# "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."

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.
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Jase Powell
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):
$z=\frac{\left(\stackrel{^}{p}-{p}_{0}\right)\sqrt{n}}{\sqrt{{p}_{0}\left(1-{p}_{0}\right)}}$
Here
${H}_{0}:p={p}_{0}=\frac{1}{2}\phantom{\rule{1em}{0ex}}{H}_{1}:p\ne \frac{1}{2}$
$\stackrel{^}{p}=\frac{50}{163}=0.307\phantom{\rule{1em}{0ex}}n=163$
so the test statistic is
$z=\frac{\left(0.307-0.5\right)\sqrt{163}}{\sqrt{0.5×0.5}}=-4.928$
The critical values are $±1.960$ as you thought. Since $-4.928$ falls outside $\left[-1.960,1.960\right]$, we reject the null hypothesis and conclude the coin is biased.
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dannyboi2006tk
So the test statistic would be -4.928