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Sagot :
Answer:
A) P(Y<70) = 0.6 or 60%
B) P(65<Y<70) = 0.5 or 50%
C) P(Y>65) = 0.9 or 90%
D) Mean = 69 minutes
SD = σ = 2.88 minutes
Explanation:
Solution:
let Y is the commuting time between Long Island Railroad from Glen Clove to New York City.
Then,
f ( y) = [tex]\frac{1}{74-64}[/tex] as, it is uniformly distributed between 64 and 74.
f (y) = 1/10
now, we have a function so, probability can be found out by the use of integration.
a) P (Y < 70 ) = [tex]\int\limits^a_b {1/10} \, dy[/tex] where, a = 70 and b = 64
by solving this integration, we will get:
P(Y<70) = [tex]\frac{70-64}{10}[/tex] = 0.6
P(Y<70) = 0.6 or 60%
b) P(65<Y<70) = again this can be solved similarly as above, but here a = 70 and b = 65
P(65<Y<70) = [tex]\int\limits^a_b {1/10} \, dy[/tex]
P(65<Y<70) = [tex]\frac{70-65}{10}[/tex] = 0.5
P(65<Y<70) = 0.5 or 50%
c) P(Y>65) = again, this can be solved similarly but here a = 74 and b = 65
P(Y>65) = [tex]\int\limits^a_b {1/10} \, dy[/tex]
P(Y>65) = [tex]\frac{74-65}{10}[/tex] = 0.9
P(Y>65) = 0.9 or 90%
d) Now, we have to calculate the mean and standard deviation of the commuting time.
So,
Mean = [tex]\frac{a + b}{2}[/tex]
Mean = [tex]\frac{64 + 74}{2}[/tex]
Mean = 69 minutes
Now, for standard deviation:
SD = σ = [tex]\sqrt{\frac{(74-64)^{2} }{12} }[/tex]
SD = σ = 2.88 minutes
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