Statistical hypothesis testing

Statistical Hypothesis Testing – A Level Year 1 Content

To start, a reminder of the comprehensive revision notes from Mathsbox

Mathsbox Statistics revision notes

The post Questions by topic – Maths A level has several sources of questions by topic including the OCR Check In tests illustrated here.

OCR Check in test – Hypothesis testing

And here’s a useful reminder from Steven Walker, OCR Maths Subject Advisor on A Level Maths – hypothesis tests and the art of being non-assertive.

It is good practice to make sure that the final conclusion refers to the parameter defined at the start. This should also be non-assertive, using phrases such as ‘There is sufficient evidence to suggest…’ or conversely ‘There is insufficient evidence to suggest…’

Steven Walker, OCR Maths Subject Advisor
Andy Lutwyche

We could use Erica’s Errors On Hypothesis Testing from Andy Lutwyche, (or on TES here).

Thinking about revision for Statistical hypothesis testing, the set of examples from Physics and Maths Tutor for Year 1 (‘Cheat sheet’) is very useful.

Physics and Maths Tutor

Looking at the second example here, we could also illustrate this with one of Mike Hadden’s Excel files. I have used BinHyp2 from the A/AS Statistics files. These are macro-enabled workbooks. Note the scroll bars to change n and p, n can be changed by changing the cell but for p the scroll bar must be used and we can only enter two decimal places. You may need to display the scroll bars and it seems the arrows are missing in later versions of Excel – these become visible if you hold down the mouse pointer near one of the ends of the scroll bar.

We could also look at a two-tailed example, from the Physics and Maths Tutor legacy questions Year 1 – questions by topic, this example is from the first question:

The mark schemes are at the end of the document.

We could use this Geogebra resource from Integral.

Integral resources

…or Mike Hadden’s Excel file.

Maths Challenge Questions

Thinking about Maths challenges, I was interested to see these Irish competitions.

The Junior competition is for Irish first year students (ages 12-13). Questions and solutions from 2011 onwards are available on self-checking Google or Microsoft Forms; you can copy the forms and use them as your own. The competition consists of 15 multiple choice problem solving questions to be answered within a 40 minute period. 

For some more problems to try, have a look at Peter’s Problem, problem solving questions set at the level of Junior Cycle Ordinary level; in terms of the GCSE the nearest Irish award is the Junior Certificate. Problems and solutions are available for 2012 onwards. All the links take you diectly to a pdf document.

Peter’s Problem 2022 extract – Irish competition
Peter’s Problem – Solutions

With the UK Junior Maths Challenge (for ages 11 – 13) on Wednesday 26th April, it’s time for some final practice papers; the 2022 paper has been published on DrFrost Maths. Students can try this paper online and get feedback on the answers.

On the UKMT site use the Junior Maths Challenge archive to freely access the 2015 – 2022 papers, solutions and investigations are also available. The solutions and investigations give full solutions, there are also additional problems for further investigation.

These are great problems for any time and really get students thinking. The 2015 – 2021 papers can be accessed online or you can download a paper. For older students use the Intermediate or Senior Mathematical Challenge archives.

For further resources try:

Challenge Prompts

See also: Maths Challenge Prompts.

For further challenging problems see the post, Polya – Problem Solving which includes many recommendations.

Zeitz, P., 2007. The art and craft of problem solving. 2nd ed. John Wiley & Sons, Inc, p.2

GCSE and A Level Mathematics Revision

With exams approaching fast I am keeping the GCSE 2023 revision collection up to date, added more recently we have:

1st Class Maths

On 1st Class Maths, new practice papers and video solutions are being created for Edexcel and AQA. Recently added we have AQA Higher Paper 1, and you can see publication dates for more papers here.

From GCSE Maths Tutor we have practice papers and worked solutions.

…and now available from CorbettMaths – The Ultimate GCSE Maths Higher Revision Video.

Turning to A Level, in my post Questions by Topic – Maths A Level, I included Doctor Oliver Mathematics you will find an extensive library of worked examples, access these via examination papers or choose questions by topic. It is so valuable that the question and solution are presented together. At A level you will find both Mathematics and Further Mathematics; recently added we have November 21 Core Pure Mathematics.

The A Level series of pages includes many resources for A Level teaching including revision resources; note Advanced Level Starters for example.

Stoke Maths MEP Starters are very attractively presented high-quality resources. Looking at the Spot the Mistake PowerPoints for example, as you can see in the image below there are a great collection of questions that include full answers. It’s great to see Mechanics and Statistics collections. The revision question starters provide very useful question sets.

From Ben Bently, comes a collection of low stakes quizzes using questions from Diagnostic questions.

