Question

In the league in which you curl, a coin is flipped prior to the game to determine which team gets the hammer in the first end. You have played 12 games in total and you’re beginning to think that the chief umpire (who does the coin flipping) is biased against you and is somehow causing you to lose the coin flips. Of the 12 games you’ve played you’ve only won the coin flip on 2 occasions. Do you have sufficient evidence (at α = .05) to show that the chief umpire is not flipping the coin fairly?

Answer 1

Let our null hypothesis be,

H₀ : The chief umpire is fair, i.e. probability of success, p = 0.5 ( as it should be for a coin toss ).

and the alternate hypothesis will be,

H₁ : The chief umpire is biased against you, i.e. probability of success, p < 0.5.

As we know that outcome of a coin toss follows binomial distribution, the test will be a one sample binomial test.

Here the total number of experiment is, n = 12.

The number of success, B = 2.

Given level of significance is $\alpha$ = 0.05.

Our test statistic is the number of successes, B = 2.

We reject H₀ if $B \leq b_\alpha$

where the constant $b_\alpha$ is chosen to make the type 1 error probability equal to $\alpha$. The number $b_\alpha$ is the lower $\alpha$ percentile point of the binomial distribution with sample size n (here, n=12) and success probability p₀ (here, p₀= 0.5).

We can find the exact value of $b_\alpha$ using R. Alternatively, for p-value approach,

Using R code

binom.test(x=2,n=12,p=0.5,alternative="less")

we can find the p-value for the test, which is 0.01929, which is less than 0.05 (or $\alpha$)

So in the light of the given data, we can reject the null hypothesis at 5% level of significance and can conclude that the chief umpire is biased against you.

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