Question

Consider a one-mean z-test. Denote z₀ as the observed value of the test statistic z. If the test is right tailed, then the P-value can be expressed as P(z$\ge$z₀). Determine the corresponding expression for the P-value if the test is

a. Left failed b. Two-failed

Short answer

(a)$$\begin{gathered} \mathrm{Left}-\mathrm{failed}\mathrm{test} \\ P-value=p(z\le z_0),wherez~N(0,1) \end{gathered}$$

(b) Therefore, one can compute the P-value of a two tailed z-test by using the expression given in (*) or by using anyone of the two expressions given in (**)

Step-by-step solution

Step 1. Given

One-mean z-test. Denote z₀ as the observed value of the test statistic z .

Step 2.  The test statistics for a mean test.

The test statistic for a one-mean z-test with null hypothesis given by H₀:μ = μ₀ is

$z=\frac{\overline{x}-{\mu }_0}{\sigma -\sqrt{n}}$

If the null hypothesis is true the test statistic has the standard normal distribution i.e.

=z$~$ N(0,1).

Part(a) Step 3. Calculation

$$\begin{gathered} \mathrm{Left}-\mathrm{failed}\mathrm{test} \\ P-value=p(z\le z_0),wherez~N(0,1) \\ =\varnothing \left(z_0\right) \end{gathered}$$

Part (b) Step 4. Calculation

$$\begin{gathered} \mathrm{Two}-\mathrm{failed}\mathrm{test} \\ \mathrm{P}-\mathrm{value}=P(z\ge z_0)+P(z\le -\overline{z_0}),\mathrm{where}\mathrm{z}~\mathrm{N}(0,1) \\ =\left\{\begin{array}{l}\mathrm{P}(\mathrm{z}\le -\overline{{\mathrm{z}}_0})+\mathrm{P}(\mathrm{z}\le -\overline{{\mathrm{z}}_0})\\ \mathrm{P}(\mathrm{z}\ge {\mathrm{z}}_0)+\mathrm{P}(\mathrm{z}\ge {\mathrm{z}}_0)\end{array}\right.\left[\begin{array}{c}\mathrm{Normal}\mathrm{curve}\mathrm{is}\mathrm{symmetric}\mathrm{about}\mathrm{is}0\\ \mathrm{P}(\mathrm{z}\le -\overline{{\mathrm{z}}_0})=\mathrm{P}(\mathrm{z}\ge {\mathrm{z}}_0)\end{array}\right] \\ \left\{\begin{array}{l}2\mathrm{P}(\mathrm{z}\le -\overline{{\mathrm{z}}_0})\\ 2\mathrm{P}(\mathrm{z}\ge {\mathrm{z}}_0)\end{array}\right....(**) \end{gathered}$$

Therefore, one can compute the P-value of a two tailed z-test by using the expression given in (*) or by using anyone of the two expressions given in (**)