Question

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.

Answer 1

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.

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%

Answer 2

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%

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%