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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?

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In response to your edit, see my answer and the comment below it. The population isn't normal. –  TooTone Feb 22 at 19:20

2 Answers 2

up vote 1 down vote accepted

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.

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But the population binomial is normal when p=.5 No? –  Joel A. Christophel Feb 22 at 19:15
    
No, it's still binomial. There are several differences: notably the binomial distribution is discrete, whereas the normal distribution is continuous. And the binomial distribution is bounded at $0$ and $n$, the number of samples you take. But for $p=0.5$, you don't have to make $n$ very large to be able to approximate the binomial distribution with the normal distribution. –  TooTone Feb 22 at 19:17
    
Also, see my edit, the population definitely isn't binomial. You take an individual from the population and they either have the characteristic or they don't. E.g. you choose a student and see if they're male or female -- draw a histogram for this and it clearly isn't normal! Compare with a typical setting for the normal distribution where you for example take a student and measure their height. Do this enough and your histogram will be bell-shaped. –  TooTone Feb 22 at 19:19
    
Thank you. This cleared things up –  Joel A. Christophel Feb 22 at 19:23

To C. According to the Central Limit Theorem you can use the normal distribution (as an approximation) as long as your sample is big enough (i.e $>>30$), independently from the initial distribution of the population that you have drawn the sample. So, no it is not required.

To B. This condition implies that your probability of success in the sample $\hat{p}$ is close to $0.5$, i.e. the normal distribution (symmetry) can be used. Otherwise, if $\hat{p}$ is close to $0$ or to $1$, your population is not symmetrical and then the normal distribution would not be as good for an approximation as the Poisson distribution. It believe that condition B. embodies both these concepts, i.e. big sample and fairly symmetrical distribution of failures and succeses in the sample (and in the population).

To A. Ok, we all agree that the sample has to be random.

In sum, I agree with the other answer posted and your teacher.

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