daktielti

Answered

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?

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?

Answer & Explanation

Elliott Gilmore

Expert

2022-07-01Added 10 answers

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 $\alpha =20\mathrm{\%}$.

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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