You test whether a signal's hit rate is 1/2 against 3/4, using 6 independent trades, and reject 1/2 when at least 5 of them hit. What is the size of this test, the chance of rejecting when the hit rate really is 1/2? Give a fraction.

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False alarm rate of a small trial

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You test whether a signal's hit rate is $\frac{1}{2}$ against $\frac{3}{4}$ , using $6$ independent trades, and reject $\frac{1}{2}$ when at least $5$ of them hit. What is the size of this test, the chance of rejecting when the hit rate really is $\frac{1}{2}$ ? Give a fraction.

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Hint

Under the null every sequence of $6$ results has the same chance.

Solution

There are $2^{6} = 64$ equally likely sequences.
Exactly $5$ hits happen in $6$ sequences and exactly $6$ hits in $1$ .
The size is $\frac{6 + 1}{64} = \frac{7}{64}$ .

Answer: $\frac{7}{64}$

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