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Say you flip a coin 10 times and observe only one heads. What would be your null hypothesis and p-value for testing whether the coin is fair or not?

This is the same question as problem #5 in the Statistics Chapter of Ace the Data Science Interview!

The null hypothesis is that the coin is fair, and the alternative hypothesis is that the coin is biased: *biased towards tails (note that this a one-sided test):*

$H_0: p_0 = 0.5$

$H_1 : p_0 < 0.5$

Since the sample size here is 10, you cannot apply the Central Limit Theorem and so cannot approximate a binomial using a normal distribution.

The p-value here is the probability of observing the results obtained given that the null hypothesis is true, i.e., under the assumption that the coin is fair. For 10 flips of a coin, there are 2^10 = 1024 possible outcomes, only 10 of which yield 9 tails and one heads.

Hence, the exact probability of the given result is the p-value, which is $\frac{10}{1024} = 0.0098$. Therefore, we can reject the null hypothesis at a 0.05 significance level.