The Solution.
Z-score formula is given as below:
[tex]Z=\frac{x-\mu}{\sigma}[/tex]
In this case,
[tex]x=70,\mu=76.4,\sigma=5[/tex]
Therefore,
[tex]Z=\frac{70-76.4}{5}=-\frac{6.4}{5}=-1.28[/tex]
Hence, using the negative z-score table, we have
[tex]Pr(Z<-1.28)=0.1003[/tex]
Hence, the probability for part A is 10.0%
For part B:
First, we find the z-score of 80
[tex]\begin{gathered} \text{ the value of x now becomes} \\ x=80 \\ \text{ thus} \\ z=\frac{80-76.4}{5}=\frac{3.6}{5}=0.72 \end{gathered}[/tex]
Hence, using the positive z-score table, we have
[tex]Pr(Z>0.72)=1-Pr(Z\le0.72)=1-0.7642=0.2358[/tex]
Therefore, the probability for part B is 23.6%
For part C:
First, we find the z-score of 72 and 78
[tex]\begin{gathered} when\text{ the value of x becomes} \\ x=72 \\ \text{ thus} \\ z=\frac{72-76.4}{5}=\frac{-4.4}{5}=-0.88 \end{gathered}[/tex]
[tex]\begin{gathered} when\text{ the value of x becomes} \\ x=78 \\ \text{ thus} \\ z=\frac{78-76.4}{5}=\frac{1.6}{5}=0.32 \end{gathered}[/tex]
Hence, using the positive z-score table, we have
[tex]Pr(-0.88
Therefore, the probability for part C is 43.6%