I also included Refreshing Revision on Transum Mathematics in the Questions by Topic – Maths A Level post, there are now several further topics available.

Time for some puzzles…

It’s holiday time so time to enjoy some more puzzles. There is a series of pages, Puzzles & Games on this blog, available from the top menu.

To highlight just a few of this large collection, Algebra includes the lovely SolveMe mobiles puzzles.

Or for a rather different approach, try Shuttle Mission Workshop from Math Playground, here students build and solve their own visual math puzzles. Play Shuttle Mission Pro first to practise.

Nrich features on many pages including reading; Nrich has many excellent articles on the use of games in the classroom; hence a Reading page.

Math Playground – PEMDAS Exhibit

On Number, Math Playground, PEMDAS Exhibit provides great practice for order of operations. Touch an operation to form that part of the operation.

On Geometry of course follow Catriona Agg who regularly posts new puzzles and has made her collection of screenshots freely available.

Catriona Agg

Following each puzzle, you can find a discussion on the puzzle. Have a look at this puzzle and discussion for example.

In response to this puzzle, you can see various solutions including an Autograph file created by Rob Smith. Rob has this up to change both squares and you can move also move a point on the smaller square.

Try these brilliant GeoGebra puzzles from Daniel Mentrard, also his 64 geometric puzzles.

Puzzles – Daniel Mentrard

Perhaps try Tangram on Mathigon.
How many of the different shapes can you make?

Nrich also have a Tangram activity for students age 7 – 11 or for 5-14 years olds try Tangram Pictures.

Nrich Tangram Browser

Another brilliant Geometry puzzle author is Ed Southall who has made 40 puzzles available to celebrate his 40th birthday; he has also provided slides for teachers. Check his Geometry Cheat Sheet.

From Probability and Statistics games…

Guess the Correlation
Guess the Correlation – Omar Wagih

The collection includes from Cambridge PhD student, Omar Wagih ‘Guess the Correlation‘, a rather addictive game with a purpose – Omar Wagih is collecting the data on the guesses collected and using it to analyse how we perceive correlations in scatter plots. Select About to read the rules and further details.

Underground Maths, this page includes several resources on puzzles and games from Underground Maths

Underground Maths – Equation Sudoku

From Underground Maths try Equation Sudoku for example.

We can write down equations:

f+g+a =19

m+k+c =16

and so on.

As always on Underground Maths we have suggestions and a complete solution as well as printable/supporting materials.


Erich’s Puzzle Palace – featured on Puzzles Page

Good Mathematicians Can Go Backwards!

There is so much opportunity for thinking backwards when we teach – a great learning opportunity and also a problem-solving strategy.

I was delighted to present a session at The Joint Conference of Mathematics Subject Associations 2023. We looked at lots of ideas and activities to get our students thinking backwards as well as forwards. The Slides from the session are available at the end of this post. The resources used are listed below.

Nrich Articles and Resources



Questions such as this can make a great starter for a lesson and provide the chance to discuss number operations and the relationships between them. Manipulating numbers like this can also help with algebraic manipulation.

Looking for some more examples of this type, I came across a really useful resource on TES, “If I know this then I also know …” by Piers Butler. This would make an ideal lesson starter. As it is an Excel spreadsheet, I thought it would be simple to add another worksheet with the answers and created the Excel file CY If_I_know_this_then_I_also_know_ which is a copy of the original, but just adds another worksheet with the answers.

Number Operations


Resource Collections

From AQA comes an outstanding resource, GCSE Mathematics: 90 maths problem solving questions. Strategies discussed include work back familiar and work back unfamiliar. Problems are indexed both by strategy and also by content.


Transum – Arithmagons

This post on Arithmagons includes the resources discussed and many more.

See also Digitisers from Jonny Griffiths.

Maths White Board

Standards Unit

A2 Creating and Solving Equations

This is an outstanding resource – many excellent activities here for the secondary classroom. Start by reading Improving Learning in Mathematics – Malcolm Swan

The resources are hosted by Nottingham University, including all the pdf files very clearly indexed. Note that this site includes the complete set of resources including the software; however, this software no longer works on modern browsers and mobile devices, but see note on my Standards Unit page for HTML5 versions.

Open Middle

Distributive Property, Open Middle

Mathsbot – Jonathan Hall
Variation Grids

Fill in the blanks

Nathan Day – Factor Trees

There are many suggestions for fill in the blanks resources in this post.

Coordinate Geometry Table, Transum, Adapted with permission from a learning resource created by Rob Southern @mrsouthernmaths.

The answer is – what was the question?

Andy Lutwyche – TES resources

Here’s the diagram – what’s the question?

Presentation Slides