A friend of yours offers to play a little game with you. The rule of the game is very simple: After a toss of a coin, if the coin lands on its head, he gives you £10; if it lands on its tail, you give him £5. You are suspicious because this sounds too good a deal to be true. If your friend is using a fair coin, it will land on its head and tail with equal probability: 50%. If that is the case, you are surely going to make loads of money after playing enough rounds of the game. So, you suspect that the coin may be biassed to land on its tail (and benefit your friend in the long run). Your friend, to give you a peace of mind, now offers you to toss the coin a few times to check for yourself whether or not the coin is in fact fair. You take the offer and toss the coin 6 times, observing the following sequence of outcomes: THTTHT, with H for head and T for tail. Can you conclude from this evidence whether the coin is going to land on its tail more often than a fair coin does?

Here is my attempt:

The null hypothesis would be: “Coin is fair.” Level of significance chosen here is 5%. So, 5% of the times we reject the fact that coin is fair when it actually is not.

Test statistic is the number of tails.

Computing p value P(Outcome with more tails than heads or in 6 throws, 4, 5 or 6 tails|Coin is fair) =C(6,4)(0.5)^4(0.5^2)+ C(6,4)(0.5)^5(0.5^1)+ C(6,6)(0.5)^6(0.5^0) =(0.5^6){C(6,4)+C(6,5)+C(6,6)} =(0.5^6)(15+6+1) =22/64 =0.34

p value=0.34>0.05 So, I do not reject the null hypothesis.

Is my analysis correct?

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  • $\begingroup$ You are correct. More broadly, six is far too small a sample. There's a $.0156$ chance of throwing all tails with a fair coin...so even that extreme outcome, while surprising, would hardly be overwhelming evidence. $\endgroup$ – lulu Feb 11 at 17:10
  • $\begingroup$ Do you want to test to see if the coin is fair? If the coin it fair you still want to play. What you want to test is to see if the coin is biased to the degree of at least 2:1 favoring heads. $\endgroup$ – Doug M Feb 11 at 17:19
  • $\begingroup$ What is the smallest possible achieved by this experiment? How do I answer this? For lowest possible p value, P(Outcome with more tails than heads or in 6 throws and 6 tails|Coin is fair) = C(6,6)(0.5)^6(0.5^0) =1/64 = 0.015625. Is this correct? $\endgroup$ – Ujjayanta Bhaumik Feb 11 at 17:24

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