Question

on san martin boulevard, embedded sensors kept track of the vehicle traffic count each hour for five weekdays, monday through friday, between 6 a.m. and 8 p.m. (5 weeks × 14 hours = 70 observations).
(a) visually estimate the quartiles (q₁, q₂, q₃), x_min and x_max.
(b) is the distribution symmetric?
yes, the distribution is symmetric
no, the distribution is skewed right
no, the distribution is skewed left

Explanation:

Step1: Identify quartiles from box - plot

In a box - plot, the left - most whisker is \(x_{min}\), the right - most whisker is \(x_{max}\), the left - side of the box is \(Q_1\), the middle line of the box is \(Q_2\), and the right - side of the box is \(Q_3\).
Looking at the box - plot:
\(x_{min}\approx2200\)
\(Q_1\approx3000\)
\(Q_2\approx3500\)
\(Q_3\approx4000\)
\(x_{max}\approx4800\)

Step2: Check for symmetry

For a symmetric distribution, the distance from \(Q_1\) to \(Q_2\) should be approximately equal to the distance from \(Q_2\) to \(Q_3\), and the distance from \(x_{min}\) to \(Q_1\) should be approximately equal to the distance from \(Q_3\) to \(x_{max}\).
\(Q_2 - Q_1=3500 - 3000 = 500\)
\(Q_3 - Q_2=4000 - 3500 = 500\)
\(Q_1 - x_{min}=3000 - 2200 = 800\)
\(x_{max}-Q_3=4800 - 4000 = 800\)

Since \(Q_2 - Q_1=Q_3 - Q_2\) and \(Q_1 - x_{min}=x_{max}-Q_3\), the distribution is symmetric.

Answer:

(a) \(x_{min}\approx2200\), \(Q_1\approx3000\), \(Q_2\approx3500\), \(Q_3\approx4000\), \(x_{max}\approx4800\)
(b) Yes, the distribution is symmetric.