To determine which scale to use for the vertical axis of a histogram such that the difference in the heights of the bars is maximized, we need to follow a systematic approach.
1. Identify the Maximum Number of People:
First, we examine the given data to find the maximum number of people in any salary range. The table shows three salary ranges and the corresponding number of people:
- [tex]$0-$[/tex]19,999: 40 people
- [tex]$20,000-$[/tex]39,999: 30 people
- [tex]$40,000-$[/tex]59,999: 35 people
Among these, the highest number of people is 40.
2. List the Possible Scales:
The possible scales provided are:
- [tex]\( 0-50 \)[/tex]
- [tex]\( 0-40 \)[/tex]
- [tex]\( 10-50 \)[/tex]
- [tex]\( 25-40 \)[/tex]
3. Determine Which Scales Include the Maximum Number of People:
We need to check which of these scales can accommodate the maximum number of people, 40:
- [tex]\( 0-50 \)[/tex]: Can include 40 since 40 is within the range.
- [tex]\( 0-40 \)[/tex]: Can include 40 since 40 is on the upper boundary of the range.
- [tex]\( 10-50 \)[/tex]: Can include 40 since 40 is within the range.
- [tex]\( 25-40 \)[/tex]: Can include 40 since 40 is on the upper boundary of the range.
4. Select the Scale with the Smallest Range to Maximize Difference:
Since all scales except [tex]\(25-40\)[/tex] and [tex]\(25-40\)[/tex], include the maximum number of people, we should select the one with the smallest range to maximize the difference in the heights of the bars. The ranges of the scales are:
- [tex]\( 0-50 \)[/tex]: Range is [tex]\( 50 - 0 = 50 \)[/tex]
- [tex]\( 0-40 \)[/tex]: Range is [tex]\( 40 - 0 = 40 \)[/tex]
- [tex]\( 10-50 \)[/tex]: Range is [tex]\( 50 - 10 = 40 \)[/tex]
- [tex]\( 25-40 \)[/tex]: Range is [tex]\( 40 - 25 = 15 \)[/tex]
The smallest range that can still include 40 is from [tex]\( 25-40 \)[/tex].
Thus, the best scale for Gemma to use for the vertical axis such that the difference in the heights of the bars is maximized is [tex]\( 25-40 \)[/tex].
