Question

The symbol (z) is often used to denote the area under the standard normal curve that lies to the left of a specified value of z. Consider a one-mean z-test. Denote z₀ as the observed value of the test statistic z. Express the P-value of the hypothesis test in terms of if the test is

a. left tailed.

b. right tailed.

c. two tailed.

Short answer

(a)$\varnothing \left(z_0\right)$

(b) $1-\varnothing \left(z_0\right)$

(c) $\left\{\begin{array}{l}2\left[1-P(z-\left|z_0\right|)\right]\\ 2\varnothing (-\left|z_0\right|)\end{array}\right.$

Step-by-step solution

Step 1. Given

Consider a 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.

=zN(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

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

Part(c) Step 5. Calculation

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