Respuesta :
Answer
(a) 63.3%
(b) 72.7%
(c) 28.7%
Explanation
Given:
[tex]\begin{gathered} Mean,\mu=42.2\text{ }inches \\ \\ Standard\text{ }deviation,\sigma=5.3\text{ }inches \end{gathered}[/tex](a) What percentage of years will have an annual rainfall of less than 44 inches?
First, we standardize 44 inches by changing x to z:
[tex]z=\frac{x-\mu}{\sigma}[/tex]Here,
[tex]\begin{gathered} x=44,\mu=42.2,and\text{ }\sigma=5.3 \\ \\ z=\frac{44-42.2}{5.3}=\frac{1.8}{5.3} \\ \\ z=0.34 \end{gathered}[/tex]So,
[tex]P(x<44)=P(z<0.34)[/tex]Interpreting the result in a normal curve, we have:
[tex]\begin{gathered} 0.5+P(z=0.34) \\ \\ 0.5+0.1331=0.6331 \end{gathered}[/tex]Therefore, the percentage to the nearest tenth will be:
[tex]0.6331\times100\%=63.3\%[/tex](b) What percentage of years will have an annual rainfall of more than 39 inches?
[tex]\begin{gathered} x=39,\mu=42.2,and\text{ }\sigma=5.3 \\ \\ z=\frac{39-42.2}{5.3}=\frac{-3.2}{5.3} \\ \\ z=-0.603 \end{gathered}[/tex]So,
[tex]P(x>5.3)=P(z>-0.603)[/tex]Interpreting the result in a normal curve, we have:
[tex]\begin{gathered} =0.5-P(z=-0.603) \\ \\ =0.5-(-0.2268) \\ \\ =0.5+0.2268 \\ \\ =0.7268 \end{gathered}[/tex]Therefore, the percentage to the nearest tenth will be:
[tex]\begin{gathered} 0.7268\times100\% \\ \\ =72.7\% \end{gathered}[/tex](c) What percentage of years will have an annual rainfall of between 38 inches and 43 inches?
We need to find P(39 ≤ x ≤ 43).
Standardizing x to z by applying:
[tex]\begin{gathered} z=\frac{x-\mu}{\sigma} \\ \end{gathered}[/tex]Here,
[tex]\begin{gathered} P(\frac{39-42.2}{5.3}\leq z\leq\frac{43-42.2}{5.3}) \\ \\ P(\frac{-3.2}{5.3}\leq z\leq\frac{0.8}{5.3}) \\ \\ P(-0.604\leq z\leq0.151) \end{gathered}[/tex]Also, interpreting the result in a normal curve, we have:
[tex]\begin{gathered} P(z=0.151)-P(z=-0.604) \\ \\ P(z=0.151)+P(z=0.604) \\ \\ =0.060+0.2271 \\ \\ =0.2871 \end{gathered}[/tex]Hence, the percentage to the nearest tenth will be:
[tex]\begin{gathered} 0.2871\times100\% \\ \\ =28.7\% \end{gathered}[/tex]In summary,
(a) 63.3%
(b) 72.7%
(c) 28.7%
