To solve this, we first need to calculate the range of the data, determine the optimal class width, and decide on the first lower class limit.

Steps:

1. Find the Range:

   • The range is calculated as: $\text{Range} = \text{Maximum value} - \text{Minimum value}$ From the dataset, the maximum value is $944$ and the minimum value is $468$: $\text{Range} = 944 - 468 = 476$

2. Optimal Class Width: To construct a grouped frequency distribution with 10 classes, the class width is: $\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}}$ $\text{Class Width} = \frac{476}{10} = 47.6$ Since class width needs to be a "nice" round number (e.g., 5, 10, 20, etc.), round it up to the nearest convenient value. Here, the optimal class width is 50.

3. Determine the First Lower Class Limit: The first lower class limit should ideally be slightly below the minimum value and be a multiple of the class width. Since $468$ is the minimum value, and the class width is $50$, the first lower class limit is 450, which is a multiple of 50.

Final Answers:

• Optimal Class Width: 50
• First Lower Class Limit: 450

Let me know if you'd like a detailed explanation of constructing the grouped frequency distribution table. Here are some related questions to consider:

1. How do we calculate the upper class limits for this data set?
2. How do we construct a complete grouped frequency distribution table?
3. What is the importance of choosing a "nice" class width?
4. How can we plot a histogram from the grouped frequency distribution?
5. What other statistics (e.g., mean, median) can we calculate from this grouped data?

Tip: Always choose a class width that simplifies interpretation while covering the entire range of data values.