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Math Help - Confidence Intervals-meaning?

  1. #1
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    Confidence Intervals-meaning?

    "...Confidence interval for population mean μ:
    Every time the measurements are repeated, there will be a different value for the mean of the sample. In 95% of the cases μ will be between the endpoints of the (random) confidence interval calculated from this mean, but in 5% of the cases it will not be. The actual confidence interval is calculated by entering the measured weights in the formula. Our 0.95 confidence interval becomes:
    (249.22,251.18)

    This interval has fixed endpoints, where μ might be in between (or not). There is no probability of such an event. We CANNOT say: "with probability (1 - α) the parameter μ lies in the confidence interval." We only know that by repetition in 100(1 - α) % of the cases μ will be in the calculated interval. In 100alpha % of the cases however it doesn't. And unfortunately we don't know in which of the cases this happens. That's why we say: with confidence level 100(1 - α) % μ lies in the confidence interval..." (Wikipedia)
    ======================================

    Now I don't understand the last paragraph.

    I think I am also confused about the concept of confidence interval in general. I know how to do the computations, but I just don't quite understand the meaning of it. For example, if we computed a confidence interval of (249.22,251.18) with NUMBERS (rather than random variables) as endpoints, what does it really mean?

    Also, why can't we say that
    (i)with probability (1 - α) the parameter μ lies in the confidence interval?
    Why must we say that
    (ii)by repetition in 100(1 - α) % of the cases μ will be in the calculated interval?
    I can't see any difference between (i) and (ii), aren't the two statements describing the EXACT same thing? They look absolutely the same to me...

    Thanks for clearing my doubts!
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  2. #2
    MHF Contributor matheagle's Avatar
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    Because once you have collected the data there is nothing random.

    P(249.22<\mu< 251.18) is either 0 or 1.

    P( \bar x-z\sigma/\sqrt{n} <\mu< \bar x+z\sigma/\sqrt{n} ) can be .95...

    The \bar X is a rv, \mu is not.

    This is why we make up this term confidence, it is NOT a probability interval.
    NOW a prediction interval is something different. That new observation is random
    and there is a probability associated with that event.

    BUT \mu is a constant. It may be unknown, but unless you're a Bayesian, ew, it's constant.

    The point of the first paragraph is the strong law of large numbers.
    IF WE resample over and over again, then approximately 95 percent (or what ever is your [tex]1-\alpha/math])
    wewill contain that unknown constant about that many times.

    I had my students do that with 100 samples of 100 exponentials last year.
    The intervals contained \mu 93-97 precent of the time quite often.
    It was a nice example.
    I first made them generate 100 U(0,1) then transform them to Exp( \theta) rvs.
    Then obtain one 95% CI for \mu and repeat.
    Finally count how many contained \mu.
    Last edited by matheagle; March 26th 2009 at 04:56 PM.
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  3. #3
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    Quote Originally Posted by matheagle View Post
    Because once you have collected the data there is nothing random.

    P(249.22<\mu< 251.18) is either 0 or 1.

    P( \bar x-z\sigma/\sqrt{n} <\mu< \bar x+z\sigma/\sqrt{n} ) can be .95...

    The \bar X is a rv, \mu is not.

    This is why we make up this term confidence, it is NOT a probability interval.
    NOW a prediction interval is something different. That new observation is random
    and there is a probability associated with that event.

    BUT \mu is a constant. It may be unknown, but unless you're a Bayesian, ew, it's constant.

    The point of the first paragraph is the strong law of large numbers.
    IF WE resample over and over again, then approximately 95 percent (or what ever is your [tex]1-\alpha/math])
    wewill contain that unknown constant about that many times.

    I had my students do that with 100 samples of 100 exponentials last year.
    The intervals contained \mu 93-97 precent of the time quite often.
    It was a nice example.
    I first made them generate 100 U(0,1) then transform them to Exp( \theta) rvs.
    Then obtain one 95% CI for \mu and repeat.
    Finally count how many contained \mu.
    But confidence interval is defined in terms of PROBABILITIES, right?
    http://en.wikipedia.org/wiki/Confide..._for_inference

    By definition, we set that probability equal to e.g. 95%
    Directly translating this into words: with probability 95% the parameter "theta" lies in the confidence interval, right???
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  4. #4
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by kingwinner View Post
    But confidence interval is defined in terms of PROBABILITIES, right?
    Confidence interval - Wikipedia, the free encyclopedia

    By definition, we set that probability equal to e.g. 95%
    Directly translating this into words: with probability 95% the parameter "theta" lies in the confidence interval, right???
    YES that interval has 1- \alpha probability.

    SINCE U(X) and V(X) are random variables.

    BUT 249.22, 251.18 and \mu are NOT random variables.
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    Quote Originally Posted by matheagle View Post
    YES that interval has 1- \alpha probability.

    SINCE U(X) and V(X) are random variables.

    BUT 249.22, 251.18 and \mu are NOT random variables.
    OK, for 249.22, 251.18, the interval is fixed.
    But, in general for confidence interval (CI) the endpoints vary from sample to sample. The endpoints of CI are random variables.
    So in this context, why is it wrong to say that "with probability (1 - α) the parameter μ lies in the confidence interval? " which means the same thing as ?
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  6. #6
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by kingwinner View Post
    OK, for 249.22, 251.18, the interval is fixed.
    But, in general for confidence interval (CI) the endpoints vary from sample to sample. The endpoints of CI are random variables.
    So in this context, why is it wrong to say that "with probability (1 - α) the parameter μ lies in the confidence interval? " which means the same thing as ?


    You can say that prior to collecting the data.
    BUT once you have, the interval either contains the parameter or it doesn't.
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  7. #7
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    Quote Originally Posted by matheagle View Post
    You can say that prior to collecting the data.
    BUT once you have, the interval either contains the parameter or it doesn't.
    OK!

    I have another question:
    Suppose we construct the confidence interval and compute it based on some observed data, and the CI for μ is found to be (249.22,251.18), with the endpoints being specific numbers rather than random variables. (this is typical of what most problems in Wackerly is asking for, i.e. computing a specific CI based on some observed sample, and we end up with a particular CI with specific numbers) Now, after the computations, what is the actual meaning of this particular CI? What is the significance of (249.22,251.18)?

    Thank you for explaining!
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