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• @MEIConference #MEIConf2019

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

You flip the coin 20 times.

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

? 18 2

Is the coin biased?

9

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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

k=5

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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

unfairness.

 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

 The significance level is how sure you are of

your accusation that the observed result is

strange.

Definition of significance level:

The probability of rejecting the null hypothesis when it

is in fact true.

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something is

biased.

Is it

definitely?

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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

https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800

https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800 https://io9.gizmodo.com/i-fooled-millions-into-thinking-chocolate-helps-weight-1707251800

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https://www.nature.com/articles/d41586-019-00857-

9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1

https://www.nature.com/articles/d41586-019-00857-9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1 https://www.nature.com/articles/d41586-019-00857-9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1 https://www.nature.com/articles/d41586-019-00857-9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1 https://www.nature.com/articles/d41586-019-00857-9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1 https://www.nature.com/articles/d41586-019-00857-9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1 https://www.nature.com/articles/d41586-019-00857-9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1 https://www.nature.com/articles/d41586-019-00857-9?utm_source=twt_nnc&utm_medium=social&utm_campaign=naturenews&sf209700813=1

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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 jrhodes@woodhouse.ac.uk

• 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.

Description

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.

𝑋~𝐵(20,0.5)

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).

Description

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

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