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LammettHash
The 20th percentile is the cutoff value [tex]k[/tex] such that [tex]\mathbb P(X\le k)=0.20[/tex].

Because [tex]X\sim\mathcal N(50,7)[/tex], you can write

[tex]\mathbb P(X\le k)=\mathbb P\left(\frac{X-50}7\le\frac{k-50}7\right)=\mathbb P\left(Z\le\frac{k-50}7\right)=0.20[/tex]

Consult a table of [tex]z[/tex] scores to find the corresponding critical value and you'll find this value to be about [tex]-0.8416[/tex]. So, you have

[tex]\frac{k-50}7=-0.8416\implies k\approx44.11[/tex]

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