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    I tell you that I have a fair coin and give it to you.

    You flip the coin 20 times.

    You get heads.

    Hands up if you think I am lying that it is a fair coin.

    ? 18 2

    Is the coin biased?


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    𝑋~𝐵(20 , 0.5)

    𝑃 𝑋 ≤ 5 = 0.0207

    𝑃 𝑋 ≤ 6 = 0.0577

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    One or two tail?

    𝐻1: 𝑝 >

    𝐻1: 𝑝 <

    𝐻1: 𝑝 ≠

    ‘is greater than’

    ‘has increased’

    ‘is less than’

    ‘has decreased’

    ‘is different’

    ‘has changed’

    Split the significance level!

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    Coin hypothesis test

    Null Hypothesis:

    The coin is fair.

    Alternate hypothesis:

    The coin is not fair

    𝐻0: 𝑝 = 0.5

    𝐻1: 𝑝 ≠ 0.5

    Where 𝑝 is actual probability of getting a head when flipping the coin.

    The question: is the observation evidence for 𝐻1 or now?

    𝑋~𝐵(20 , 0.5)

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    Testing and concluding  Observed result is 5.

     Two tail test so 2.5% at each end.

     𝑃 𝑋 ≤ 5 = 0.0207 < 0.025.

     The result is significant. Reject 𝐻0.

     There is sufficient evidence to suggest the coin

    is biased.

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    Useful  Before you teach binomial hypothesis testing do

    questions of the form:

    𝑋~𝐵(20 , 0.5)

    𝑃 𝑋 ≤ 5 = 0.0207

    𝑃 𝑋 ≤ 6 = 0.0577

    𝑃 𝑋 ≤ 4 = 0.0059


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    Are they cheating?

     In pairs.

     One person chooses a number of

    pieces of paper, folds them and puts

    them in the plastic sleeve.

     Either fair or cheat!

     The other person takes 10 pieces out

    (with replacement) and tests for


     Your choice of significance level.

    (5%,10%, even 20% if you like)

     Were you correct about whether they

    were cheating?

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    Significance level  Flip your coin 10 times and count how many

    heads you got.

     The significance level is how sure you are of

    your accusation that the observed result is


    Definition of significance level:

    The probability of rejecting the null hypothesis when it

    is in fact true.

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    Plinko We decide

    something is


    Is it


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    Try some hypothesis tests for yourself

     In the booklets I have given some ideas.

     I have props!

     Also, can you come up with some different tests

    that you could do in a classroom with some

    easy to find objects?

     Do you already do some interesting tests?

     We will share ideas at the end.

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    Hypothesis test slider

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    The problem of p-hacking

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    About MEI  Registered charity committed to improving

    mathematics education

     Independent UK curriculum development body

     We offer continuing professional development

    courses, provide specialist tuition for students

    and work with employers to enhance

    mathematical skills in the workplace

     We also pioneer the development of innovative

    teaching and learning resources

  • Statistical experiments

    for hypothesis testing

    in the classroom

    John Brennan-Rhodes

  • Hypothesis test 1 Coin flips out of 20

    Distribution: Binomial (AS level maths)

    When to use

    Useful in the first lesson of teaching hypothesis tests as it can be easily understood, and

    students can get a good sense of what might be meant by the critical region.


    Test for whether a coin is biased or not.

    𝐻0: 𝑝 = 0.5,

    𝐻1: 𝑝 ≠ 0.5

    Where p is the true probability of getting a head when flipping the coin.


    How to use

    Ask students, when flipping a coin 20 times, what would be a suspicious number of heads?

    You might be surprised at how close to guessing the critical region students get.

    Would 9 heads seem right? Would 6? Would 4?

    At what point do you think the coin is biased? This should, roughly, be the critical region.

    Other similar ideas

     One successful alternative is to create (in secret) a biased pack of cards with, say,

    three times as many red cards as black. Tell the students it is fair and then play a

    game (albeit a bit of a boring one) where you say that for red cards you get a point,

    for black cards they get a point. Draw 20 cards (with replacement). When you

    (hopefully) win, ask them if they would accuse you of cheating.

     Can also get students to flip fair coins 20 times repeatedly to demonstrate how, at a

    5% significance level, you would incorrectly reject the null hypothesis 5% of the time.

  • Hypothesis test 2 Shoe size versus height

    Distribution: PMCC (A level maths)

    When to use

    When teaching the product moment correlation coefficient (and regression).


    Test for whether there is a positive correlation between shoe size and height.

    𝐻0: 𝜌 = 0.5,

    𝐻1: 𝜌 ≠ 0.5

    Where 𝜌 is the product moment correlation coefficient between shoe size and height in the

    population of A level maths students.

    How to use

    Collect the students shoe size and height. Plot the data on Excel (or get students to do it by

    hand). Students calculate the PMCC on their calculators and compare to the tables.

    Questions to ask the students (when combining with teaching regression):

     Is the data actually linear? (otherwise you couldn’t use PMCC).

     Could the line of regression be used to approximate the height of the teacher? (no,

    outside of population)

     Could you use the above regression line to approximate someone’s shoe size from

    their height? (no, regression lines only work from x to y)

    Other similar ideas

     Use the large data set instead of student collected data.

     Use the data collected in other lessons e.g. Yo