It looks like you're working on creating a grouped frequency distribution table (GFDT) for a dataset. The task involves dividing the data into 10 classes with an appropriate class width, ensuring that the class intervals are clearly defined using closed intervals. Let’s solve this step-by-step:

Step 1: Determine the Range

The range of the data is calculated as: $\text{Range} = \text{Maximum Value} - \text{Minimum Value}.$

From the image:

• Minimum value: $137$
• Maximum value: $598$

$\text{Range} = 598 - 137 = 461.$

Step 2: Calculate Class Width

To create 10 classes, divide the range by the number of classes, rounding up to the nearest whole number: $\text{Class Width} = \lceil \frac{\text{Range}}{\text{Number of Classes}} \rceil = \lceil \frac{461}{10} \rceil = 47.$

Step 3: Define the Class Intervals

Using the class width of 47 and the minimum value as the starting point, the intervals are:

1. $[130, 179]$
2. $[180, 229]$
3. $[230, 279]$
4. $[280, 329]$
5. $[330, 379]$
6. $[380, 429]$
7. $[430, 479]$
8. $[480, 529]$
9. $[530, 579]$
10. $[580, 629]$

Step 4: Tally the Frequencies

Count how many data points fall into each interval. From your data, the frequency tally is already provided, and it matches the intervals above.

• $[130, 179]$: 4
• $[180, 229]$: 5
• $[230, 279]$: 6
• $[280, 329]$: 5
• $[330, 379]$: 3
• $[380, 429]$: 2
• $[430, 479]$: 4
• $[480, 529]$: 2
• $[530, 579]$: 1
• $[580, 629]$: 2

Final GFDT Table

Let me know if you'd like further details or clarification! Here are some follow-up questions:

1. Would you like a step-by-step explanation of how the frequencies were counted?
2. Do you need assistance visualizing this data in a histogram?
3. Would you like me to calculate cumulative frequencies for this data?
4. Should we discuss how the class width impacts the distribution table?
5. Do you want help interpreting the results of the GFDT?

Tip: Always use a calculator or spreadsheet to double-check frequencies when dealing with larger datasets!