Answer:
70th percentile for the amount of time between customers entering is 30.10
Step-by-step explanation:
given data
mean = 25
to find out
What is the 70th percentile for the amount of time between customers entering Clay's store
solution
we know that here mean is
mean = [tex]\frac{1}{\lambda}[/tex] ..................1
so here 25 = [tex]\frac{1}{\lambda}[/tex]
and we consider time value corresponding to 70th percentile = x
so we can say
P(X < x) = 1 - [tex]e^{- \lambda *x}[/tex]
P(X < x) = 70 %
1 - [tex]e^{\frac{-x}{25}}[/tex] = 70 %
1 - [tex]e^{\frac{-x}{25} }[/tex] = 0.70
[tex]e^{\frac{- x}{25} }[/tex] = 0.30
take ln both side
[tex]\frac{-x}{25}[/tex] = ln 0.30
[tex]\frac{x}{25}[/tex] = 1.203973
x = 30.10
70th percentile for the amount of time between customers entering is 30.10
The 70th percentile for the amount of time between customers entering Clay's store is 30
The mean of an exponential distribution is:
[tex]E(x) = \frac 1\lambda[/tex]
The mean is 25.
So, we have:
[tex]\frac 1\lambda = 25[/tex]
Solve for [tex]\lambda[/tex]
[tex]\lambda = \frac 1{25}[/tex]
[tex]\lambda = 0.04[/tex]
An exponential function is represented as:
[tex]P(x < x) = 1 - e^{-\lambda x}[/tex]
For the 70th percentile, we have:
[tex]1 - e^{-\lambda x} = 0.70[/tex]
This gives
[tex]e^{-\lambda x} = 1 - 0.70[/tex]
[tex]e^{-\lambda x} = 0.30[/tex]
Substitute [tex]\lambda = 0.04[/tex]
[tex]e^{-0.04x} = 0.30[/tex]
Take the natural logarithm of both sides
[tex]-0.04x = \ln(0.30)[/tex]
This gives
[tex]-0.04x = -1.20[/tex]
Divide both sides by -0.04
[tex]x = 30[/tex]
Hence, the 70th percentile for the amount of time between customers entering Clay's store is 30
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