Minimum Expected Length Confidence Interval

In summary, the conversation is about proving that the Student's t based confidence interval has the minimum expected length for a given confidence level (1-a). This can be done using the Neyman-Pearson Lemma, which states that the most powerful test among all intervals with the same coverage probability (1-a) must have the smallest expected length. Since the Student's t interval is the most powerful among all intervals with the same coverage probability (1-a), it must also have the smallest expected length.
  • #1
hunraj
2
0
(Moving this to another section, because it is not for a particular course ...)

I'd greatly appreciate any direction toward establishing the result below.
I am studying statistics on my own, and found this in some online materials,
but it didn't come with a solution, and I've been a bit stuck ...

HR

Homework Statement



Given a sample [ X(1), X(2), ... X(n) ] from a Normal population with
unknown mean and variance, show that the Student's t based
confidence interval has the minimum expected length for a given
confidence level (1-a).


Homework Equations





The Attempt at a Solution

 
Last edited:
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  • #2
This result can be proved using the Neyman-Pearson Lemma. The idea is that the Student's t interval is the most powerful test among all intervals with the same coverage probability (1-a). The Neyman-Pearson Lemma states that if two tests have the same significance level (1-a) and one has greater power than the other, then the test with greater power must have a smaller expected length. Therefore, since the Student's t interval is the most powerful among all intervals with the same coverage probability (1-a), it must also have the smallest expected length.
 

1. What is a Minimum Expected Length Confidence Interval?

A Minimum Expected Length Confidence Interval (MECI) is a statistical technique used to estimate the true value of a population parameter based on a sample. It provides a range of values within which the true population parameter is likely to fall, along with a level of confidence for this estimate.

2. How is a MECI calculated?

A MECI is calculated by taking the sample mean and confidence level, and then using the formula: MECI = sample mean ± (critical value * standard error). The critical value is dependent on the sample size and desired confidence level, while the standard error is a measure of the variability of the data.

3. What is the significance of a MECI?

A MECI is significant because it helps to quantify the uncertainty in a sample and provide a more accurate estimate of the true population parameter. It also allows for comparisons between different populations or groups.

4. What are the assumptions of a MECI?

The main assumptions of a MECI include: a random and representative sample, normal distribution of the data, independence of observations, and homogeneity of variances. Violations of these assumptions can lead to inaccurate estimates and confidence intervals.

5. How is a MECI interpreted?

A MECI is typically interpreted as follows: "We are [confidence level] confident that the true population parameter falls within the range of [lower limit, upper limit]." For example, if a MECI for the mean age of a population is calculated to be 40-50 with a 95% confidence level, it can be interpreted as: "We are 95% confident that the true mean age of the population is between 40 and 50 years old."

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