Question

Which of the following is NOT a condition for performing a significance test about an unknown population proportion p?

a) If you are sampling without replacement from a finite population, then you should sample no more than 10% of the population.

b) The data should come from a random sample or randomized experement

c) Individual measurements should be independent of one another

d) The population distribution should be approximately Normal, unless the sample size is large

e) Both np and n(1-p) should be at least 10

PLease explain why the answer is correct or incorrect Thank you!

Answer 1

Here' the answer to the question. Please don't hesitate to give a "thumbs up" in case you're satisfied with the answer

Answer is A. If you are sampling without replacement from a finite population, then you should sample no more than 10% of the population.

When the sample gets too large compared to the population, consider the extreme case in which the sample size is the same as the population size: then there is only one possible sample mean, so the sampling distribution isn't really normal in any meaningful sense.

As for what goes wrong: CLT is an asymptotic result that holds when the population that you are sampling from is infinite (so that your sample size can grow unboundedly). For finite populations, we can sort of wave our hands and pretend that we're actually sampling with replacement, which effectively gives us an infinite population to sample with (since we can sample each individual in the population unboundedly many times)--so the CLT holds. If the sample size is small relative to the population size then sampling with and without replacement is almost the same, so we can pretend like we're sampling with replacement even if we're not. But as the sample size grows, sampling without replacement becomes very different from sampling with replacement