Appendix B: Confidence Interval and Hypothesis Test Reference Tables

Confidence Interval

Interval	\(E[Y]\) Increases with \(\theta\)?	\(q\) for Lower Bound	\(q\) for Upper Bound
two-sided	yes	\(1-\alpha/2\)	\(\alpha/2\)
	no	\(\alpha/2\)	\(1-\alpha/2\)
one-sided lower	yes	\(1-\alpha\)	\(-\)
	no	\(\alpha\)	\(-\)
one-sided upper	yes	\(-\)	\(\alpha\)
	no	\(-\)	\(1-\alpha\)

Hypothesis Test: Rejection Region(s)

Type	\(E[Y]\) Increases with \(\theta\)?	Rejection Region Boundary	Reject If…
two-tail	yes	\(y_{\rm RR,lo} = F_Y^{-1}(\alpha/2 \vert \theta_o)\)	\(y_{\rm obs} < y_{\rm RR,lo}\)
		\(y_{\rm RR,hi} = F_Y^{-1}(1-\alpha/2 \vert \theta_o)\)	\(y_{\rm obs} > y_{\rm RR,hi}\)
	no	\(y_{\rm RR,lo} = F_Y^{-1}(1-\alpha/2 \vert \theta_o)\)	\(y_{\rm obs} < y_{\rm RR,lo}\)
		\(y_{\rm RR,hi} = F_Y^{-1}(\alpha/2 \vert \theta_o)\)	\(y_{\rm obs} > y_{\rm RR,hi}\)
lower-tail	yes	\(y_{\rm RR} = F_Y^{-1}(\alpha \vert \theta_o)\)	\(y_{\rm obs} < y_{\rm RR}\)
	no	\(y_{\rm RR} = F_Y^{-1}(1-\alpha \vert \theta_o)\)	\(y_{\rm obs} > y_{\rm RR}\)
upper-tail	yes	\(y_{\rm RR} = F_Y^{-1}(1-\alpha \vert \theta_o)\)	\(y_{\rm obs} > y_{\rm RR}\)
	no	\(y_{\rm RR} = F_Y^{-1}(\alpha \vert \theta_o)\)	\(y_{\rm obs} < y_{\rm RR}\)

Hypothesis Test: p-Value and Power

Type	\(E[Y]\) Increases with \(\theta\)?	\(p\)-Value	Test Power
two-tail	yes	2 \(\times\) min[\(F_Y(y_{\rm obs} \vert \theta_o)\),	\(F_Y(y_{\rm RR,lo} \vert \theta)\) +
		\(1-F_Y(y_{\rm obs} \vert \theta_o)\)]	\(1 - F_Y(y_{\rm RR,hi} \vert \theta)\)
	no	same as above	same as above
lower-tail	yes	\(F_Y(y_{\rm obs} \vert \theta_o)\)	\(F_Y(y_{\rm RR} \vert \theta)\)
	no	\(1-F_Y(y_{\rm obs} \vert \theta_o)\)	\(1-F_Y(y_{\rm RR} \vert \theta)\)
upper-tail	yes	\(1-F_Y(y_{\rm obs} \vert \theta_o)\)	\(1-F_Y(y_{\rm RR} \vert \theta)\)
	no	\(F_Y(y_{\rm obs} \vert \theta_o)\)	\(F_Y(y_{\rm RR} \vert \theta)\)