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Confidence Intervals for the mean, and One confidence interval within another.

  1. May 15, 2012 #1
    1. The problem statement, all variables and given/known data
    1a) Data x1, x2, ...xn are modelled as observations of independent random variables each having the Gaussian distribution with unknown mean  μ and unknown variance σ2.
    Suppose that n = 10, Ʃxi = 5 and Ʃxi2= 42:5.
    Look up the value of the 95% percentile of a t- distribution with appropriate degrees of freedom, and hence determine numerically a 90% confidence
    interval for μ .
    Now suppose that  instead of σ2 being unknown, is known to be equal to 4. Look up the value of the 95% percentile of the standard normal distribution and hence determine numerically a 90% confidence
    interval for μ.

    1b)Continue to assume the variance is known to equal 4, and consider hypothetical repetitions of the experiment from which the data is generated. Suppose that new intervals are to be constructed from the new data using the two methods. What is the frequency with which the interval generated by
    the first method contains the true value of the mean? What is the frequency with which the interval generated by the second method contains the true value of the mean?
    Is the interval constructed by the first method, still a 90% con dence interval, even though you know the value of σ2.?

    1c)Let L1 be a random variable that corresponds to the length of the interval constructed by the first method, and L2 be a random variable that corresponds to the length of the interval constructed by the second method. By considering the distribution of L1/L2 determine how frequently, in repetitions of the experiment, the second interval is contained within the first.


    3. The attempt at a solution
    1a) For the first method I used fishers theorem to say that xbar - μ / √(S2/n) has a t distribution of n-1 degrees of freedom where S2 is the sample variance. Putting the numbers in, and rearranging I get (-.722,1.722) for the 90% CI.
    For the second method I use xbar - μ / √(σ2/n) has standard normal distribution. Putting the numbers in and rearranging I get (-0.540, 1.540) as the 90% CI.

    1b) In hypothetical repetitions, arent the frequencies for both just 90%, since we fixed them this way!?!?
    And surely knowing the variance doesnt affect the first method, because the point of using the t-distribution for Gaussian Random variables to estimate μ is that they dont depend on variance?

    1c) This is where I get stuck..
    for L1, firstly the interval in the first case is;
    ( -t.95,n-1√(S2/n) + XBAR, t.95,n-1√(S2/n) + Xbar) .
    So L1 = 2t.95,n-1√(S2

    And for L2, the interval in the second case is;
    (-Z.95√(σ2/n) + XBAR, Z.95√(σ2/n) + XBAR)

    So L2 = 2Z.95√(σ2/n)

    So L1/L2 = t.95,n-1√(Ʃ(Xi-XBAR)2/2(n-1)Z.95
    By using variance is 4.

    But I'm not sure what to do now, or if the above is right.
    Thanks for any help
     
  2. jcsd
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