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jlimahaverford
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You had me at the first paragraph of your reconciliation and lost me in the second. Let me try to be clear.

When we say we are 95% confident that the parameter is in the interval we are (as you said) not saying that the probability that the parameter is in the interval is 95%, but that the probability that this procedure yields an interval containing the parameter is 95%.

The distribution that this probability is coming from is the sampling distribution. 95% of samples will lead to an interval containing the (completely fixed) parameter.

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Everything about the parameter is deterministic. However, when you relate it to a random variable you can make probabilistic statements that are based on the distribution of that random variable.

The sample you draw is a random variable. Therefore the confidence interval you get is a random variable. So the answer to the question "Will the parameter be in the interval?" is a random variable. But the answer to "is the parameter in this interval?" is not.

To make an analogy. If I produce a coin that is heads 95% and tails 5%. And tell you it is 95-5 but I don't tell you which side is more likely. The following statement is true "If I flip the coin the probability it will land on the face it is biased toward is 95%." Once you flip the coin, the following is false "There is a 95% chance it is biased toward this face." You could declare "I say, with 95% confidence that the coin is biased to this face" because the word confidence in this field means precisely that!

You had me at the first paragraph of your reconciliation and lost me in the second. Let me try to be clear.

When we say we are 95% confident that the parameter is in the interval we are (as you said) not saying that the probability that the parameter is in the interval is 95%, but that the probability that this procedure yields an interval containing the parameter is 95%.

The distribution that this probability is coming from is the sampling distribution. 95% of samples will lead to an interval containing the (completely fixed) parameter.

You had me at the first paragraph of your reconciliation and lost me in the second. Let me try to be clear.

When we say we are 95% confident that the parameter is in the interval we are (as you said) not saying that the probability that the parameter is in the interval is 95%, but that the probability that this procedure yields an interval containing the parameter is 95%.

The distribution that this probability is coming from is the sampling distribution. 95% of samples will lead to an interval containing the (completely fixed) parameter.

edit

Everything about the parameter is deterministic. However, when you relate it to a random variable you can make probabilistic statements that are based on the distribution of that random variable.

The sample you draw is a random variable. Therefore the confidence interval you get is a random variable. So the answer to the question "Will the parameter be in the interval?" is a random variable. But the answer to "is the parameter in this interval?" is not.

To make an analogy. If I produce a coin that is heads 95% and tails 5%. And tell you it is 95-5 but I don't tell you which side is more likely. The following statement is true "If I flip the coin the probability it will land on the face it is biased toward is 95%." Once you flip the coin, the following is false "There is a 95% chance it is biased toward this face." You could declare "I say, with 95% confidence that the coin is biased to this face" because the word confidence in this field means precisely that!

Source Link
jlimahaverford
  • 3.6k
  • 12
  • 23

You had me at the first paragraph of your reconciliation and lost me in the second. Let me try to be clear.

When we say we are 95% confident that the parameter is in the interval we are (as you said) not saying that the probability that the parameter is in the interval is 95%, but that the probability that this procedure yields an interval containing the parameter is 95%.

The distribution that this probability is coming from is the sampling distribution. 95% of samples will lead to an interval containing the (completely fixed) parameter.