daktielti

2022-06-30

I am struggling with the following question:
A test is constructed to see if a coin is biased. It is tossed 10 times and if there are 10 heads, 9 heads, 1 head or 0 heads, it is declared to be biased. Can 20 be the significance level of this test?
My thinking is as follows: ${H}_{0}:p=0.5$
${H}_{1}:p\ne 0.5$
Let X be the number of heads, under ${H}_{0}$, X$\sim$B(10,0.5).
If we look at a table of values, we get:
X=0,P=0.00098
X=1,P=0.0107
X=9,P=0.999
X=10,P=1
Since it is declared biased for these values, P<0.1 or P>0.9. Therefore, shouldn’t we be able to use 20% as the significance level, yet my book says we can’t. Any clarification?

Elliott Gilmore

Expert

If the null hypothesis of the probability of a head being 0.5 is true then you have almost calculated that the probability of seeing 0, 1, 9 or 10 heads from 10 tosses is about 0.0215.
Meanwhile the probability of seeing 0, 1, 2, 8, 9 or 10 heads from 10 tosses is about 0.1094.
So with the test described seeing 1 or 9 has a p-value just over 2% (seeing 0 or 10 has a p-value just under 0.2%) and you might have come up with the same test if you had been aiming for a test with significance level of $\alpha =5\mathrm{%}$ or 10%.
You would have not used the described test if you had been aiming for a test with significance level of $\alpha =20\mathrm{%}$.

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