To determine which vertical axis scale will maximize the differences in the heights of the bars in Gemma's histogram, we need to consider the different scales available and see how the heights of the bars (number of people) fit within each scale.
The given table is:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Salary Range} & \text{Number of People} \\
\hline
\$0-\$19,999 & 40 \\
\hline
\$20,000-\$39,999 & 30 \\
\hline
\$40,000-\$59,999 & 35 \\
\hline
\end{array}
\][/tex]
Let's analyze each of the scales:
1. 0-50 Scale:
- The range is from 0 to 50.
- All the values (40, 30, 35) fit within this range.
- The difference between the maximum (40) and the minimum (30) number of people is [tex]\(40 - 30 = 10\)[/tex].
2. 0-40 Scale:
- The range is from 0 to 40.
- While the values 30 and 35 fit within this range, the value 40 is at the upper limit.
- The difference between the values 40 (max) and 30 (min) is [tex]\(40 - 30 = 10\)[/tex].
3. 10-50 Scale:
- The range is from 10 to 50.
- All the values (40, 30, 35) fit within this range.
- However, since the starting point is 10, we need to consider how values are represented relative to 10.
- The difference between the maximum (40) and the minimum (30) in terms of their relative difference is [tex]\(40 - 30 = 10\)[/tex].
4. 25-40 Scale:
- The range is from 25 to 40.
- Only the values 30 and 35 fit within this range, with 40 being at the upper limit and below 25 not counted.
- The difference between the maximum (35) within the range and the minimum (30) is [tex]\(35 - 30 = 5\)[/tex].
After analyzing these scales:
- 0-50 and 0-40 scales both give a maximum difference of 10.
- 10-50 gives a difference of 10 as well.
- 25-40 gives the smallest difference of 5.
However, to maximize the visual representation of differences, where each increment counts significantly, using the 0-50 scale is most suitable. This scale will ensure all values are displayed within the range and offer the broadest visual differentiation while maintaining inclusiveness from 0 to the highest value of 40.
Hence, Gemma should choose the 0-50 scale for the vertical axis to maximize the difference in the heights of the bars.
