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For quality control purposes we have to accept or reject a large shipment of items. In order to do so we collect a sample of 200 items, 24 of which are defective. The manufacturer claims that at most 1 in 10 items in a shipment are defective. At the 4% level of significance, is there significant evidence to disprove the claim?

Though exact answers are useful I am more of looking for how to do this. My current thinking is to find the standard normal deviation and find whether it is within the top 4%, which I believe is 1.75 deviations from 0.

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up vote 2 down vote accepted

We are presumably using as null hypothesis that the probability $p$ of a defective is $1/10$, likely versus the alternative that $p\gt 1/10$.

The sample size of $200$ seems large enough, even with the relatively small $p$, to make the normal approximation good enough.

Assuming the null hypothesis, the standard deviation of the number of defectives in a sample of size $200$ is $\sqrt{(1/10)(9/10)(200)}$. This simplifies to $3\sqrt{2}$, about $4.24$.

On the null hypothesis, the mean number of defectives is $20$. Note that $24$ is therefore somewhat less than $1$ standard deviation unit above the mean. That is not far enough from the mean to reject the null hypothesis, at significance level $4\%$, or indeed any reasonable significance level.

And indeed, as you wrote, for any normal, the probability that we are more than $1.75$ standard deviation units above the mean is about $4\%$.

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