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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?

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

heads you got.

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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Plinko We decide

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

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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

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We offer continuing professional development

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and work with employers to enhance

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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. You can use the height data for the

sampling distribution hypothesis testing also.

guessthecorrelation.com – trust me, look it up!

Hypothesis test 3 Throwing skill game

Distribution: Geometric distribution (A level further maths)

When to use

This is technically a topic for further maths but could be adapted for an A level maths class

as they do effectively learn the geometric distribution in the probability chapter.

Description

Test for whether students are as good as they think they are at shooting basketball hoops

(test against p = their declared skill)

𝐻0: 𝑝 = ?,

𝐻1: 𝑝 > ?

Where 𝑝 is the true probability of student successfully shooting a hoop in basketball.

How to use

Students are to test whether they are as good as they think they are! I took students to the

sports hall and got them to shoot basketball hoops.

First, students say how good at shooting they think they are (i.e. they might think they score

60% of the time). The students keep throwing until they score. Test this against the

geometric distribution.

Other similar ideas

Can be easily redesigned around any ‘success or fail’ skill game. Throwing

whiteboard pens that have run out into the bin?!

Can be easily changed into a negative binomial test if changed to testing for how

many shots before scoring a set number of hoops.

Hypothesis test 4 Coke or Pepsi?

Distribution: Binomial distribution

When to use

This is useful as an experiment to do later in the teaching of binomial hypothesis testing as it

takes some thought on how to set up the experiment. In fact, the question of how to test it

itself is an excellent one. Below is just one possible way to set up the experiment.

Description

Test for whether students think they can correctly identify the odd one out of three samples

of Coke or Pepsi.

𝐻0: 𝑝 =1

3,

𝐻1: 𝑝 >1

3

Where 𝑝 is the probability of correctly selecting the odd one out given three samples of Coke

or Pepsi.

How to use

Students randomly set up n groups of three cups, possibly by rolling a dice and using the

following table below (keeping a note of which cup contains what).

Die Roll Cup A Cup B Cup C

1 Coke Pepsi Pepsi

2 Pepsi Coke Pepsi

3 Pepsi Pepsi Coke

4 Pepsi Coke Coke

5 Coke Pepsi Coke

6 Coke Coke Pepsi

Other student(s) try to guess which is the odd one out in each group of three. Success or

failure in each attempt is recorded. Number of successes out of attempts tested.

Other similar ideas

bit.ly/2KC0V8E – credit for this experiment setup and more details.

Other, healthier alternatives are available!

Hypothesis test 5 Weight of a cookie

Distribution: Sampling distribution (with some Central Limit Theorem)

When to use

When teaching the sampling distribution after teaching the Normal distribution. Or to further

mathematicians when teaching the Central Limit Theorem.

Description

Test for whether the average weight of a cookie is as stated on the packet.

𝐻0: 𝜇 = 10.5g

𝐻1: 𝜇 ≠ 10.5g

Where 𝜇 is the mean weight of a cookie in a sample of 20 cookies.

Note: We will have to use the standard deviation of the sample as an estimate of the

population estimate and really, this would require a larger sample size, which is worth

discussing with students.

𝑋~𝑁 (10.5, (𝑠

√20)

2

)

How to use

The packet will be our sample. Students weigh all of the cookies in the packet and find the

mean and find the sample mean (and standard deviation). Test against the sample mean

said on the packet.

It is worth discussing how really the sample is too small. But, more importantly, discuss why

the standard deviation of sample means would be less than the standard deviation of cookie

weights. The video below is excellent for this.

Other similar ideas

https://youtu.be/olK80ngCbXc (or type ‘puppet sampling distribution’ into YouTube) -

An excellent (and slightly odd) video explaining the sampling distribution.

Other healthier products to test are available! Fruit versus average species weight?

Can also test for the proportional area of chocolate chips on the top of the cookie

versus the picture on the packet.

Hypothesis test pro forma

State the distribution.

