# Finding one sided confidence intervals

• eruth
In summary, the conversation discusses a problem involving a sample of iid random variables with a known constant and unknown parameter. The first part involves showing that P=X(n)/θ is a pivot for θ, and the second part involves finding the right-sided 95% confidence interval for θ by calculating the upper and lower quantiles of the distribution of P and constructing the interval as [X(n)/Q(αU), X(n)/Q(αL)].
eruth
1. Let X1,...,Xn be iid with cdf Fθ, where Fθ(x) = (x/θ)^β for x in [0, θ]. Here β>0 is a known constant and θ>0 is an unknown parameter. Let X(n)= max (X1,...,Xn). f(x|θ)=nβ(x^(nβ-1))/(θ^(nβ)) when x is in [0, θ].
Part one was to show that P= X(n)/θ is a pivot for θ. Which I did by showing its distribution doesn't depend on any unknowns.
Part two is to find the right-sided 95% confidence interval for θ.

2.
I know to make 1-alpha= Pr(Q(alphaL) ≤ P ≤ Q(alphaU))
= Pr(X(n)/Q(alphaU) ≤θ≤ X(n)/Q(alphaL))

But then I'm not sure where to go from there. I know it needs to be in the form of [L(X1,...,Xn), c] where c is some constant. I know I need to make alphaL and alphaU either .05 and 1 or 0 and .95...
Thanks so much any help would be great, as the next problem builds on this one.

The solution is to calculate the upper and lower quantiles, Q(αL) and Q(αU), of the distribution of P. From this, you can construct the 95% confidence interval for θ as [X(n)/Q(αU), X(n)/Q(αL)].

## What is a one-sided confidence interval?

A one-sided confidence interval is a type of statistical estimate that provides a range of values within which the true value of a population parameter is likely to lie with a certain level of confidence. Unlike a two-sided confidence interval, which provides a range of values above and below a point estimate, a one-sided interval only provides a range of values on one side of the estimate.

## When should I use a one-sided confidence interval?

A one-sided confidence interval is typically used when you have a specific hypothesis or direction of effect in mind. For example, if you are testing a new drug and you are only interested in whether it is better than the current standard treatment, a one-sided interval can be used to determine if the new drug has a higher success rate than the standard treatment.

## How do I calculate a one-sided confidence interval?

To calculate a one-sided confidence interval, you will need to know the point estimate (such as a mean or proportion), the standard error of the estimate, and the desired level of confidence. The formula for a one-sided interval will vary depending on the type of estimate being used, but it typically involves multiplying the standard error by a critical value from a t-distribution or z-distribution.

## What is the difference between a one-sided and two-sided confidence interval?

The main difference between a one-sided and two-sided confidence interval is the direction of the range of values provided. A one-sided interval only provides a range of values on one side of the estimate, while a two-sided interval provides a range of values both above and below the estimate. Additionally, a two-sided interval is typically used when there is no specific hypothesis or direction of effect in mind.

## How can I interpret a one-sided confidence interval?

The interpretation of a one-sided confidence interval is similar to that of a two-sided interval. It provides a range of values within which the true value of a population parameter is likely to lie with a certain level of confidence. The only difference is that a one-sided interval only provides this range on one side of the estimate, rather than both sides.

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