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A two-sided test detects departures in either direction from the null hypothesis, for example $\theta \ne {\theta }_0$. With a continuous symmetric null statistic centred at zero, its p-value is $P_0(|T|\ge |t_{\mathrm{obs}}|)=2P_0(T\ge |t_{\mathrm{obs}}|)$. This is the usual coefficient test printed by Gaussian regression summaries, unlike a directional one-sided comparison.

For a null hypothesis $H_0$, a type I error rejects $H_0$ when it is true, while a type II error does not reject $H_0$ when the alternative is true. If a test does not reject $H_0$ on an event $A$, their conditional probabilities are $\alpha =P_0(A^c)$ and $\beta =P_1(A)$. Nonrejection does not establish the truth of the null hypothesis.

For an independent statistical sample from a gamma distribution with known shape $\lambda >0$ and unknown rate $\theta >0$, test $\theta ={\theta }_0$ against $\theta >{\theta }_0$. The alternative-to-null likelihood ratio for any larger rate ${\theta }_1$ is $({\theta }_1/{\theta }_0{)}^{n\lambda }\exp \{-({\theta }_1-{\theta }_0)S\}$, where $S={\sum }_i{X}_i$. This decreases in $S$, so the Neyman-Pearson lemma gives one common lower-tail ordering for all alternatives.

Let $G_a$ be the unit-rate gamma distribution cumulative distribution function with shape $a$. The uniformly most powerful test of exact test size $\alpha \in (0,1)$ rejects when $S<c_{\alpha }$, where $c_{\alpha }=G_{n\lambda }^{-1}(\alpha )/{\theta }_0$. Its power function is displayed above. If $n\lambda =1$, the sum has an exponential distribution, so $c_{\alpha }=-\log (1-\alpha )/{\theta }_0$ and $\pi (\theta )=1-(1-\alpha {)}^{\theta /{\theta }_0}$. Monotonicity of $G_a(\theta c_{\alpha })$ also makes this test valid and uniformly most powerful for the larger null hypothesis $\theta \le {\theta }_0$.

For two independent samples from approximately normal distributions, Welch's t-test compares their means without assuming equal unknown variances. The displayed test statistic uses sample variances $s_X^2,s_Y^2$. Under the equal-mean null hypothesis, its reference Student t-distribution has approximate statistical degrees of freedomA p-value from this approximation addresses the specified statistical error model; it cannot distinguish a material effect from an unmeasured experimenter bias when sample and experimenter are confounded.

For independent paired observations, form differences $D_i=U_i-V_i$. Assume the differences are identically distributed, continuous and symmetric about a location $\delta$. Under the zero-location null hypothesis, the signs are independent fair signs conditional on their absolute values. Rank the absolute differences and add ranks with positive signs. A negative-location alternative rejects for small $W_{+}$. Its exact conditional distribution is that of a sum of an independently selected subset of ranks, each of the $2^n$ subsets equally likely. Thus its mean is $n(n+1)/4$ and its variance is $n(n+1)(2n+1)/24$. Symmetry is essential: a zero median alone does not justify this signed-rank null law. Zero differences are omitted in the usual version, and tied absolute differences receive average ranks.