Hypothesis statements. Do not forget to state what the tested variable is.

𝐻0:

𝐻1:

Run the test. This is the maths bit. Clearly compare the observed value to the

significance level to decide whether there is evidence for 𝐻0 or 𝐻1.

Conclusion. This is where you report what you have found.

Remember:

Use ‘non-assertive’ language as you don’t know for sure your conclusion is correct (start with the phrase “There is evidence to suggest…”).

Answer the question. The answer should be understandable to someone who doesn’t understand the actual test (your boss maybe?)

Write in full sentences.

where

The result significant.

H0.

There sufficient evidence to suggest that…

(The H1 thing, in words, in context)

is

is

Reject

is not

is not

Accept

Instructions:

1. Cut out the two shapes above, including the rectangles on the top one.

2. Fold back the flaps on the top piece and slide the bottom piece in.

3. Moving the bottom piece back and forth switches between the conclusions.

CUT OUT

CUT OUT

CUT OUT

Hypothesis test:

Hypothesis statement:

How to do the test:

Don’t forget:

PMCC

Find critical values in formula book.

Make values negative if testing for

negative correlation.

Compare to the sample PMCC (r).

That the table is in

the formula book!

H0: ρ = 0

H1: ρ < 0 (or > or ≠)

Where ρ is the…

… in the population.

Binomial Distribution

Assume 𝑋~𝐵(𝑛, 𝑝)

Use calculator to find:

Critical region (Use List in Binomial

CD) then compare to observed.

Or p-value (Variable in Binomial

CD) and compare to sig. level.

𝑃(𝑋 ≥ 13) = 1 − 𝑃(𝑋 ≤ 12)

To be careful with upper tail

critical regions as

H0: p =

H1: p < (or > or ≠)

Where p is the…

… in the population.

Sampling Distribution

Assume �̅�~𝑁 (𝜇, (𝜎

√𝑛)

2)

Use calculator to find:

Critical region (Inverse Normal)

then compare to sample mean.

Or p-value (Normal CD) and

compare to sig. level.

To divide the standard

deviation by √𝑛

H0: µ =

H1: µ < (or > or ≠)

Where µ is…

…in the population.

…then conclude in context.

(say sufficient or insufficient

evidence to suggest…)

Start test with:

CU

T OU

T

Instructions:

1. Cut out the two shapes above, including the rectangle on the top one.

2. Fold back the flaps on the top piece and slide the bottom piece in.

3. Moving the bottom piece back and forth gives a summary of each hypothesis test.

The result significant.

H0.

There sufficient evidence to suggest that…

(The H1 thing, in words, in context)

is

is

Reject

is not

Is not

Accept

Hypothesis test:

Hypothesis statement:

How to do the test:

Don’t forget:

PMCC

Find critical values in formula book.

Make values negative if testing for

negative correlation.

Compare to the sample PMCC (r).

That the table is in

the formula book!

H0: ρ = 0

H1: ρ < 0 (or > or ≠)

Where ρ is the…

… in the population.

Binomial Distribution

Assume 𝑋~𝐵(𝑛, 𝑝)

Use calculator to find:

Critical region (Use List in Binomial

CD) then compare to observed.

Or p-value (Variable in Binomial

CD) and compare to sig. level.

𝑃(𝑋 ≥ 13) = 1 − 𝑃(𝑋 ≤ 12)

To be careful with upper tail

critical regions as

H0: p =

H1: p < (or > or ≠)

Where p is the…

… in the population.

Sampling Distribution

Assume �̅�~𝑁 (𝜇, (𝜎

√𝑛)2)

Use calculator to find:

Critical region (Inverse Normal)

then compare to sample mean.

Or p-value (Normal CD) and

compare to sig. level.

To divide the standard

deviation by √𝑛

H0: µ =

H1: µ < (or > or ≠)

Where µ is…

…in the population.

…then conclude in context.

(say sufficient or insufficient

evidence to suggest…)

Start test with:

CU

T OU

T