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The Meaning of a 95% T Confidence Interval for the Mean

  1. Dec 31, 2014 #1
    1. The problem statement, all variables and given/known data
    A diet pill is given to 9 subjects over six weeks. The average difference in weight (follow up - baseline) is -2 pounds. What would the standard deviation have to be for the 95% T confidence interval to lie entirely below 0?

    ANSWER: Around 2.6 pounds or less

    Refer to the previous question. The interval would up being [-3.5, -0.5] pounds. What can be said about the population mean weight loss at 95% confidence?

    A: We can not rule out the possibility of no mean weight loss at 95% confidence.

    B: There is support of mean weight gain at 95% confidence.

    C: There is support at 95% confidence of mean weight loss.

    D: We can not rule out the possibility of mean weight gain at 95% confidence.

    2. Relevant equations


    3. The attempt at a solution
    I have three attempts to answer this question.

    On my first attempt, I said D, thinking that since 5% of intervals do not contain the population mean, there is a chance that the population mean could be positive (and there a mean weight gain). But I got the wrong answer, and I don't understand why.

    On my second attempt, I thought that "weight gain" wasn't necessarily going to occur but "no mean weight loss" was, so I chose A, and I still got the wrong answer.

    I am surprised that confidence intervals will guarantee a support (as the remaining answers of the question suggest), so I now think the answer is C. B sounds ridiculous, because it doesn't make sense why there is support for mean weight gain at 95% confidence.

    Can someone help me understanding the question? I hope I finally get this question right.

    Thank you.
     
    Last edited: Dec 31, 2014
  2. jcsd
  3. Dec 31, 2014 #2

    Ray Vickson

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    For D: it said "We can not rule out the possibility of mean weight gain at 95% confidence". Of course we cannot rule out the possibility of mean gain altogether, but with 95% confidence, we can. In other words, 95% confidence is not 100% confidence.

    Look at your other answers in the same way.
     
  4. Dec 31, 2014 #3
    I see.

    So (theoretically), a 100% confidence interval would be the interval (-∞, +∞). But in this case, a 95% confidence interval is [-3.5, -0.5]. As you said, the important thing is that the question asks what can be said at 95% confidence.

    Since the interval does not contain any positive numbers, at 95% confidence, there is no possibility that there can be a mean weight gain (and for that matter, no mean weight loss), so D and A must be incorrect.

    Using the same positive-numbers argument, presumably, there is no support for a mean weight gain at 95% confidence, so B must also be incorrect. Therefore, C must be correct.

    How is that?

    What puzzles me still is why any confidence interval can guarantee a support within that interval. Isn't there the slightest chance that the function is 0 within the interval?

    Thank you.
     
  5. Dec 31, 2014 #4

    Ray Vickson

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    (1) A 100% confidence interval need not be the whole line; it is just an interval we are SURE contains the true mean---not just "almost sure", but absolutely sure. It could be a point (which it would be in the limit of an infinite sample size, example).
    (2) I think you have identified the true/false answers correctly.
    (3) Your statement "why any confidence interval can guarantee a support within that interval" is false: there is no such guarantee. We can only be more-or-less sure, but not absolutely, 100% sure. Every once in a while we will be wrong (about 5% of the time if we are looking at a 95% confidence interval).
     
  6. Jan 1, 2015 #5
    That makes sense. Thank you very much. I did get the right answer, by the way. :)
     
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