Confidence intervals (CI) are useful tools for quantifying the uncertainty associated with a point estimate. The amount of uncertainty depends on a associated with du
What does it mean to say "...the $100\times(1-\alpha)\%$ CI for $\theta =(\underline{\widehat{\theta}},\overline{\widehat{\theta}})$..."
For $\alpha=0.05,$ a $95\%$ CI on $\theta$ this means
If an experiment were run $i$ times (each run producing a distinct data set)
The true (but unknown) value of $\theta$ would be in the interval $(\underline {\widehat {\theta_{i}}},\overline{\widehat{\theta_{i}}})$ in $95\%$ of those runs.
A $95\%$ CI on $\theta$
For each experiment the probability $P(\underline{\widehat{\theta_{i}}} <\theta<\overline{\widehat{\theta_{i}}})$ is either $1$ or $0$
This is because $\theta$ is either in the interval, or it isn't
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