The Washington, DC, Metro has some long and deep escalators. The longest continu

Question

The Washington, DC, Metro has some long and deep escalators. The longest continuous escalator is at the Wheaton (Red Line) stop. It is 230 feet long and approximately 140 feet deep. For those who simply ride the escalator (without extra steps), it takes just over 2.5 minutes to complete the journey. However, because many people walk while on the escalator, the amount of time to travel the 230 feet is approximately normal with mean 1.9 minutes and standard deviation 0.3 minute.

Suppose an escalator rider is selected at random.

(a) What is the probability (±0.0001) that it takes less than 2.3 minutes to ride the escalator?

P(X<2.3) =

(b) What is the probability (±0.0001) that it takes between 1.4 and 2.6 minutes to ride the escalator?

P(1.4X2.6) =

(c) Suppose that five escalator riders are selected at random. What is the probability (±0.0001) that three of the five take at least 2 minutes and 20 seconds to ride the escalator?

P(X>2 minutes and 20 seconds) =

Explanation / Answer

mean = 1.9 and standard deviation = 0.3

a)

For x = 2.3, the z-value z = (2.3 - 1.9) / 0.3 = 1.33

Hence P(x < 2.3) = P(z < 1.33) = [area to the left of 1.33] = 0.9082

b)

For x = 1.4 , z = (1.4 - 1.9) / 0.3 = -1.66 and for x = 2.6, z = (2.6 - 1.9) / 0.3 = 2.33

Hence P(1.4 < x < 2.6) = P(-1.66 < z < 2.33) = [area to the left of z = 2.33] - [area to the left of -1.66]

= 0.9901 - 0.0495 = 0.9406

c) as 2 minutes 20 seconds are equal to 2.33

here we have to find just the reverse of part A

HENCE the answer will be 1- 0.9082 = 0.0918

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