To solve this problem, let’s go through the steps needed to find the range and the optimal class width for the data set:

Step 1: Calculate the Range of the Data Set

The range is calculated by subtracting the minimum value from the maximum value in the data set.

• Maximum value in the data set: $64.7$
• Minimum value in the data set: $40.1$

$\text{Range} = 64.7 - 40.1 = 24.6$

So, the range of this data set is 24.6.

Step 2: Determine the Optimal Class Width

To create 10 classes with an even distribution, the class width can be calculated as:

$\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}}$

Substituting the range (24.6) and the desired number of classes (10):

$\text{Class Width} = \frac{24.6}{10} = 2.46$

Since we need a "nice" class width, we should round this number to a convenient value, typically to the nearest whole or commonly used decimal value. In this case, rounding 2.46 to 2.5 would be appropriate as it provides a rounded and manageable interval.

Therefore, the optimal class width is 2.5.

Would you like further clarification on any part of this process?

Here are some related questions to expand on this topic:

1. How do we select "nice" class widths when rounding the calculated class width?
2. Why is it essential to choose the number of classes appropriately in a frequency distribution?
3. How would the process differ if we were constructing a cumulative frequency distribution?
4. What are some common methods for determining class intervals in grouped data?
5. How would changing the number of classes impact the distribution and interpretation of the data?

Tip: Choosing "nice" class widths helps make the frequency distribution easier to interpret and analyze, often aligning with rounded numbers for better readability.