It's a question derived from my previous post's thread.

Ninety-nine percent (99%) of all the batteries made at the Pineville factory meet the manufacturer’s specifications. A random sample of 400 batteries is selected for testing.

Let's say a battery is invalid if it doesn't meet the manufacturer's specification and valid if it does.

We can think about the population with a proportion p=0.99 where p represents the probability batteries are valid. We don't know the shape of the population but by the Central Limit Theorem, we know the shape of the sampling distribution for proportion with n=400 is a normal curve. When n=400, we can tell the standard error of p is sqrt(p*(1 - p)/n) =sqrt(0.99 * 0.01/400) = 0.00497. With these information we can calculate P(p>1) = normCDF(1, 2, 0.99, 0.00497) = 0.02221.

Because the proportion represents the probability batteries are valid, it can't be greater than 1 in any samples. But in the normal curve we know there's about 2.221% probability the proportion is bigger than 1. Because that about 2%, normCDF(0, 1, 0.99, 0.00497) = 1 - 0.02221 = 0.97779, while the probability that the proportion is with 0 and 1 must be 1.

So i think we have to remove that about 2% probability from 100% to get a more accurate probability.

So let's say f(x) = P(p < x)/0.97779 where f(x) gives the more accurate proabibility and the domain is [0,1]. Fox example, f(1) = 0.97779/0.97779 = 1. Works as we wanted. It seems f(x) really gives the accurate probability.

But the problem occurs. f(0.99) = 0.5/0.9779. it's not equal to 0.5 while intuitively thinking, the probability less than(or more than) 99% of 400 batteries are valid is 50% because the proportion we were given is 0.99.

My question is, i thought f(x) outputs the accurate probability, but i can't explain why f(0.99) != 0.5. So i can't tell gives the accurate probability anymore. If it's not f(x) i've built, what function would give the actually accurate probability in this particular battery scenario when given the proportion?

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Im sorry i'm not good at describing the mathematical(or any other) problem. If you have a question or didn't understand the problem, feel free to ask me.