Confidence Intervals - Doubts on InterpretationsSuppose we use a sample mean X ¯ to construct a 95% confidence interval [a,b].I was told that it is incorrect to say that there is a 95% probability that the population mean lies between a and b. Because the population mean is a constant and not a random variable. The probability that a constant falls within any given range is either 0 or 1.However, from the textbook it is said that we expect 95% of the confidence intervals to include the population mean.If 95% of the confidence intervals are expected to include the population mean, then each confidence interval has 95% probability to include the population mean. Therefore I think it is correct to say there is a 95% probability that the population mean lies between a and b.Where did I make mistakes?

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Confidence Intervals - Doubts on InterpretationsSuppose we use a sample mean             X      ¯       to construct a 95% confidence interval [a,b].I was told that it is incorrect to say that there is a 95% probability that the population mean lies between a and b. Because the population mean is a constant and not a random variable. The probability that a constant falls within any given range is either 0 or 1.However, from the textbook it is said that we expect 95% of the confidence intervals to include the population mean.If 95% of the confidence intervals are expected to include the population mean, then each confidence interval has 95% probability to include the population mean. Therefore I think it is correct to say there is a 95% probability that the population mean lies between a and b.Where did I make mistakes?

Miya Swanson · Accepted Answer

Step 1You&#039;re right that the interpretation of 95% confidence is as follows: if you collected many samples, and from each one generated a different confidence interval, then 95% of the intervals generated would capture the true mean μ inside them.So, why is the other interpretation incorrect? It&#039;s tricky to see, in part because of the use of the placeholders a,b. Let&#039;s make this concrete and suppose your 95% interval for   μ is specifically [2.6,8.3]. If someone asked you what the probability that   μ was in [2.6,8.3] was, your answer should be: &quot;That question makes no sense.&quot; ** Remember that   μ is a fixed number, and you just don&#039;t have the privilege of knowing which specific number it is. You would never ask something like the probability that 2 is in the interval [2.6,8.3], or the probability that   π  =  3.14159... is in the interval [2.6,8.3]. Either the numbers are in the interval, or they aren&#039;t.Step 2That&#039;s the issue with the interpretation at the top of the question. Once you actually commit to a sample and its resulting confidence interval, it either has   μ inside it, or it doesn&#039;t. But the endpoints of the interval are no longer random variables (because you&#039;ve realized them into actual numbers), and   μ was never a random variable in the first place. It&#039;s easy to obfuscate this nuance when you use placeholders like a,b for the endpoints of the interval.

I was told that it is incorrect to say that there is a 95% probability that the population mean lies between a and b. Because the population mean is a constant and not a random variable. The probability that a constant falls within any given range is either 0 or 1.

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