Ben has determined that the length of time it takes him to commute from his residence to school is normally distributed with a mean of 45 minutes and a standard deviation of 5 minutes. How much time must he allow if he wishes to be in his first class on time 90% of the time?
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Answer:
51.4 min
Step-by-step explanation:
Let X=X= the length of time: X\sim N(\mu, \sigma^2).X∼N(μ,σ
2
).
Given \mu=45\ min, \sigma=5\ min.μ=45 min,σ=5 min.
P(X<x)=P(Z<\dfrac{x-\mu}{\sigma})=P(Z<\dfrac{x-45}{5})=0.9P(X<x)=P(Z<
σ
x−μ
)=P(Z<
5
x−45
)=0.9
=P(Z<\dfrac{x-45}{5})=0.9=P(Z<
5
x−45
)=0.9
\dfrac{x-45}{5}\approx1.2816
5
x−45
≈1.2816
x=51.4\ minx=51.4 min