Sunday, March 10, 2019

New Gcse: Capture-Recapture

If your schoolhouse plans to work Edexcel for GCSE from 2017 in addition to then you lot may get got spotted the capture-recapture method inwards the Higher Tier specification:
Source: Edexcel GCSE (9 - 1) Mathematics Teaching Guidance

Here's the sampling content inwards to a greater extent than detail:

Source: Edexcel Content Exemplification FAQs

It's worth noting that both Edexcel in addition to AQA listing stratified sampling equally a topic that has been removed from GCSE. However, this comes alongside a caveat - both sampling in addition to proportional reasoning create characteristic inwards the ix - 1 GCSE, hence it would live on reasonable for exams to include a stratified sampling inquiry fifty-fifty if students haven't been explicitly taught this topic (as long equally the inquiry doesn't work the give-and-take 'stratified' without explaining its meaning). So my advice - whatever board you're using - is to await at a few stratified sampling questions alongside your GCSE class, whether inwards a statistics lesson or a ratio in addition to proportion lesson.

Stratified sampling is a groovy chance to work proportional reasoning, equally is the capture-recapture method. If you've taught Edexcel's GCSE Statistics in addition to then you'll already live on familiar alongside capture-recapture, but I'll explicate it hither inwards instance you've non seen it before.

A elementary example
Try this question... it volition alone get got you lot a second.
I captured 50 fish from a lake. I marked a big cross on the dorsum of each fish alongside a permanent marker*... 
I seat the marked fish dorsum inwards the lake in addition to they happily swam away to bring together their friends. 
The adjacent twenty-four hr catamenia I captured 20 fish from the same lake. 10 of them had a cross on their back. 
Can you lot gauge the total population of fish inwards the lake?
*no fish were hurt, promise.

I'm sure you lot spotted that the proportion of marked fish inwards the minute sample was 0.5, in addition to nosotros tin assume the same proportion of marked fish inwards the whole population. Given that I marked 50 fish, nosotros tin gauge that at that topographic point are 100 fish inwards the lake.

A formula
If the numbers are less straightforward hence the estimation can't live on done mentally, it's slowly to laid a formula to operate out the population. This is sure non a formula that students volition quest to memorise - it tin live on deduced using proportional reasoning.
You tin come across that the formula on the left merely shows that proportion of marked fish inwards the population is equal to the proportion of marked fish inwards the sample. The formula on the correct has been rearranged to brand due north the subject.

Here's an illustration from the Biology department of BBC Bitesize. It would live on improve if they had shown the proportional reasoning in addition to rearrangement procedure rather than exactly give a terminal formula.
Source: BBC Bitesize

The illustration goes on to listing closed to assumptions - these are sure worth discussing alongside your students.
  • There is no death, immigration or emigration (ie the population is closed)
  • The sampling methods used are identical
  • The marking has non affected the survival charge per unit of measurement of the animals 

We besides assume that animals create non lose their marks, that marking does non touching the likelihood of recapture, in addition to that sufficient fourth dimension is left betwixt marking in addition to recapture for all marked individuals to live on randomly dispersed throughout the population.

An bear witness question
So what mightiness this topic await similar inwards a GCSE exam? Here's a question from Edexcel's 2014 Higher GCSE Statistics paper.
A scientist wants to gauge the let on of fish inwards a disused canal. He catches a sample of thirty fish from the canal. He marks each fish alongside a dye in addition to and then puts them dorsum inwards the canal. The adjacent twenty-four hr catamenia the scientist catches 20 fish from the canal. He finds that 4 of them are marked alongside the dye. 
(a) Estimate the total let on of fish inwards the canal. (2)
(b) Write downwards whatsoever assumptions you lot made. (2)

For purpose b, candidates get got to squall 2 ideas, including something almost the population beingness unchanged, or the thought of randonmess, or that markings stay unchanged.

For to a greater extent than bear witness questions, see Edexcel's Emporium in addition to await under GCSE 1MA1 Practice Papers > Themed Papers.

Teaching ideas
I squall back this volition live on quite a overnice topic to teach. Here are a few useful links:

Finally, here's a overnice video to present inwards your lesson - Johnny Ball estimates the let on of dark cabs inwards London.






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