Question

The scheduled commuting time on the Long Island Railroad from Glen Cove to New York City is 65 minutes. Suppose that the actual commuting time is uniformly distributed between 64 and 74 minutes. What is the probability that the commuting time will be a. Less than 70 minutes? b. Between 65 and 70 minutes? c. Greater than 65 minutes? d. What are the mean and standard deviation of the commuting time?

Answer 1

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

Step-by-step explanation:

Solution:

let Y is the commuting time between Long Island Railroad from Glen Clove to New York City.

Then,

f ( y) =
`(1)/(74-64)` 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 ) =
`\int\limits^a_b {1/10} \, dy` where, a = 70 and b = 64

by solving this integration, we will get:

P(Y<70) =
`(70-64)/(10)` = 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) =
`\int\limits^a_b {1/10} \, dy`

P(65<Y<70) =
`(70-65)/(10)` = 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) =
`\int\limits^a_b {1/10} \, dy`

P(Y>65) =
`(74-65)/(10)` = 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 =
`(a + b)/(2)`

Mean =
`(64 + 74)/(2)`

Mean = 69 minutes

Now, for standard deviation:

SD = σ =
`\sqrt{((74-64)^(2) )/(12) }`

SD = σ = 2.88 minutes