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This question should be a very common one between beginners because the introductory books in statistics don't seem to answer it clearly.

We know when we do a sampling distribution, we have the parameter (in the case of the picture, the mean $\mu$) in the center of the distribution

See the picture:

Let's call a population proportion as $p$ and $\hat p$ a sample proportion. Then, we have the 95% confidence interval:

$$\hat p=p\pm1.96\times SE$$

However the 95% confidence interval is known as

$$p=\hat p\pm1.96\times SE$$

I don't understand why $\hat p=p\pm1.96\times SE$ implies $p=\hat p\pm1.96\times SE$ with a 95% level of confidence.

So why we have $p=\hat p\pm1.96\times SE$ instead of $\hat p=p\pm1.96\times SE$? for it's not clear at all.

user45523
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    You lost me at the phrase "unbiased parameter," because in the usual statistical senses of "unbiased" and "parameter" this makes no sense. Could you explain what you mean by these terms and tell us what your symbols "$p$" and "$\hat p$" refer to? (Your use of these symbols appears inconsistent with their uses in the caption to Figure 8.3.) Have you searched our site for explanations of confidence intervals? Have you seen https://stats.stackexchange.com/questions/26450? – whuber Nov 27 '20 at 19:27
  • @whuber I edited the question. Let's forget what I said about the unbiased parameter, it's not important to my question (I think I meant unbiased estimator) and $\hat p$ is the sample proportion – user45523 Nov 27 '20 at 19:41
  • Thank you. However, your first formula ("thus we have a...") for a confidence interval is incorrect because it is impossible: it constructs a confidence interval using the value of the parameter itself, which is unknown! Your question appears to be asking what a confidence interval is -- and that is answered in many places here on CV. If that's not what you're asking, then could you edit your post to show how it is a different question? – whuber Nov 27 '20 at 20:16

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