From the uploaded image, you are asked to determine the optimal class width and the first lower class limit to construct a grouped frequency distribution table (GFDT) with the following instructions:

1. There should be 10 classes.
2. Each class width should be a "nice" number.
3. The lower class limits must all be multiples of the class width.

Step-by-step Process:

1. Find the Range:

   • Range = $\text{Maximum Value} - \text{Minimum Value}$.
   • From the table, the maximum value appears to be $3.47$ and the minimum value appears to be $2.56$.
   • $\text{Range} = 3.47 - 2.56 = 0.91$.

2. Calculate Class Width:

   • Class Width $= \frac{\text{Range}}{\text{Number of Classes}}$.
   • $\frac{0.91}{10} = 0.091$.
   • Round this up to a convenient "nice" number. A good choice would be 0.1.

3. Choose the First Lower Class Limit:

   • The lower class limit should be a multiple of the class width.
   • A reasonable choice here is 2.5, as it is a multiple of 0.1 and is slightly below the minimum value of the data set.

Final Answers:

• Optimal class width = 0.1
• First lower class limit = 2.5

If you have further questions or need additional clarifications, feel free to ask!

Here are 5 related questions to expand your understanding:

1. What happens if I choose a smaller or larger number of classes than 10?
2. Why do we round the class width to a "nice" number instead of using the exact calculation?
3. How do you verify that the chosen lower class limit captures all data points?
4. What would the boundaries of the first class look like with this setup?
5. How does the distribution of data affect the shape of the histogram generated from the GFDT?

Tip: Always ensure your class intervals fully capture the range of your data to avoid losing any data points during the grouping process.