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Let's say one day my sister tells me she has psychic powers and can help me predict the winners in horse racing games and for whatever reason, we only have 2 horse racers in a game. She tells me that if she chooses 'blue', then the blue horse wins (painted blue okay?) and if she chooses 'red', then the red horse wins. She does this $8$ times and it turns out she was correct on all of them, making me very wealthy. Now is this a fluke or not? Let's test this at a significance level of $\alpha = 0.05$

Assuming my sister isn't really a psychic, then I expect she would be correct half the time. This yields $\mu = 4$. So my null hypothesis is $\mu = 4$ and my alternative would be $\mu = 8 > 4$. Let $X$ be the random variable denoting the "number of correct predictions my sister makes".

So $X \sim Bin(8,1/2)$

and I did $P(X = 8| \mu = 4) = (1/2)^8 \approx 0 < 0.05$. This leads me to conclude I should reject my initial null hypthosis and conclude my sister may be a psychic.

However this seems to go against any intuition since my sister is actually only 3 years old, and I probably made a Type I Error. Is this a correct analysis?

Lemon
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    Is rejection of the null hypothesis $\mathcal{H}_0: \theta= 1/2$ really the same thing as concluding that she's a psychic? See this, for example: http://xkcd.com/1132/ :-) – hejseb Aug 06 '13 at 06:27
  • I am not smart enough to understand the comic... – Lemon Aug 06 '13 at 06:35
  • Is it because I can never "prove" she really is a psychic just because the null is wrong? – Lemon Aug 06 '13 at 06:54
  • The alternative is not that she is a psychic. The alternative is that the parameter $\theta$ is larger than 1/2 (or, equivalently, that the mean $\mu=8\theta>4$). In general, you should be cautious when drawing conclusions form hypothesis tests. You reject the null - that is, you did not find enough evidence to accept the null. That does not mean you concluce that the alternative is necessarily true. And most importantly in this case, $\mu>4$ does not imply that she is a psychic. – hejseb Aug 06 '13 at 07:08

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