To construct a grouped frequency distribution table (GFDT) with 10 classes, we need to determine the optimal class width and the first lower class limit.

Step-by-Step Solution

1. Identify the Range:

   • Find the minimum and maximum values in the dataset to determine the range.
   • By inspecting the values in the image, the minimum is $342$ and the maximum is $797$.
   • Range $= \text{Maximum} - \text{Minimum} = 797 - 342 = 455$.

2. Determine Class Width:

   • To get 10 classes, we divide the range by the number of classes and round up to a "nice" number if necessary.
   • Class Width $= \frac{\text{Range}}{\text{Number of Classes}} = \frac{455}{10} = 45.5$.
   • Rounding to a more convenient class width, $50$ would be a suitable choice.

   So, the optimal class width is $50$.

3. Determine the First Lower Class Limit:

   • Typically, the first lower class limit should be a convenient number that is less than or equal to the minimum value in the dataset and a multiple of the class width.
   • The minimum value is $342$, so a suitable lower class limit could be $300$ (a multiple of 50).

   Thus, the first lower class limit is $300$.

Summary

• Optimal Class Width: $50$
• First Lower Class Limit: $300$

Would you like further details on how to construct the entire frequency distribution table, or do you have any questions about this solution?

Here are some questions to extend this topic:

1. How would you adjust the class width if you needed fewer or more classes?
2. What would be the impact of choosing a different starting lower class limit?
3. How can you calculate the midpoint of each class in the distribution table?
4. How do you determine the upper class limit for each class?
5. How would you construct a histogram based on this frequency distribution?

Tip: When constructing a frequency distribution table, always verify that each class width is consistent to maintain uniformity in the distribution.