# Fairness of a coin that gives 50 heads from 163 trials

You toss a coin 163 times and get 50 heads. Test the hypothesis that it is a fair coin, using a two-tailed test and 5% level of significance.

The answer wants each of the following: test statistic, critical value, and accept or reject the hypothesis.

It is quite obvious to me that we will reject the null hypothesis since the 2 z-scores associated with getting less than 50 tails (50.5 using continuity correction) are -4.86 and 4.86, while at a 5% confidence interval, the two z-scores are -1.96 and 1.96, but I am not sure what he wants for the test statistic or critical value, or what those are exactly.

• Hint: what distribution does the coin follow? The theorem you want to use afterward is central to your success. – Sean Roberson Oct 9 '16 at 5:08
• normal? And wouldn't the critical value be 1.96 since thats where the 2nd deviation is? – Remy Oct 9 '16 at 5:15
• binomial actually. I know how to calculate that probability, I'm just not sure what a test statistic or critical value is, exactly. – Remy Oct 9 '16 at 5:17

## 1 Answer

Since 50 heads out of 163 is a proportion, we should use the relevant one-sample $z$-test (we can do so since 163 is a large number): $$z=\frac{(\hat p-p_0)\sqrt n}{\sqrt{p_0(1-p_0)}}$$ Here $$H_0:p=p_0=\frac12\quad H_1:p\ne\frac12$$ $$\hat p=\frac{50}{163}=0.307\quad n=163$$ so the test statistic is $$z=\frac{(0.307-0.5)\sqrt{163}}{\sqrt{0.5×0.5}}=-4.928$$ The critical values are $\pm1.960$ as you thought. Since $-4.928$ falls outside $[-1.960,1.960]$, we reject the null hypothesis and conclude the coin is biased.

• So the test statistic would be -4.928? – Remy Oct 9 '16 at 5:21
• @JohnH Yes.${}$ – Parcly Taxel Oct 9 '16 at 5:21
• Why isn't it 50-81.5/sqrt(163*.5*.5). They are close to each other, and both nearly impossible, but still. – Remy Oct 9 '16 at 5:23
• Oh and just realized you said that that was the test statistic in your original answer, oops. – Remy Oct 9 '16 at 5:26
• @JohnH We are testing a proportion that is restricted to $[0,1]$ here, not a continuous value like 50 heads or 17 cars passing by. This difference is subtle and I learned it in my statistics class. – Parcly Taxel Oct 9 '16 at 5:27