Confidence interval interpretation difficultyWhen we are dealing with 95% confidence interval we mean that if we repeat process of collecting samples of same size and calculate 95% intervals for those samples then 95% of those intervals will contain the true population parameter.Let the infinite number of intervals be represented by 100 for simplicity. Then 95 of these intervals will contain true population parameter.Suppose we got an interval at the starting of the above process (L,U). Then if I ask what is the probability that this interval (L,U) contains the true population parameter then shouldn't it be 95 / 100 = 0.95? (Because this interval (L,U) can be anyone of 100 and it would contain true population parameter of its one of those 95).But this interpretation of confidence interval is considered incorrect. Can someone explain me why is this so?

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Confidence interval interpretation difficultyWhen we are dealing with 95% confidence interval we mean that if we repeat process of collecting samples of same size and calculate 95% intervals for those samples then 95% of those intervals will contain the true population parameter.Let the infinite number of intervals be represented by 100 for simplicity. Then 95 of these intervals will contain true population parameter.Suppose we got an interval at the starting of the above process (L,U). Then if I ask what is the probability that this interval (L,U) contains the true population parameter then shouldn&#039;t it be   95      /    100  =  0.95? (Because this interval (L,U) can be anyone of 100 and it would contain true population parameter of its one of those 95).But this interpretation of confidence interval is considered incorrect. Can someone explain me why is this so?

Anabelle Hicks · Accepted Answer

Step 1For an analogy, consider the following game. Alice pays Bob five dollars to flip a fair coin. If the coin lands heads, Alice wins ten dollars; if the coin lands tails, Alice wins nothing. Let W be the random variable representing Alice&#039;s winnings. Consider the question, &quot;Did Alice win five dollars?&quot; (i.e. &quot;Is   W  =  +  5?&quot;)Now:before Bob flips the coin, we have:  P  (  W  =  +  5  )  =  P  (  W  =  −  5  )  =  0.5.So the answer is Yes with probability 0.5.But, after Bob flips it, the coin either came up heads, or it came up tails. So W is now either equal to +5, or not. The answer is now Yes either with probability 1, or probability 0.This is the case generally: the act of performing an experiment changes probabilities to certainties. Whatever likelihood we assign to an event happening or not happening beforehand, ceases to matter after the experiment has been performed, and the event either did actually happen, or did not actually happen.Step 2Similarly for your question about 95% confidence intervals. When we ask the question, &quot;Does the 95% confidence interval (L,U) contain the true population parameter?&quot; where L,U are the random variables representing the lower and upper endpoints of the interval, then before we take our sample, the answer is Yes with probability 0.95.But after we take our sample, L and U are no longer random variables, but have taken specific numerical values. Once the sample is taken and the endpoints are calculated, either (L,U) actually contains the true population parameter, or does not actually contain the true population parameter. So the probability of the answer being Yes is now either 1 (if it does contain the true parameter) or 0 (if it does not).

acapotivigl · Accepted Answer

Step 1Look at the problem this way: L and U are fixed numbers you know, since you have calculated them. They are non-random given your sample. Now the true population parameter, let&#039;s call it   θ may be unknown to you, but it is again a fixed non-random number. Step 2So you have three fixed numbers L, U and   θ and you ask: What is the probability that   θ  ∈  [  L  ,  U  ]? This is really easy to answer: Either   θ is in the interval, then the probability is one or it isn&#039;t, then the probability is zero.

Confidence interval interpretation difficulty. When we are dealing with 95% confidence interval we mean that if we repeat process of collecting samples of same size and calculate 95% intervals for those samples then 95% of those intervals will contain the true population parameter.

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