Question

Suppose that T and Z are random variables.

(a) If $P(T>2.02)=0.05$and $P(T<-2.02)=0.05,$determine $P(-2.02\le T\le 2.02)$

(b) Suppose that $P(-1.64\le Z\le 1.64)=0.90$and suppose that $P(Z>1.64)=P(Z<-1.64).$

Find$P(Z>1.64).$

Short answer

Part (a) $P(-2.02\le T\le 2.02)=0.9$.

Part (b)$P(Z>1.64)=0.05$.

Step-by-step solution

Part (a) Step 1. Given information

$P(T>2.02)=0.05\mathrm{and}P(T<-2.02)=0.05$

Part (a) Step 2. Solution

$$\begin{gathered} P(-2.02\le T\le 2.02)=1-\left[P\right(T>2.02) \\ =0.05+P(T<-2.02\left)\right] \\ =1-(0.05+0.05) \\ =1-0.1 \\ =0.9 \end{gathered}$$

Part (b) Step 1. Given information 

$$\begin{gathered} P(Z>1.64)=P(Z<-1.64)\mathrm{and} \\ P(-1.64\le Z\le 1.64)=0.90 \end{gathered}$$

Part (b) Step 2. Solution.

$$\begin{gathered} P(Z>1.64)=1-\left[P\right(-1.64\le Z\le 1.64)+P(Z<-1.64\left] \\ P\right(Z>1.64)=1-[0.90+P(Z>1.64)] \\ 2·P(Z>1.64)=0.10 \\ P(Z>1.64)=0.05 \end{gathered}$$