Alright, let's take this step-by-step to understand the distribution and calculate the necessary values:
## Problem Breakdown:
The table shows the distribution of hours students spend on homework in a week and its frequencies. We need to determine the class widths and class midpoints.
## Classes and Frequencies:
1. Class: 17-21, Frequency: 6
2. Class: 22-26, Frequency: 8
3. Class: 27-31, Frequency: 9
4. Class: 32-36, Frequency: 12
5. Class: 37-41, Frequency: 6
## Step 1: Class Widths Calculation
The class width is the difference between the upper and lower boundaries of each class, plus one. This is because class limits are inclusive.
- For class [tex]\(17-21\)[/tex]:
[tex]\[
\text{Class Width} = 21 - 17 + 1 = 5
\][/tex]
- For class [tex]\(22-26\)[/tex]:
[tex]\[
\text{Class Width} = 26 - 22 + 1 = 5
\][/tex]
- For class [tex]\(27-31\)[/tex]:
[tex]\[
\text{Class Width} = 31 - 27 + 1 = 5
\][/tex]
- For class [tex]\(32-36\)[/tex]:
[tex]\[
\text{Class Width} = 36 - 32 + 1 = 5
\][/tex]
- For class [tex]\(37-41\)[/tex]:
[tex]\[
\text{Class Width} = 41 - 37 + 1 = 5
\][/tex]
So, the class widths for all intervals are uniformly 5.
## Step 2: Class Midpoints Calculation
The midpoint of each class is calculated by taking the average of the lower and upper boundaries of that class:
- For class [tex]\(17-21\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{17 + 21}{2} = 19.0
\][/tex]
- For class [tex]\(22-26\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{22 + 26}{2} = 24.0
\][/tex]
- For class [tex]\(27-31\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{27 + 31}{2} = 29.0
\][/tex]
- For class [tex]\(32-36\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{32 + 36}{2} = 34.0
\][/tex]
- For class [tex]\(37-41\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{37 + 41}{2} = 39.0
\][/tex]
## Results:
- Class Widths: [5, 5, 5, 5, 5]
- Class Midpoints: [19.0, 24.0, 29.0, 34.0, 39.0]
Hence, we have successfully calculated the class widths and the class midpoints for the given frequency table representing the distribution of hours students spend on homework in a week.
