Question

Find a value of the standard normal random variable z, call it${\mathbf{z}}_{\mathbf{0}}$ such that

$$\begin{gathered} \mathbf{a}\mathbf{.}\mathbf{}\mathbf{P}\left(-z_0\le z\le z_0\right)\mathbf{=}\mathbf{.}\mathbf{98}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{b}\mathbf{.}\mathbf{}\mathbf{P}\left(z\ge z_0\right)\mathbf{=}\mathbf{.}\mathbf{05} \\ \mathbf{c}\mathbf{.}\mathbf{}\mathbf{P}\left(z\ge z_0\right)\mathbf{=}\mathbf{.}\mathbf{70}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{d}\mathbf{.}\mathbf{}\mathbf{P}\left(z\ge z_0\right)\mathbf{=}\mathbf{.}\mathbf{025} \\ \mathbf{e}\mathbf{.}\mathbf{}\mathbf{P}\left(z\le z_0\right)\mathbf{=}\mathbf{.}\mathbf{70}\mathbf{} \end{gathered}$$

Short answer

$$\begin{gathered} \mathrm{a}.{\mathrm{z}}_0=2.326 \\ \mathrm{b}.{\mathrm{z}}_0=1.645 \\ \mathrm{c}.{\mathrm{z}}_0=-0.524 \\ \mathrm{d}.{\mathrm{z}}_0=1.96 \\ e.{\mathrm{z}}_0=0.524 \end{gathered}$$

Step-by-step solution

Given information

z is a standard normal variable, which is called${\mathrm{z}}_0$.

Finding the value of z0 when P(-z0≤z≤z0)=.98 

$$\begin{gathered} \mathrm{a}. \\ P\left(-{\mathrm{z}}_0\le \mathrm{z}\le {\mathrm{z}}_0\right)=0.98 \\ 2P\left(0\le \mathrm{z}\le {\mathrm{z}}_0\right)=0.98\left[Duetosymmetricdistrubution\right] \\ P\left(0\le z\le {\mathrm{z}}_0\right)=0.49 \\ P\left(z\le {\mathrm{z}}_0\right)-P\left(z<0\right)=0.49 \\ P\left(z\le {\mathrm{z}}_0\right)-0.50=0.49 \\ P\left(z\le {\mathrm{z}}_0\right)=0.99 \\ \Phi \left({\mathrm{z}}_0\right)=0.99 \\ {\mathrm{z}}_0={\mathrm{\Phi }}^{-1}\left(0.99\right) \\ {\mathrm{z}}_0=2.326 \end{gathered}$$

Therefore, the z₀the score is 2.326.

Finding the value of z0 when P(z≥z0)=.05

b.

$$\begin{gathered} P\left(z\ge z_0\right)=0.05 \\ 1-P\left(z<z_0\right)=0.05 \\ P\left(z<z_0\right)=0.95 \\ \Phi \left(z_0\right)=0.95 \\ {z}_0={\Phi }^{-1}\left(0.95\right) \\ {z}_0=1.645 \end{gathered}$$

Therefore, the z₀ score is 1.645.

Finding the value of z0 when P(z≥z0)=.70

c.

$$\begin{gathered} P\left(z\ge z_0\right)=0.70 \\ 1-P\left(z\ge z_0\right)=0.70 \\ P\left(z\ge z_0\right)=0.30 \end{gathered}$$

Therefore, the score is -0.524.

Finding the value of z0 when P(z≥z0)=.70

$$\begin{gathered} \mathrm{d}. \\ P\left(z\ge z_0\right)=0.025 \\ 1-P\left(z<z_0\right)=0.025 \\ P\left(z<z_0\right)=0.975 \\ \Phi \left(z_0\right)=0.975 \\ {z}_0={\Phi }^{-1}\left(0.975\right) \\ {z}_0=1.96 \end{gathered}$$

Therefore, the z₀ score is 1.96.

Finding the value z0 of when P(z≤z0)=.70

$$\begin{gathered} \mathrm{e}. \\ \mathrm{P}\left(\mathrm{z}\le {\mathrm{z}}_0\right)=0.70 \\ \mathrm{\Phi }\left({\mathrm{z}}_0\right)=0.70 \\ {\mathrm{z}}_0={\mathrm{\Phi }}^{-1}\left(0.70\right) \\ {\mathrm{z}}_0=0.524 \end{gathered}$$

Therefore, the z₀ score is 0.524.