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# Powerful Hypothesis Testing

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### Chi-squared Faker

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Age 16 to 18

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Robin has a bag containing red and green balls. Robin wants to test the following hypotheses, where $\pi$ is the proportion of green balls in the bag:

Robin is allowed to take out a ball at random, note its colour and then replace it: this is called a trial. Robin can do as many trials as desired.

Robin uses the following approach:

If the null hypothesis is**false**, what is the probability that the null hypothesis will be rejected?

You can explore this question with the following simulation.

*Warning - the computer needs a little bit of thinking time to do the simulations!*

In this simulation, you can:

Now try changing the settings. Can you predict what will happen as a result of your changes?

Here are some further questions you could consider:

If Robin wants to be 90% certain of rejecting the null hypothesis if it is wrong, how many trials are needed?

You may want to ask and explore other questions as well.

*The probability of correctly rejecting $H_0$ when it is false is called the ***power** of the test. Accepting $H_0$ when it is false is called a Type II error.

*This resource is part of the collection Statistics - Maths of Real Life*

$H_0\colon \pi=\frac{1}{2}$ and $H_1\colon \pi\ne\frac{1}{2}$

Robin is allowed to take out a ball at random, note its colour and then replace it: this is called a trial. Robin can do as many trials as desired.

Robin uses the following approach:

"I will do exactly 50 trials. If the p-value* is less than 0.05, then I will reject the null hypothesis at the 5% significance level, otherwise I will accept it."

If the null hypothesis is

You can explore this question with the following simulation.

In this simulation, you can:

- specify the number of green and red balls actually in the bag - note that in a real experiment we would not know this!
- specify the number of trials per experiment (up to 200)
- specify the proportion for the null hypothesis (which we took to be $\frac{1}{2}$ above)
- repeat the experiment

Now try changing the settings. Can you predict what will happen as a result of your changes?

Here are some further questions you could consider:

- What is the probability of $H_0$ being rejected?
- If $H_0$ is rejected, how likely is it that the alternative hypothesis $H_1$ is true?

- the true proportion of greens in the bag changes?
- the significance level changes?
- the hypothesised proportion $\pi$ changes?

If Robin wants to be 90% certain of rejecting the null hypothesis if it is wrong, how many trials are needed?

You may want to ask and explore other questions as well.

* If you want to read about what p-values are, have a look at What is a Hypothesis Test?. In this case, the p-value is calculated like this: after all of the trials, we find twice the probability of obtaining this number of greens or a more extreme number, assuming that $H_0$ is true. For more on the effect of different ways
of choosing the number of trials to perform, see Robin's Hypothesis Testing.

How would you massage the data in this Chi-squared test to both accept and reject the hypothesis?