Conditions for a Proportion Test

A test's p-value comes from a normal curve, and that curve only describes the sampling distribution of p-hat accurately when certain things are true about the data. So before computing anything, you check three conditions. Random: the data must come from a random sample or random assignment, or the test's logic about what results would look like by chance falls apart. Independence: one observation can't influence another, which you justify either because the sampling was done without replacement from a population at least 10 times the sample size, or because of how a random process was run. Large counts: the sample needs to be big enough that the sampling distribution of p-hat is approximately normal, checked by requiring n times p0 and n times (1 minus p0) to both be at least 10.

The key twist for tests, compared to confidence intervals, is that large counts uses p0, the null hypothesis value, not p-hat. A test is built on the assumption that the null hypothesis is true — that's the whole premise of asking 'how surprising would this sample be if p0 were correct?' So every calculation under that assumption, including whether the normal approximation is valid, has to use p0 instead of the sample proportion.

If the sample wasn't randomly selected, you don't get to use this machinery at all in a trustworthy way: any conclusion only applies to the specific sample, and inference to a wider population is no longer statistically justified, no matter how large n is. This sets up the test itself in the next topic, where these three checks become the required first step you show before calculating a test statistic and p-value.