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I have a question below :

The 99% confidence interval is always smaller than the 95% confidence interval and the answer for this is FALSE.

Can someone explain to me this? Also can you explain through a diagram?

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  • $\begingroup$ If you want to have your random variable inside an interval with 99% probability, the interval has to be at least as large as the "smallest" interval such that the r.v. is in that interval with 95% probability $\endgroup$
    – Maximilian Janisch
    Nov 3, 2019 at 16:13
  • $\begingroup$ See this graphic $\endgroup$
    – Maximilian Janisch
    Nov 3, 2019 at 16:13
  • $\begingroup$ I suppose this should have a self-study tag... But wouldn't a couple of examples convince you that the 99% ci is wider than the 95% ci? $\endgroup$
    – Sal Mangiafico
    Nov 3, 2019 at 16:13

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Not a diagram, but an analogy: If you have a net that catches the magical goldfish with probability 95%, then to make the net even better (catching rate of 99%) you have to make it larger, not smaller.

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I don't have a diagram but ...

If you want to be 99% sure of something, you need a wider interval than if you only want to be 95% sure of it.

The actual definition of confidence interval is a bit tricky and I don't want to confuse things further, so don't take the following too far (it's more an analogy) but suppose you had 1000 people and their weights. Now, if you want an interval that include 950 of them, it will be narrower than one that includes 990 of them.

So the statement in your question is just the reverse of what is actually the case and, therefore, it is false.

EDIT: Let me stress that this is NOT technically correct, I am just trying to get at the idea of "wider" and "narrower".

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  • $\begingroup$ I think your analogy is a little bit misleading, since it suggests, that the confidence interval tries to catch the many observations rather than a single entity - the true parameter. $\endgroup$
    – ghlavin
    Nov 3, 2019 at 17:01
  • $\begingroup$ @ghlavin I recognize that .... but, like I said, I wanted to keep it as simple as possible given the question. $\endgroup$
    – Peter Flom
    Nov 3, 2019 at 18:52

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