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Is the interval too narrow?

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A model publishes an $80%$ interval every day. Over $400$ independent days, the outcome falls outside the interval on $96$ days. If the model is right, how many standard deviations above its expected value is that count? Give a whole number.

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Hint

The count of misses is binomial with $p = \frac{1}{5}$ .

Solution

The expected count is $400 \times \frac{1}{5} = 80$ .
Its variance is $400 \times \frac{1}{5} \times \frac{4}{5} = 64$ , so the standard deviation is $64 = 8$ .
The distance is $\frac{96 - 80}{8} = 2$ .

Answer: $2$

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