To determine the appropriate vertical axis scale for Gemma's histogram such that the difference in heights of the bars is maximized, we need to evaluate the given scales in relation to the number of people in each salary range. Here's a step-by-step guide to finding the best scale:
1. Identify the Number of People in Each Salary Range:
- Salary range \[tex]$0-\$[/tex]19,999: 40 people
- Salary range \[tex]$20,000-\$[/tex]39,999: 30 people
- Salary range \[tex]$40,000-\$[/tex]59,999: 35 people
2. Calculate the Absolute Differences in the Number of People Between Ranges:
- Difference between the first and second ranges (40 - 30) = 10
- Difference between the second and third ranges (35 - 30) = 5
- Difference between the first and third ranges (40 - 35) = 5
The differences in the number of people are: [tex]\[10, 5, 5\][/tex]
3. Evaluate Each Given Vertical Axis Scale:
- Scale [tex]\(0-50\)[/tex]: This accommodates all values (0 to 40 people) comfortably. The full range of the vertical axis is 50 units.
- Scale [tex]\(0-40\)[/tex]: This accommodates the values, but the range of the vertical axis is only 40 units.
- Scale [tex]\(10-50\)[/tex]: Starting at 10 means that values will be plotted between 10 and 50. It effectively transforms the data range into [tex]\[(30-50), (40-50), (45-50)\][/tex]. However, the differences between the heights of the bars remain the same [tex]\[10, 5, 5\][/tex] as the scale span is still 40 units.
- Scale [tex]\(25-40\)[/tex]: The scale here spans from 25 to 40, giving us a span of 15 units. This would not be appropriate as the differences between the bars (10, 5, 5) could become extremely small within this narrow range.
4. Choose the Scale with the Maximum Height Difference:
Given that the differences in the number of people are consistently [tex]\[10, 5, 5\][/tex] for each scale:
- The [tex]\(0-50\)[/tex] scale allows the full range of values while maintaining the actual differences in people's heights more visibly due to the broader span.
- Also, the differences are in a domain giving sufficient visual range across the vertical axis, without overly compressing or stretching the intervals.
Conclusion:
The best vertical axis scale for Gemma's histogram, which maximizes the differences in the heights of the bars, is:
[tex]\[0-50.\][/tex]
