). Total \(=20\). If the answers do not come out as whole numbers, round and check that the total is still correct.

Worked example 7: Using the capture-recapture formula

A zoologist marks 24 hares. In a second sample of 30 she finds 4 marked. Estimate the population.

The proportion marked in the second sample, \(\dfrac4{30}\), is taken to equal the proportion marked in the population, \(\dfrac{24}N\). Hence \(N=\dfrac{24\times30}4=180\). The estimate depends on the second sample being random and on the population being the same at both times.

Worked example 8: Interpolation and extrapolation

A scatter graph of revision time \(x\) (1 to 8 hours) and mark \(y\) has line \(y=4x+10\). Predict the mark for 6 hours and for 15 hours, and comment.

For 6 hours, \(y=34\): this is interpolation, within the range of the data, so reasonably reliable. For 15 hours, \(y=70\) is extrapolation, outside the range. The marks may not keep increasing at the same rate (and might exceed the maximum possible mark), so this prediction is unreliable.

Worked example 9: Describing correlation fully

Describe the relationship shown when points go from top left to bottom right and lie close to a straight line.

This is a strong negative correlation: as one variable increases the other decreases. The word strong refers to how close the points are to a line, while negative refers to the downward direction. If the points are widely scattered it is weak, and if there is no pattern then there is no correlation.

Worked example 10: A good line of best fit

List the features of a good line of best fit drawn by eye.

It passes through the mean point of the data, roughly half of the points lie above it and half below it, and it follows the general direction of the points. It does not need to pass through the origin or through any particular point. Ignore obvious outliers when drawing it.

Worked example 11: Questionnaire design

Improve the question "Don\'t you agree that school lunches are terrible?"

The question is leading because it suggests the answer. A better question is "How would you rate school lunches?" with response boxes: very good, good, average, poor, very poor. Response boxes should not overlap and should cover all possible answers. Keep questions short, clear and about one thing at a time.