# Interval Estimation

A $100∗p$% likelihood Interval is ${θ:R(θ)≥p}$

- where $R(θ)$ is the Relative Likelihood Function

For instance, if we want to the 50% likelihood interval, we would find ${θ:R(θ)≥0.5}$. When we solve $θ$, since this in an inequality, we will get a range $[a,b]$, which gives us our interval.

#### Example

Example: Suppose a coin is tossed $200$ times with $θ=P(H)$. Suppose that we observe $y=80$ successes. Find the 10% ($p=0.1$) likelihood interval for $θ$.

We first find $θ$, which we know is $ny $ since this is data follows a Binomial Distribution. $θ=20080 =0.4$ Find all $θ$ s.t. $R(θ)≥0.1$ $R(θ)=L(θ)L(θ) ≥0.1$ $(80200 )0.4_{80}0.6_{120}(80200 )θ_{80}(1−θ)_{120} ≥0.1$

By the magic of computers, we find the interval $θ∈[0.33,0.47]$, which is at a minimum 10% as likely as $θ=0.4$.

Using this interval, we can then conclude for example that $θ=0.36$ is plausible, since $R(0.36)≥0.1$.