Question

practice midterm exam
score: 6.5/17 answered: 7/17
question 8
the boxplot below shows salaries for cpas and actuaries in a town.
if a person is making the first quartile salary for a cpa, they are making less than or equal to % of actuaries.
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Explanation:

Step1: Analyze the box - plots

The first quartile (\(Q_1\)) of the CPA's salary is the value that divides the lower \(25\%\) of the CPA's data from the upper \(75\%\). For the Actuary's box - plot, we need to find the proportion of Actuary's salaries that are less than or equal to the CPA's \(Q_1\).

Step2: Use the properties of box - plots

The box - plot for Actuaries: The median divides the data into two halves (\(50\%\) below and \(50\%\) above). The first quartile (\(Q_1\)) of Actuaries is at \(60\) (approximate) and the second quartile (\(Q_2\) or median) is at \(70\) (approximate). The first quartile of CPAs (let's assume from the box - plot) is at \(50\).
Looking at the Actuary's box - plot, the value \(50\) is below the first quartile of Actuaries. The proportion of data less than or equal to the first quartile of Actuaries is \(25\%\), but since \(50\) (CPA's \(Q_1\)) is even lower, we consider the entire lower part of the Actuary's data. The Actuary's box - plot has a minimum value (left - most whisker). But if we assume a normal distribution - like interpretation (in terms of quartiles), the first quartile of CPAs (\(Q_1\) of CPA) is less than the first quartile of Actuaries. The proportion of Actuary's data less than or equal to the first quartile of Actuaries is \(25\%\), and since the value (CPA's \(Q_1\)) is even lower, we can think of it in terms of the cumulative distribution.
The Actuary's box - plot: The left - most part (before the first quartile of Actuaries) is \(25\%\) of the data. But if we consider the fact that the first quartile of CPAs is lower than the first quartile of Actuaries. The Actuary's data: The first quartile (\(Q_1 = 60\)), median (\(Q_2=70\)), third quartile (\(Q_3 = 80\)). If the CPA's \(Q_1\) is \(50\), then looking at the Actuary's data, \(25\%\) of Actuary's data is between the minimum and \(Q_1\) (60). But since \(50<60\), and assuming the data is spread (evenly enough for quartile - based interpretation), the proportion of Actuary's data less than or equal to \(50\) (CPA's \(Q_1\)) is \(25\%\) (the lower quartile of Actuaries is \(60\), but if we consider the fact that the first \(25\%\) of Actuary's data is from the minimum to \(60\), and \(50\) is within that lower \(25\%\) range.

Answer:

\(25\)