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My classmates and I are doing a coin toss experiment (i.e. toss coin 100 times). I have already determined that I have a fair coin, since I tossed $43$ heads, and this falls into a $95$% confidence interval of $40-60$ heads.

Now, I am to assume that everyone in my class, and including myself, have a fair coin (i.e. $p=0.5$). How do I determine whether everyone in the class will find that they do in fact have a fair coin, based on their calculations? I think that most people will find that their experiment will support the fact they do have a fair coin. However, I also think there has to be some who will not. How do I determine what percentage of the class conclude that they have a fair coin?

There are 20 people in the class.

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You misunderstand what a 95% confidence interval means, it means that if the coin is fair and you performed the experiment many times, 95% of your results will be within the confidence interval.

The confidence interval describes expectations about the data assuming your assumption is correct. The assumption either is or isn't; the data does not tell you which.

You can never find evidence that a coin is fair - if you toss it enough it will eventually come up heads 1,000 times in a row (or a million or a trillion or $10^{4563554}$). You can only find evidence suggesting that it isn't.

For 20 trials, your expectation is that 1 will be outside the 95% confidence interval but 0, 1 and even 2 or 3 would not be out of the question. It does not mean that those coins are fair or unfair.

Also, if you are each using different coins then you are not actually repeating the experiment - you are in fact making 20 different assumptions of fairness.

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  • $\begingroup$ Is there a method of calculating the probability that each person will estimate that they have a fair coin? Also, I was hypothesizing that as the number of tosses increase from 100 to beyond, the probability that each person will estimate that they have a fair coin will decrease. $\endgroup$ – vanHohenheim Aug 1 '15 at 2:20
  • $\begingroup$ Well, it depends if they do or don't have a fair coin that they test. $\endgroup$ – Dale M Aug 1 '15 at 2:31
  • $\begingroup$ Wouldn't $95\%$ of the fair coins pass the $95\%$ test? $\endgroup$ – Empy2 Aug 1 '15 at 5:52
  • $\begingroup$ More or less but you don't know the coins are fair $\endgroup$ – Dale M Aug 1 '15 at 5:54
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You are asking 'how many students' have a 'yes' answer, so it becomes a binomial question. How many students, and what is the probability of 'yes'?

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