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## Homework Statement

1a) Data x1, x2, ...xn are modeled as observations of independent random variables each having the Gaussian distribution with unknown mean μ and unknown variance σ

^{2}.

Suppose that n = 10, Ʃxi = 5 and Ʃxi

^{2}= 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 σ

^{2}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% condence interval, even though you know the value of σ

^{2}.?

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 thesecond method. By considering the distribution of L1/L2 determine how frequently, in repetitions of the experiment, the second interval is contained within the first.

## The Attempt at a Solution

1a) For the first method I used fishers theorem to say that xbar - μ / √(S

^{2/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 doesn't affect the first method, because the point of using the t-distribution for Gaussian Random variables to estimate μ is that they don't 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}