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Math Help - Central Limit Theorem

  1. #1
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    Central Limit Theorem

    Hi,

    I have a problem regarding CLT:

    "The lengths of mass produced nails are normally distributed with a mean of 3cm and variance of 0.01 cm^2. 16 nails are randomly chosen and laid end to end.

    What is the probability that its length exceeds 48.5cm?"

    This is my working out, I'm not sure what i am doing wrong:

    Let  T_{16} = X_1 + X_2 + .... + X_{16}, where X_i are identically and independently distributed variables.

    So, each of X_i ~  N(3,0.01^2)
    Thus, T_n ~ N(48, 0.04^2)

    We need to find P(T_{16} > 48.5). So
     P(T_{16} > 48.5) = 1 - P( T_{16} < 48.5) = 1 - P( Z < (48.5 - 48) / (0.04) ) = 1 - P( Z < 12.5)

    Then, I'm stuck here. Any help will be greatly appreciated thanks
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  2. #2
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    Quote Originally Posted by shinn View Post
    Hi,

    I have a problem regarding CLT:

    "The lengths of mass produced nails are normally distributed with a mean of 3cm and variance of 0.01 cm^2. 16 nails are randomly chosen and laid end to end.

    What is the probability that its length exceeds 48.5cm?"

    This is my working out, I'm not sure what i am doing wrong:

    Let  T_{16} = X_1 + X_2 + .... + X_{16}, where X_i are identically and independently distributed variables.

    So, each of X_i ~  N(3,0.01^2)
    Thus, T_n ~ N(48, 0.04^2)

    We need to find P(T_{16} > 48.5). So
     P(T_{16} > 48.5) = 1 - P( T_{16} < 48.5) = 1 - P( Z < (48.5 - 48) / (0.04) ) = 1 - P( Z < 12.5)

    Then, I'm stuck here. Any help will be greatly appreciated thanks
    The sum follows a normal distribution.

    The mean of the sum is 48. The variance of the sum is (16)(0.01) = 0.16 therefore the sd of the sum is 0.4.
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  3. #3
    MHF Contributor matheagle's Avatar
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    There are a few errors here.
    1. There isn't a CLT here, the sample is quite small.
    So no limit theorem whatsoever.
    BUT you are told that the 16 X's are normal, hence their sum is normal.
    2. The variance is .01 not  (.01)^2.
    3. The st deviation of T_n is also wrong, it is .4.

     P(T_{16} > 48.5) = 1 - P( T_{16} < 48.5) = 1 - P( Z < (48.5 - 48) / .4 ) = 1 - P( Z < 1.25)
    is what you want to look up.
    Last edited by matheagle; June 17th 2009 at 10:58 PM.
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  4. #4
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    thanks for the above replies.
    Last edited by mr fantastic; June 17th 2009 at 11:36 PM. Reason: Question moved to new thread
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