To determine the appropriate scale for the vertical axis of the histogram that maximizes the differences in the heights of the bars, we need to examine the differences in the number of people in each salary range.
Let's outline the differences in the height of the bars (number of people) for each salary range:
1. The number of people in the [tex]$0-$[/tex]19,999 range is 40.
2. The number of people in the [tex]$20,000-$[/tex]39,999 range is 30.
3. The number of people in the [tex]$40,000-$[/tex]59,999 range is 35.
Next, let's calculate the differences between these values:
- The difference between the [tex]$0-$[/tex]19,999 and [tex]$20,000-$[/tex]39,999 ranges is:
[tex]\( |40 - 30| = 10 \)[/tex]
- The difference between the [tex]$0-$[/tex]19,999 and [tex]$40,000-$[/tex]59,999 ranges is:
[tex]\( |40 - 35| = 5 \)[/tex]
- The difference between the [tex]$20,000-$[/tex]39,999 and [tex]$40,000-$[/tex]59,999 ranges is:
[tex]\( |30 - 35| = 5 \)[/tex]
We find that the differences are 10, 5, and 5.
Given this information, let's review the proposed scales:
1. [tex]$0-50$[/tex]: The maximum difference is 50, which accommodates all values without being excessively large.
2. [tex]$0-40$[/tex]: This scale might not be sufficient since the highest bar reaches 40.
3. [tex]$10-50$[/tex]: This scale does not include zero, which can be problematic since some bars start at 0.
4. [tex]$25-40$[/tex]: This scale also does not start at zero and might not adequately represent bars starting from zero.
The most appropriate scale that maximizes the differences in height while being representative of the data is [tex]$0-50$[/tex]. This scale allows all values to be shown clearly with the differences being observable.
Therefore, Gemma should use the [tex]$0-50$[/tex] scale for the vertical axis of her histogram.
