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: $H_0: p=0.5$
$H_1: p\neq0.5$
Let $X$ be the number of heads, under $H_0$, $X$~$B(10, 0.5)$.
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?
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 $\alpha=5\%$ or $10\%$.
You would have not used the described test if you had been aiming for a test with significance level of $\alpha=20\%$.