ANSWER:
a. 0.4207
b. 0.2578
c. 0.5468
STEP-BY-STEP EXPLANATION:
Given:
μ = 85
σ = 20
Now, the probabilities are obtained, after obtaining the z-score for the given corresponding sample data points. We calculate the value of z as follows:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
We calculate for each case:
a. P( x > 89)
[tex]\begin{gathered} P\left(x>89\right)=1-P\left(x<89\right) \\ \\ z=\frac{89-85}{20}=\frac{4}{20}=0.2 \end{gathered}[/tex]
We locate this value in the normal table:
Therefore:
[tex]\begin{gathered} P(x\gt89)=1-P(x\lt89)=1-0.5793 \\ \\ P(x\gt89)=0.4207 \end{gathered}[/tex]
b. P (x < 72)
[tex]\begin{gathered} P\left(x<72\right) \\ \\ z=\frac{72-85}{20}=\frac{-13}{20}=0.65 \end{gathered}[/tex]
We locate this value in the normal table:
Therefore:
[tex]P(x<72)=0.2578[/tex]
c. P (70 < x < 100)
[tex]\begin{gathered} P(70We locate this value in the normal table:
Therefore:
[tex]\begin{gathered} P(70\lt x\lt100)= P(x\lt100)- P(x\lt70) \\ \\ P(70\lt x\lt100)=0.7734-0.2266 \\ \\ P(70\lt x\lt100)=0.5468 \end{gathered}[/tex]
