To determine which set of intervals is appropriate for creating a histogram of the given data, we need to consider the range and distribution of the data points.
The given data points are:
[tex]\[7, 18, 23, 41, 34, 12, 48, 65, 57, 52\][/tex]
1. Sort the Data:
- First, we sort the data in ascending order:
[tex]\[7, 12, 18, 23, 34, 41, 48, 52, 57, 65\][/tex]
2. Determine the Range of Data:
- The range is the difference between the maximum and minimum values.
- Maximum value = 65
- Minimum value = 7
- Range = 65 - 7 = 58
3. Evaluate the Intervals:
- We need to choose intervals that cover the entire range of the data without leaving large gaps. Let's evaluate each option:
Option 1: [tex]\(0-9\)[/tex], [tex]\(10-19\)[/tex], [tex]\(20-29\)[/tex], [tex]\(30-39\)[/tex], [tex]\(40-50\)[/tex]
- This set of intervals only goes up to 50, which does not cover the maximum value of 65 in our data. Hence, it is not appropriate.
Option 2: [tex]\(0-14\)[/tex], [tex]\(15-29\)[/tex], [tex]\(30-44\)[/tex], [tex]\(45-59\)[/tex], [tex]\(60-74\)[/tex]
- The intervals cover the range up to 74, encompassing our maximum value of 65. However, the intervals are uneven (some are 15 units wide, and others are 14 units wide). This unevenness can make the histogram less intuitive.
Option 3: [tex]\(0-15\)[/tex], [tex]\(15-20\)[/tex], [tex]\(20-35\)[/tex], [tex]\(35-60\)[/tex], [tex]\(60-75\)[/tex]
- These intervals are irregular and mismatched. The sizes of the intervals vary significantly, making it inappropriate for a clear and consistent histogram.
Option 4: [tex]\(0-20\)[/tex], [tex]\(20-40\)[/tex], [tex]\(40-60\)[/tex], [tex]\(60-80\)[/tex]
- This set of intervals evenly covers the entire range from 0 to 80. The intervals are consistent (each 20 units wide) and encompass our data points, including the maximum value of 65.
Given the need for appropriate and consistent intervals that cover the range of the data, Option 4 is the most suitable choice.
Therefore, the appropriate intervals to use when creating a histogram of the data are:
[tex]\[0-20, 20-40, 40-60, 60-80\][/tex]
The correct answer is:
4
