Supposed in a population, I take a sample with a size of 1500 to build a 96% confidence interval for the true mean $\mu$. Then such an interval is assumed to range from 6.8 to 8.0. Assume the sample mean is $\overline{x}$. Then I conclude that:

if a sample with a size of 1500 were repeatedly constructed, then over a long run 96% of the confidence interval formed would contain the true mean $\mu$. (Another trivial interpretation is that I am 96% confident that the true mean will lie between 6.8 and 8.0 so I don't want to bother to mention this interpretation).

Am I correct with my above conclusion? Thank you!

  • $\begingroup$ @DavidKozak But the statistics textbook states that for the second interpretation. This is an introduction course so I think they try to make it as simple and straightforward as possible. However, I am sure when I go deeper in the Stat major, I will understand more of your comment above. So I apologize in advance for my ignorance. And of course we will never touch Bayesian Paradigm in this intro course. Thanks David! I appreciate it. $\endgroup$ – Joseph Nov 22 '17 at 3:39
  • $\begingroup$ which textbook are you using? This post will likely soon be flagged as a duplicate question and closed. I strongly urge you to click on some of the related questions and see more thorough explanations. $\endgroup$ – David Kozak Nov 22 '17 at 3:42
  • $\begingroup$ @DavidKozak Statistics: The Art and Science of Learning from Data (4th Edition) [link] (amazon.com/Statistics-Art-Science-Learning-Data/dp/0321997832/…) $\endgroup$ – Joseph Nov 22 '17 at 3:43
  • $\begingroup$ @DavidKozak can you please try to post a quick response to my question so then I can mark your answer as accepted? This way, I don't need more contribution to the topic and it can be closed. Thanks! $\endgroup$ – Joseph Nov 22 '17 at 3:45

In a frequentist paradigm we treat the true mean $\mu$ as fixed. Since we are taking a sample from the generating distribution the randomness comes from this sample. In this way we construct a confidence interval. In this way, your first interpretation is correct: if we were to sample 1500 observations from the same distribution repeatedly then a confidence level of $\alpha$ implies that the true parameter $\mu$ would fall within the confidence interval $\alpha$% of the time.

The second, seemingly more natural interpretation is incorrect from a frequentist perspective. It is more akin to the credible interval of the Bayesian paradigm. Recall that a Bayesian treats the parameter $\mu$ as random. An $\alpha$% credible interval of [a, b] suggests that the subjective (due to the observed data) probability that $\mu \in [a, b]$ is $\alpha$%.

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