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The American Community Survey showed that residents of New York City have the longest travel times to get to work compared to residents of other cities in the United States. According to the latest statistics available, the average travel time to work for residents of New York City is 38.3 minutes. Assume the variable is exponentially distributed. What is the probability that it will take a resident of this city between 20 and 40 minutes to travel to work

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Answer:

[tex]P(20<X<40)=0.24[/tex]

Step-by-step explanation:

Given that  our variable is exponentially distributed with [tex]\mu=38.3 \ mins[/tex], let X be the time to travel to work defined as:

[tex]f(X)=\frac{1}{\mu}e^-^x^/^u;\ \ \ \ \ \ \ \ \mu=38.3\\f(X)=\frac{1}{38.3}e^-^x^/^3^8^.^3[/tex]

To find [tex]P(20<X<40)[/tex];

[tex]P(20<X<40)=(1-e^-^4^0^/^3^8^.^3)-(1-e^-^2^0^/^3^8^.^3)\\=-e^-^1^.^0^4+-e^-^0^.^5^2^2\\=0.24[/tex]

The probability that it takes between 20 to 40 minutes to get to work is 0.24

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