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Since $H_0$ and $H_1$ are contradictory, one (and only one) of them must be true.
\begin{itemize}
\item If $H_0$ is true and we fail to reject it, we've done the right thing.
\item Similarly, if $H_0$ is false and we reject it, we've also done the right thing.
\item If, on the other hand, $H_0$ is true and we reject it, we've made a mistake called a \textbf{Type I Error}.
\item If $H_0$ is false and we fail to reject it, we've made a mistake called a \textbf{Type II Error}.
\end{itemize}
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Medical test analogy: $H_0$ is that you don't have a disease
\begin{itemize}
\item \textbf{Type I Error:} False positive
\item \textbf{Type II Error:} False negative (more serious)
\end{itemize}
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Trial analogy: $H_0$ is that you're innocent (innocent until proven guilty)
\begin{itemize}
\item \textbf{Type I Error:} Convict an innocent person (more serious)
\item \textbf{Type II Error:} Let a guilty person go free
\end{itemize}
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\paragraph{Summary:} Type I means incorrectly rejecting $H_0$; Type II means incorrectly NOT rejecting it.
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\begin{example}{Error Types}
Suppose Frank tests his rock-climbing equipment, and $H_0$ is that his equipment is safe.\\
\paragraph{Type I Error:} Thinking his equipment isn't safe when it actually is.
\paragraph{Type II Error:} Thinking his equipment is safe when it actually isn't. \textbf{(more serious)}
\end{example}
\begin{example}{Error Types}
The victim of a car accident is brought to the emergency room, and $H_0$ is that she is alive when she comes in.\\
\paragraph{Type I Error:} Thinking she is dead when she is actually alive. \textbf{(more serious)}
\paragraph{Type II Error:} Thinking she is alive when she is actually dead.
\end{example}
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