QUESTION A
Calculate the z-score of 57.5.
The z-score formula is given to be:
[tex]z=\frac{x-\bar{x}}{\sigma}[/tex]
where
[tex]\begin{gathered} x=57.5 \\ \bar{x}=\operatorname{mean}=42.3 \\ \sigma=S\mathrm{}D\mathrm{}=13.8 \end{gathered}[/tex]
Therefore, we have:
[tex]z=\frac{57.5-42.3}{13.8}=1.101[/tex]
The probability is given to be:
[tex]P(x<57.5)=P(z\leq0)+P(0From the z-score table:[tex]\begin{gathered} P(z\leq0)=0.5 \\ P(0Therefore, we have[tex]\begin{gathered} P(x<57.5)=0.5+0.3646 \\ P(x<57.5)=0.8646 \end{gathered}[/tex]
QUESTION B
We will calculate two z scores.
Z-score of 40:
[tex]z_1=\frac{40-42.3}{13.8}=-0.1667[/tex]
Z-score of 47.7:
[tex]z_2=\frac{47.7-42.3}{13.8}=0.391[/tex]
Therefore,
[tex]\begin{gathered} P(40
