Interval Estimation

A % likelihood Interval is

For instance, if we want to the 50% likelihood interval, we would find . When we solve , since this in an inequality, we will get a range , which gives us our interval.

title: Plausibility
We use the [[Relative Likelihood Function]] to quanitfy plausibility.
- $R(\theta) \geq 0.5 \rightarrow$  very plausible
- $R(\theta) \geq 0.1 \rightarrow$ plausible
- $R(\theta) \lt 0.1 \rightarrow$ implausible
- $R(\theta) \lt 0.01 \rightarrow$  very implausible


Example: Suppose a coin is tossed times with . Suppose that we observe successes. Find the 10% () likelihood interval for .

We first find , which we know is since this is data follows a Binomial Distribution. Find all s.t.

By the magic of computers, we find the interval , which is at a minimum 10% as likely as .

Using this interval, we can then conclude for example that is plausible, since .