Test of Means

For samples following a normal distribution, use Z statistic or the T statistic if $σ$ is unknown

Two-sided alternative

A two-tailed test

p-value = P (Z > | Z_{o b s} | \times 2) < α ⟶ Reject H_{0}

Only one case:

H_{0} : μ = μ_{0} vs H_{a} : μ \neq μ_{0}

Using the Likelihood ratio test statistic, the critical region for an $α$ level test is given by

\begin{aligned} | \bar{x} - μ_{0} | & \geq Z_{1 - \frac{α}{2}} \frac{σ}{\sqrt{n}} \\ C & = {(x_{1}, \dots, x_{n}); | \bar{x} - μ_{0} | \geq Z_{1 - \frac{α}{2}} \frac{σ}{\sqrt{n}}} \\ C & = {(x_{1}, \dots, x_{n}); z \geq Z_{1 - \frac{α}{2}}, z \leq - Z_{1 - \frac{α}{2}}} \end{aligned}

A one tailed test

p-value = P (Z > Z_{o b s}) < α ⟶ Reject H_{0}

Case 1:

H_{0} : θ = θ_{0} vs H_{a} : θ < θ_{0}

We would like to reject $H_{0}$ if $\hat{θ}$ is much smaller than $θ_{0}$

With samples from a normal distribution:

\begin{aligned} α & = P (\bar{x} - μ_{0} \leq - Z_{1 - α} \frac{σ}{\sqrt{n}}) \\ C & = {(x_{1}, \dots, x_{n}); z \leq - Z_{1 - α}} \end{aligned}

Case 2:

H_{0} : θ = θ_{0} vs H_{a} : θ > θ_{0}

We would like to reject $H_{0}$ if $\hat{θ}$ is much larger than $θ_{0}$

\begin{aligned} α & = P (\bar{x} - μ_{0} \geq Z_{1 - α} \frac{σ}{\sqrt{n}}) \\ C & = {(x_{1}, \dots, x_{n}); z \geq Z_{1 - α}} \end{aligned}