 # 100 random samples are taken to estimates the mean mu. A 95% confidence interval on the mean is 0.49 <= mu <= 0.82. Consider the following statement: There is a 95% chance that mu is between 0.49 and 0.82. Is the statement correct? Explain your answer. Terry Briggs 2022-09-12 Answered
100 random samples are taken to estimates the mean $\mu$. A 95% confidence interval on the mean is $0.49\le \mu \le 0.82$. Consider the following statement:
There is a 95% chance that $\mu$ is between 0.49 and 0.82.
Is the statement correct? Explain your answer.
I suppose the statement is wrong and the right one must be: There is a 95% chance that $\mu$ is actually in some interval that are found.For example, by researching, I found 100 different intervals and 95 of them contains $\mu$ in average.
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Step 1
$\mu$ for the population either is or is not in the interval - it's a binary thing, there is no "chance" of it being anywhere. On the other hand, what this 95% confidence basically means is that, if you were to generate many, many samples each with their own mean, and used those sample means $\overline{x}$ to generate likewise 95% confidence intervals, about 95% of those generated intervals would in fact contain the population mean $\mu$.
For any particular interval, $\mu$ is either in it or it's not. There is no chance of it, it's only a single interval after all. But if you were to generate many confidence intervals, on average, you would expect about 95% of them to contain $\mu$.
Step 2
Or phrased differently yet again, from a huge bunch of confidence intervals generated this way, you could uniformly randomly pick one interval containing $\mu$ with 95% probability. The chosen interval either does or doesn't. But you'd have an about 95% chance at picking such an interval.

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