Answer :

Answer:

the 13th percentile = 35.238

Step-by-step explanation:

We can find the 13th percentile by using the Z-score formula:

[tex]\boxed{Z=\frac{x-\mu}{\sigma} }[/tex]

where:

  • [tex]Z=\text{Z-score}[/tex]
  • [tex]x=\text{observed value}[/tex]
  • [tex]\mu=\text{mean}[/tex]
  • [tex]\sigma=\text{standard deviation}[/tex]

By using the normal distribution table, we can convert the 13th percentile (13%) into z-score:

[tex]P(X)=0.13[/tex]

[tex]Z(X)=-1.127[/tex]

Hence:

[tex]\begin{aligned}\\Z(X)&=\frac{x-\mu}{\sigma} \\\\-1.127&=\frac{x-42}{6} \\\\x-42&=6(-1.127)\\\\x&=-6.762+42\\\\x&=\bf 35.238\end{aligned}[/tex]

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