Answer :
In this problem, we are required to determine which of the given scales for the vertical axis results in the maximum difference in the heights of the bars in a histogram.
First, let's summarize the information given:
Salary Ranges:
1. [tex]$0 - $[/tex]19,999[tex]$ : 40 people 2. $[/tex]20,000 - [tex]$39,999$[/tex] : 30 people
3. [tex]$40,000 - $[/tex]59,999[tex]$ : 35 people We have four different scales to consider for the vertical axis: 1. 0 - 50 2. 0 - 40 3. 10 - 50 4. 25 - 40 We need to calculate the heights of each bar under each scale and see which scale provides the maximum difference in heights among the bars. 1. Scale 0 - 50: - The bar heights are directly the number of people in each salary range: - $[/tex]0 - [tex]$19,999$[/tex]: 40
- [tex]$20,000 - $[/tex]39,999[tex]$: 30 - $[/tex]40,000 - [tex]$59,999$[/tex]: 35
- Difference in heights:
- Max height = 40, Min height = 30
- Difference = 40 - 30 = 10
2. Scale 0 - 40:
- As the maximum range is 40, each bar is limited to a maximum height of 40:
- [tex]$0 - $[/tex]19,999[tex]$: min(40, 40) = 40 - $[/tex]20,000 - [tex]$39,999$[/tex]: min(40, 30) = 30
- [tex]$40,000 - $[/tex]59,999[tex]$: min(40, 35) = 35 - Difference in heights: - Max height = 40, Min height = 30 - Difference = 40 - 30 = 10 3. Scale 10 - 50: - This scale only considers values from 10 to 50: - $[/tex]0 - [tex]$19,999$[/tex]: 40 (since 40 is within 10-50 range, we use 40 - 10 since it must reach at least 10 to count)
- [tex]$20,000 - $[/tex]39,999[tex]$: 30 (same logic, use 30 - 10) - $[/tex]40,000 - [tex]$59,999$[/tex]: 35 (same logic, use 35 - 10)
- Adjusted bar heights:
- [tex]$0 - $[/tex]19,999[tex]$: 40 - 10 = 30 - $[/tex]20,000 - [tex]$39,999$[/tex]: 30 - 10 = 20
- [tex]$40,000 - $[/tex]59,999[tex]$: 35 - 10 = 25 - Difference in heights: - Max height = 30, Min height = 20 - Difference = 30 - 20 = 10 4. Scale 25 - 40: - This range starts from 25: - $[/tex]0 - [tex]$19,999$[/tex]: 40 (use 40 - 25 if more than 25, else 0)
- [tex]$20,000 - $[/tex]39,999[tex]$: 30 (use 30 - 25 below 40, else 30) - $[/tex]40,000 - [tex]$59,999$[/tex]: 35 (use 35 - 25)
- Adjusted bar heights:
- [tex]$0 - $[/tex]19,999[tex]$: 40 - 25 = 15 - $[/tex]20,000 - [tex]$39,999$[/tex]: 30 - 25 = 5
- [tex]$40,000 - $[/tex]59,999$: 35 - 25 = 10
- Difference in heights:
- Max height = 15, Min height = 5
- Difference = 15 - 5 = 10
After assessing each scale, we find that all scales provide a maximum difference of 10 in the bar heights. Therefore, the choice can be generalized to:
- 0-50
- 0-40 is the preferable scales.
Thus, the scale that maximizes the difference in the heights of the bars in Gemma's histogram can be either 0-50 or 0-40.
First, let's summarize the information given:
Salary Ranges:
1. [tex]$0 - $[/tex]19,999[tex]$ : 40 people 2. $[/tex]20,000 - [tex]$39,999$[/tex] : 30 people
3. [tex]$40,000 - $[/tex]59,999[tex]$ : 35 people We have four different scales to consider for the vertical axis: 1. 0 - 50 2. 0 - 40 3. 10 - 50 4. 25 - 40 We need to calculate the heights of each bar under each scale and see which scale provides the maximum difference in heights among the bars. 1. Scale 0 - 50: - The bar heights are directly the number of people in each salary range: - $[/tex]0 - [tex]$19,999$[/tex]: 40
- [tex]$20,000 - $[/tex]39,999[tex]$: 30 - $[/tex]40,000 - [tex]$59,999$[/tex]: 35
- Difference in heights:
- Max height = 40, Min height = 30
- Difference = 40 - 30 = 10
2. Scale 0 - 40:
- As the maximum range is 40, each bar is limited to a maximum height of 40:
- [tex]$0 - $[/tex]19,999[tex]$: min(40, 40) = 40 - $[/tex]20,000 - [tex]$39,999$[/tex]: min(40, 30) = 30
- [tex]$40,000 - $[/tex]59,999[tex]$: min(40, 35) = 35 - Difference in heights: - Max height = 40, Min height = 30 - Difference = 40 - 30 = 10 3. Scale 10 - 50: - This scale only considers values from 10 to 50: - $[/tex]0 - [tex]$19,999$[/tex]: 40 (since 40 is within 10-50 range, we use 40 - 10 since it must reach at least 10 to count)
- [tex]$20,000 - $[/tex]39,999[tex]$: 30 (same logic, use 30 - 10) - $[/tex]40,000 - [tex]$59,999$[/tex]: 35 (same logic, use 35 - 10)
- Adjusted bar heights:
- [tex]$0 - $[/tex]19,999[tex]$: 40 - 10 = 30 - $[/tex]20,000 - [tex]$39,999$[/tex]: 30 - 10 = 20
- [tex]$40,000 - $[/tex]59,999[tex]$: 35 - 10 = 25 - Difference in heights: - Max height = 30, Min height = 20 - Difference = 30 - 20 = 10 4. Scale 25 - 40: - This range starts from 25: - $[/tex]0 - [tex]$19,999$[/tex]: 40 (use 40 - 25 if more than 25, else 0)
- [tex]$20,000 - $[/tex]39,999[tex]$: 30 (use 30 - 25 below 40, else 30) - $[/tex]40,000 - [tex]$59,999$[/tex]: 35 (use 35 - 25)
- Adjusted bar heights:
- [tex]$0 - $[/tex]19,999[tex]$: 40 - 25 = 15 - $[/tex]20,000 - [tex]$39,999$[/tex]: 30 - 25 = 5
- [tex]$40,000 - $[/tex]59,999$: 35 - 25 = 10
- Difference in heights:
- Max height = 15, Min height = 5
- Difference = 15 - 5 = 10
After assessing each scale, we find that all scales provide a maximum difference of 10 in the bar heights. Therefore, the choice can be generalized to:
- 0-50
- 0-40 is the preferable scales.
Thus, the scale that maximizes the difference in the heights of the bars in Gemma's histogram can be either 0-50 or 0-40.
