# How Effective Is Inverting a Test Statistic for New Confidence Intervals?

• ghostyc
In summary, the conversation discusses using a powerful method to construct a confidence interval for an unknown parameter, particularly the mean of a normal distribution with a known variance. The project focuses on understanding this method and constructing one-sided and two-sided confidence intervals. There is a prize for those who can create a new confidence interval using this method. The conversation also mentions a possible connection to the pivotal interval and asks for related resources.
ghostyc
A very powerful method of construction of a confidence interval/set for an unknown parameter, the mean of a normal distribution for example, is to invert a family of hypotheses tests about that parameter. This project is firstly to demonstrate the understanding of this method by using it to construct the usual one-sided and two-sided confidence intervals for the unknown mean of a normal distribution with a known variance. But the prize is given if a new confidence interval can be constructed by using this method for the unknown mean of a normal distribution with a known variance.

Here, `A new confidence interval' means a confidence interval that is
different from the usual one-sided and two-sided confidence intervals
for the unknown mean of a normal distribution with a known variance.

Hi,

Can someone kindly explain a bit on the 'red' part.
I have tried to search for it, but nothing much related seem to come out.
You may not tell me what it is, all I need is some related resource ( section in a book, or a website, or an article? )

Thanks.

Hi there,

I finally see something here.

Is there any possibility that the 'red' part is referering to the pivotal interval?

Thanks.

I am a bit desperate now.

slowly giving up on this one

## 1. What is the concept of inverting a test statistic?

Inverting a test statistic refers to the process of finding the values of the independent variable that would result in a certain value of the test statistic, based on a given significance level. This is often used in hypothesis testing to determine the critical region and make decisions about rejecting or failing to reject the null hypothesis.

## 2. How is inverting a test statistic related to the p-value?

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. Inverting a test statistic involves finding the values of the independent variable that would result in a specific value of the test statistic, which can then be used to calculate the corresponding p-value. This allows for comparison to the chosen significance level to determine if the null hypothesis should be rejected or not.

## 3. What are the steps involved in inverting a test statistic?

The steps involved in inverting a test statistic include: identifying the null and alternative hypotheses, choosing a significance level, calculating the test statistic, finding the critical region by inverting the test statistic, and making a decision about the null hypothesis based on the comparison of the test statistic to the critical region.

## 4. Can inverting a test statistic be used for any type of hypothesis test?

Yes, inverting a test statistic can be used for any type of hypothesis test, including one-sample, two-sample, and ANOVA tests. It is a fundamental concept in hypothesis testing and is applicable to various types of statistical analyses.

## 5. Are there any limitations to inverting a test statistic?

While inverting a test statistic is a useful tool in hypothesis testing, it is important to note that it relies on certain assumptions, such as the data being normally distributed and the sample size being sufficiently large. Additionally, inverting a test statistic may not always provide a unique solution, leading to potential issues with interpreting the results. It is important to carefully consider the assumptions and limitations before using inverting a test statistic in statistical analyses.

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