To determine which scale Gemma should use for the vertical axis so that the difference in the heights of the bars is maximized, let's follow a step-by-step process:
1. Understand the data:
- The given salary ranges and the corresponding number of people are:
- [tex]$0 - \$[/tex]19,999[tex]$: 40 people
- $[/tex]20,000 - \[tex]$39,999$[/tex]: 30 people
- [tex]$40,000 - \$[/tex]59,999$: 35 people
2. Identify the maximum number of people:
- The maximum number of people in any given salary range is 40.
3. Consider the provided scale options:
- The scale options for the vertical axis are:
- 0-50
- 0-40
- 10-50
- 25-40
4. Calculate the range covered and the height difference for each scale option:
- For each scale, we calculate the difference between the maximum and minimum values of the scale to determine the range covered. We then look at the difference between the maximum height (40 people) and just below the maximum (i.e., 30 or 35 people) relative to the range.
- 0-50 scale:
- Range covered = 50 - 0 = 50
- Height difference for 40 - 30: [tex]\(\frac{40}{50} - \frac{30}{50} = 0.2\)[/tex]
- 0-40 scale:
- Range covered = 40 - 0 = 40
- Height difference for 40 - 30: [tex]\(\frac{40}{40} - \frac{30}{40} = 0.25\)[/tex]
- 10-50 scale:
- Range covered = 50 - 10 = 40
- Height difference for 40 - 30: [tex]\(\frac{40-10}{40} - \frac{30-10}{40} = 0.75 - 0.5 = 0.25\)[/tex]
- 25-40 scale:
- Range covered = 40 - 25 = 15
- Height difference for 40 - 30: [tex]\(\frac{40-25}{15} - \frac{30-25}{15} = 1 - 0.3333 = 0.6667\)[/tex]
5. Compare the height differences for each scale:
- We want to maximize the difference in the heights of the bars (i.e., maximize the calculated difference):
- For 0-50 scale: 0.2
- For 0-40 scale: 0.25
- For 10-50 scale: 0.25
- For 25-40 scale: 0.6667
6. Result:
- The scale option of 25-40 produces the maximum difference in the heights of the bars.
Therefore, Gemma should use the vertical axis scale of 25-40 to maximize the difference in the heights of the bars in her histogram.
