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To study a certain characteristic about a population of people we take a sample of $100$ individuals. The $80$ percent confidence interval for the mean is $(0.9,1.1)$.

Part I: Find the sample mean and standard deviation (easy). $\bar x = 1$ and $\hat \sigma = 0.7752$

Part II: What is the probability that $(0.9,1.1)$ doesn't contain the mean.

What about Part II? Is it just $1 - 0.8 = 0.2$, or is it a trick question?

George
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    Seems right to me, but I'm only about 80% confident in my answer :P – Patrick Stevens Aug 22 '15 at 12:00
  • @PatrickStevens it seems to me that confusion is normal in such situations – George Aug 22 '15 at 12:03
  • Maybe they are after a technical point: There is a standard abuse of language involved here, discussed in (eg) https://en.wikipedia.org/wiki/Confidence_interval . Once the interval is chosen, there is no probability involved. The mean is either in it or it isn't. The "$80%$" means that the method of choosing the band results in a band which will contain the mean "$80%$" of the time. – lulu Aug 22 '15 at 12:04
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    @George this question is just mean. – Patrick Stevens Aug 22 '15 at 12:04
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    @George Do you see the possible misunderstanding here? To take a clearer instance: if you roll a fair die there is a $\frac 16$ chance of getting a $6$. So, you roll it and you get a $4$. This does not mean that there is a $\frac 16$ chance that $4=6$. – lulu Aug 22 '15 at 12:38
  • i now see what you mean @lulu – George Aug 22 '15 at 12:40

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Part 2 isn't a trick question. The answer is 0.2. The trick question which tends to be asked in these situations is "what is the probability $\mu$ lies in the confidence interval?" Then the answer is "either $1$ or $0$, depending on whether it does or does not."

David Quinn
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  • thank you, but how does the question you suggested differ from that in my post? – George Aug 22 '15 at 12:34
  • or what is the kind of situation that you are supposing so that this question means something else – George Aug 22 '15 at 12:36
  • It's all in the wording. In your question, they are referring to the confidence interval as a variable, and the mean is fixed, which is correct. In the trick question, they are implying that the mean is a variable, which it is not. – David Quinn Aug 22 '15 at 12:39