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

[stukbv]

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 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.

The Attempt at a Solution

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

 

[mighty2000]

!

Hello, thank you for your post. It seems like you have made a good attempt at solving this problem. Here are some suggestions for how to proceed:

1a) Your calculations for the confidence intervals look correct. Just be sure to mention the degrees of freedom when discussing the t-distribution.

1b) You are correct that the frequencies for both methods should be 90% in hypothetical repetitions. However, for the first method, since we know the variance, we could also use the standard normal distribution instead of the t-distribution. This would result in a slightly different confidence interval. Additionally, it is important to note that even though we know the variance, the t-distribution still provides a more accurate estimate of the confidence interval compared to the standard normal distribution.

1c) Your calculations for L1 and L2 look correct. To determine the frequency with which the second interval is contained within the first, you can use the definition of the confidence interval and the properties of the t-distribution and standard normal distribution. Specifically, you can use the fact that the t-distribution with n-1 degrees of freedom approaches the standard normal distribution as n gets larger. This will give you an expression for the probability that the second interval is contained within the first, which you can then use to determine the frequency.