MATH235 Week 9 - Workshop problems

Show that if $X_1,X_2,\mathrm{\dots },X_n$ are iid $X_i\sim N(\theta ,1)$ the deviance function reduces to

$$D(\theta )=n{(\overline{x}-\theta )}^2.$$

If $n=20$ and $\overline{x}=6.5$ show that the limits of a $95\%$ confidence interval for $\theta$ based on the deviance are solutions to the equation

$$20{(6.5-\theta )}^2=3.84,$$

and hence find the interval.

The figure below shows the deviance function, $D(\theta )$, associated with a set of data $x_1,\mathrm{\dots },x_n$ which are modelled as a random sample from a probability density function $f(x|\theta )$. By annotating the sketch, obtain a 95% confidence interval for $\theta$.

Unnumbered Figure: Link

Deviance for a sample of data from $f(x|\theta )$.