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The proportion of pink candies in a bag is supposed to be $50\%$. The filling machine is to be tested to see if it fills with the right proportion. A random sample of $50$ candies is made. The machine is determined to work properly if the number of pink candies is $8$ away from the expected value. Assuming the machine is working properly, what is the probability that it is determined that the machine is not working properly.

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Hint: Student law. – Yves Daoust May 30 '14 at 17:06
up vote 3 down vote accepted

The problem asks us to suppose that the true mean is $50\%$ ("Assuming that the machine is working properly $\dots$").

If that is the case, then the number $X$ of pink in a sample of $50$ has binomial distribution, mean $25$, variance $(50)(0.5)(0.5)$.

So for our problem, the mean and standard deviation can be considered as known.

We are asked to find $\Pr(|X-25|\ge 9$, or perhaps to estimate it using a normal approximation.

Remark: Here "$npq$" is on the small side, and the normal approximation may not be very reliable. One can use the continuity correction and cross one's fingers. But nowadays many pieces of software, and even some calculators, will compute binomial cumulative distribution functions "exactly."

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