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Fill rate interval

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A market maker's quotes were filled in $200$ of $n = 400$ attempts, so the estimated fill rate is $p = 0.5$ . The confidence interval half-width is $h = 1.96 \times \frac{p ( 1 - p )}{n}$ . What is $h$ ?

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Hint

Evaluate the value under the root before taking it.

Solution

Compute $p (1 - p) = 0.5 \times 0.5 = 0.25$ .
Divide by the sample size: $\frac{0.25}{400} = 0.000625$ .
The root is $0.000625 = 0.025$ .
Multiply: $1.96 \times 0.025 = 0.049$ .
So the half-width is $0.049$ .

Answer: $0.049$

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