Question

Imagine that we know the following information for populations living in cities and rural areas. In big cities, average commutes are 40 minutes with a standard deviation of 15 minutes. In rural areas, the average commute is 25 minutes with a standard deviation of 6 minutes. Who has a longer commute, Mary from the city who travels for 35 minutes or Jean from the country who travels 30 minutes? a. Jean, because her commute is in the country. b. Mary, because city traffic moves slowly c. Mary's commute is 5 minutes longer d. Jean, because she travels 30 minutes on slow country roads

Answer 1

Answer:

a. Jean, because her commute is in the country.

Step-by-step explanation:

Z-score:

In a set with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

In this question:

Whoever has the higher z-score will have the longer commute.

Mary:

Travels for 35 minutes. In big cities, average commutes are 40 minutes with a standard deviation of 15 minutes. This means that $X = 35, \mu = 40, \sigma = 15$. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{35 - 40}{15}$

$Z = -0.33$

Jean:

Travels for 30 minutes. In rural areas, the average commute is 25 minutes with a standard deviation of 6 minutes. This means that $X = 30, \mu = 25, \sigma = 6$. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{30 - 25}{6}$

$Z = 0.83$

Due to being in the Country, Jean's z-score is higher, so her commute is longer. The correct answer is given by option A.