Use z confidence interval to estimate population proportion

Question

Which of the following must be true of a sample in order for it to be appropriate to use a $z$ confidence interval to estimate the population proportion?

(A) The sample is a random sample from the population of interest.

(B) $n\hat{p}\ge10$ and $n(1-\hat{p})\ge10$

(C) The population distribution is normal.

(D) All of (A), (B), and (C) are required for appropriate use of the $z$ confidence interval to estimate the population proportion.

(E) Only (A) and (B) are required for appropriate use of the $z$ confidence interval to estimate the population proportion.

I chose (A) with the reasoning that either (B) or (C) must be true but that neither had to be true. If the population distribution is normal, the sampling distribution is normal. Likewise, if (B) is true, the sampling distribution is normal. So either (B) or (C) would independently ensure the sampling distribution to be normal.

My teacher marked it incorrect and gave the answer to be (E). Who is right?

Answer 1

You both agree that A has to hold: you need independence to avoid bias, reduce your true standard error -- and in this case to be able to approximate using the normal distribution (see B, below).

Now to C. The population distribution isn't normal: your model is that there is a certain proportion of the population that has a characteristic. So you take an individual from the population and they have a characteristic $C=1$ or they don't $C=0$, with proportion $p$. This gives $P(C=1)=p$ which is a Bernoulli distribtion. If you take a sample from such a population you get a Binomial distribution which leads us to...

B. As a rule of thumb it is reasonable to approximate a binomial distribution $B(n,p)$ with a normal distribution $N(np, npq)$ under the conditions given, which amount to saying that you if you don't have an extremely large number or an extremely small number of individuals in the population with or without the characteristic, then the number of successes drawn from the binomial distribution can be approximated with the continuous bell curve of the normal distribution.