Question

Average commuting times (in minutes) for people from Pleasantville, PA who drive to work are normally distributed with mean 30 minutes with a standard deviation of 6 minutes. The average commuting times (in minutes) for people in Pleasantville who bicycle to work are also normally distributed, and the meantime for cycling to work is 30 minutes with a standard deviation of 8 minutes. Which of the following situations is you least likely to come by (i.e., which is most unusual)?

There are at least two different, but equally valid ways to do this problem.
a. A driver with a 45-minute commute
b. A driver with an 8-minute commute
c. A bicyclist with a 44-minute commute
d. A bicyclist with a 6-minute commute

Answer 1

b. A driver with an 8-minute commute

d. A bicyclist with a 6-minute commute

Explanation:

X = driver to work time (Normal)

`\mu_x = 30`

`\sigma _x = 6`

`Z_x = (x- \mu)/(\sigma)`

`= (x-30)/(6)`

Y = bicycle to work time (Normal)

`\mu_y = 30`

`\sigma _y = 8`

`Z_y = (y- \mu)/(\sigma)`

`= (y-30)/(8)`

Driving

1) If x = 45

`Z_x = (45- 30)/(6)\\\\=2.50`

ii) If x = 8

`Z_x = (8- 30)/(6)\\\\=-3.67`

Bicycle

i) If y = 44

`Z_y = (44- 30)/(8)\\\\=1.75`

ii) If y = 6

`Z_x = (6 - 30)/(8)\\\\=-3.00`

The event at driving commute time x = 8 and time bicycle y = 6 are the most unusual time. this is because they are 3 standard deviation below the mean time of respective events

• driver with an 8-minute commute

• A bicyclist with a 6-minute commute