Passengers using New York's MetroCard system must swipe the card at a rate between 10 and 40 inches per second, or else the card must be re-swiped through the card reader. Research shows that actual swipe rates by subway riders are uniformly distributed between 5 and 50 inches per second.

A) Mean swipe rate=27.5

B) Standard deviation=12.99

C) 1st quartile=16.25

D) 3rd quartile=38.75

E) Percentage of subway riders must re-swipe the card because they were outside the acceptable range=33.33%

Refer below for explanation.

Step-by-step explanation:

A) 27.5

B) [[ (50-5) + 1]^2 - 1]/12] = squareroot[[46^2-1]/12] = 12.99

C) For 1st quartile:

Base (1/45) = 0.25

= 45*0.25

= 11.25

1st quartile=5+11.25 = 16.25

D) For 3rd quartile

Base (1/45) = 0.75

= 45*0.75

= 33.75

3rd quartile=5+33.75 = 38.75

E) What percentage of subway riders must re-swipe the card because they were outside the acceptable range?

1 - Percentage (10< = a <=40) = 1/3 = 33.33%

E (X) = 27.5 in/s

s. d (X) = 13.28 in/s

1st quartile = 16.25 in/s

3rd quartile = 38.75 in/s

33.33%

Step-by-step explanation:

Solution:-

- The Research shows that actual swipe rates by subway riders are uniformly distributed between 5 and 50 inches per second.

- The uniform distribution of the swipe rate can be expressed by the limits of swipe rates as [ a, b ] as follows:

- The random variable X follows a uniform distribution for the rate of card swiped expressed by parameters [ 5, 50 ].

X ~ U (5, 50)

a) The mean value E (X) for the distribution is:

E (X) = (b - a) / 2

E (X) = (50-5) / 2

E (X) = 27.5 in/s

b) The standard deviation s. d for the distribution is as follows:

s. d (X) = sqrt ({ [ (b-a) + 1 ]^2 - 1 } / 12)

s. d (X) = sqrt ({ [ (5-5) + 1 ]^2 - 1 } / 12)

s. d (X) = 13.28 in/s/

c) The first quartile is given by the 25% of the swipe rates lie under:

1st quartile = a + 0.25 * (b-a)

1st quartile = 5 + 0.25 * (50 - 5)

1st quartile = 5 + 11.25 = 16.25 in/s

d) The third quartile is given by the 75% of the swipe rates lie under:

3rd quartile = a + 0.75 * (b-a)

3rd quartile = 5 + 0.75 * (50 - 5)

3rd quartile = 5 + 33.75 = 38.75 in/s

e) what percentage of subway riders must re-swipe the card because they were outside the acceptable range?

1 - P (10< = x <=40) =

1 - (30/45) = 1 - 2/3 = 1/3 = 33.33%