Question

Question: Find the following probabilities for the standard normal random variable z:

$$\begin{gathered} \mathbf{}\mathit{a}\mathbf{.}\mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathit{z}\mathbf{>}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{46}\mathbf{)}\mathbf{} \\ \mathit{b}\mathbf{.}\mathbf{}\mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathbf{}\mathit{z}\mathbf{}\mathbf{<}\mathbf{}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{56}\mathbf{)}\mathbf{}\mathbf{} \\ \mathit{c}\mathbf{.}\mathbf{}\mathit{P}\mathbf{}\left(.67\le z<2.41\right) \\ \mathit{d}\mathbf{.}\mathbf{}\mathit{P}\mathbf{(}\mathbf{}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{96}\mathbf{\le }\mathbf{}\mathit{z}\mathbf{}\mathbf{-}\mathbf{.}\mathbf{33}\mathbf{)} \\ \mathit{e}\mathbf{.}\mathit{P}\mathbf{(}\mathbf{}\mathit{Z}\mathbf{\ge }\mathbf{0}\mathbf{)} \\ \mathit{f}\mathbf{.}\mathit{P}\left(-2.33<z<1.50\right) \end{gathered}$$

Short answer

Answer

A random variable is a mathematical expression of a statistical study's result.

Step-by-step solution

(a) The data is given below

The calculation is given below:

$$\begin{gathered} \mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathit{z}\mathbf{>}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{46}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{-}\mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathit{z}\mathbf{\le }\mathbf{}\mathbf{1}\mathbf{.}\mathbf{46}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{9279} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{0721} \end{gathered}$$

(b) The data is given below

The calculation is given below:

$\mathbf{}\mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathbf{}\mathit{z}\mathbf{}\mathbf{<}\mathbf{}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{56}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0594}\mathbf{}\mathbf{}$

(c) The data is given below

The calculation is given below:

$$\begin{gathered} \mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathbf{.}\mathbf{67}\mathbf{\le }\mathbf{z}\mathbf{}\mathbf{<}\mathbf{2}\mathbf{.}\mathbf{41}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathbf{.}\mathbf{67}\mathbf{\le }\mathbf{z}\mathbf{}\mathbf{<}\mathbf{2}\mathbf{.}\mathbf{41}\mathbf{)}\mathbf{-}\mathbf{}\mathit{P}\mathbf{}\mathbf{(}\mathbf{z}\mathbf{<}\mathbf{}\mathbf{.}\mathbf{0}\mathbf{.}\mathbf{67}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99920}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{7486} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2434} \end{gathered}$$

(d) The data is given below

The calculation is given below:

$$\begin{gathered} \mathbf{}\mathit{P}\mathbf{(}\mathbf{}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{96}\mathbf{\le }\mathbf{}\mathit{z}\mathbf{}\mathbf{-}\mathbf{.}\mathbf{33}\mathbf{)}\mathbf{=}\mathit{P}\mathbf{(}\mathbf{}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{96}\mathbf{\le }\mathit{Z}\mathbf{<}\mathbf{0}\mathbf{)}\mathbf{-}\mathit{P}\mathbf{(}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{33}\mathbf{\le }\mathit{Z}\mathbf{<}\mathbf{0}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{4750}\mathbf{}\mathbf{-}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{1293} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{3457} \end{gathered}$$

(e) The data is given below

The calculation is given below:

$$\begin{gathered} \mathit{P}\mathbf{(}\mathbf{}\mathit{Z}\mathbf{\ge }\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{1}\mathbf{-}\mathit{P}\mathbf{}\mathbf{(}\mathit{z}\mathbf{}\mathbf{<}\mathbf{0}\mathbf{)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{.}\mathbf{5} \end{gathered}$$

(f) The data is given below

The calculation is given below:

$\mathit{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{33}\mathbf{}\mathbf{<}\mathbf{}\mathbf{z}\mathbf{}\mathbf{<}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{)}\mathbf{=}\mathit{P}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{33}\mathbf{<}\mathit{z}\mathbf{<}\mathbf{0}\mathbf{)}\mathbf{+}\mathit{P}\mathbf{(}\mathbf{0}\mathbf{<}\mathit{Z}\mathbf{<}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{)}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}$

$$\begin{gathered} \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{4901}\mathbf{}\mathbf{+}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{4332} \\ \mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9233} \end{gathered}$$