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 ≠ 0.5Let 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=0.00098X=1,P=0.0107X=9,P=0.999X=10,P=1Since it is declared biased for these values, P&lt;0.1 or P&gt;0.9. Therefore, shouldn’t we be able to use 20% as the significance level, yet my book says we can’t. Any clarification?

Question

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  ≠  0.5Let 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=0.00098X=1,P=0.0107X=9,P=0.999X=10,P=1Since it is declared biased for these values, P&amp;lt;0.1 or P&amp;gt;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 · Accepted Answer

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   α  =  5  % or 10%.You would have not used the described test if you had been aiming for a test with significance level of   α  =  20  %.