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Math Help - Probability an Average is Below 25 if 3 Samples are Taken from a Normal Distribution

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    Probability an Average is Below 25 if 3 Samples are Taken from a Normal Distribution

    Three independent samples are taken from the normal random variable of part B (μ = 60, σ = 20). Determine the probability that the average of the three samples is less than 25.

    I'm reviewing for the FE exam, and this one showed up. I simply have no clue where to begin. The hint says to look at Central Limit Theorem, which shows what should happen for extremely large n. I don't understand how that helps, though, since 3 is not extremely large. I've taken an introductory course on probability, so I know some of the basics (such as distributions, means, standard deviations, look up tables, etc.), but this type of problem never surfaced, nor was it taught. Compounding the issue, I don't even know what to google to find related information -- a name for this type of problem (you can probably tell by my bloated title).

    Feel free to give a solution or a nudge in the right direction
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  2. #2
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    Quote Originally Posted by 1005 View Post
    Three independent samples are taken from the normal random variable of part B (μ = 60, σ = 20). Determine the probability that the average of the three samples is less than 25.

    I'm reviewing for the FE exam, and this one showed up. I simply have no clue where to begin. The hint says to look at Central Limit Theorem, which shows what should happen for extremely large n. I don't understand how that helps, though, since 3 is not extremely large. I've taken an introductory course on probability, so I know some of the basics (such as distributions, means, standard deviations, look up tables, etc.), but this type of problem never surfaced, nor was it taught. Compounding the issue, I don't even know what to google to find related information -- a name for this type of problem (you can probably tell by my bloated title).

    Feel free to give a solution or a nudge in the right direction
    The mean of a sample of size 3 taken from N(\mu,\sigma^2) has distribution N(\mu,\sigma^2/3)

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    The mean of a sample of size 3 taken from N(\mu,\sigma^2) has distribution N(\mu,\sigma^2/3)

    CB
    Ok, so I now have a new probability density function -- but of what? Does integrating it from -infinity up to x represent the probability that the sum of X_1 X_2 and X_3 equals x, meaning my answer would be to integrate from -infinity to n*average => 3*25 = 75, or do I integrate the new PDF up to 25. If it is the former, great, but if it is the latter, please explain why. That would blow my mind.

    And just a curiosity: does the central limit theorem result from the fact that if Z = X + Y then Z_pdf = X_pdf*Y_pdf where * is the convolution operator?

    Thank you,
    1005
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    Quote Originally Posted by 1005 View Post
    Ok, so I now have a new probability density function -- but of what? Does integrating it from -infinity up to x represent the probability that the sum of X_1 X_2 and X_3 equals x, meaning my answer would be to integrate from -infinity to n*average => 3*25 = 75, or do I integrate the new PDF up to 25. If it is the former, great, but if it is the latter, please explain why. That would blow my mind.

    And just a curiosity: does the central limit theorem result from the fact that if Z = X + Y then Z_pdf = X_pdf*Y_pdf where * is the convolution operator? Mr F says: Google

    Thank you,
    1005
    \overline{X} ~ \displaystyle N\left(\mu, \frac{\sigma^2}{3}\right) and you calculate \Pr(\overline{X} < 25) in the usual way that you have been taught (tables, calculator, etc.)
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    Quote Originally Posted by mr fantastic View Post
    \overline{X} ~ \displaystyle N\left(\mu, \frac{\sigma^2}{3}\right) and you calculate \Pr(\overline{X} < 25) in the usual way that you have been taught (tables, calculator, etc.)
    Why is it 25 instead of 75? Would PR(x<25) not represent the probability that the sum of the three independent samples equals 25 or less, so if I were to find the probability that the average is below 25, I would actually find p(x<75) since 75/3 = 25?
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    Quote Originally Posted by 1005 View Post
    Why is it 25 instead of 75? Would PR(x<25) not represent the probability that the sum of the three independent samples equals 25 or less, so if I were to find the probability that the average is below 25, I would actually find p(x<75) since 75/3 = 25?
    You're told "the average of the three samples is less than 25" and \overline{X} is the random variable 'mean of sample', already given to you by CaptainBlack.
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