To determine the optimal class width and the first lower class limit for a grouped frequency distribution table (GFDT), we'll follow these steps:

Step 1: Find the Range of the Data

The range is calculated as: $\text{Range} = \text{Maximum value} - \text{Minimum value}$ From the data set:

• Maximum value = 276
• Minimum value = 183

$\text{Range} = 276 - 183 = 93$

Step 2: Determine the Class Width

To get a "nice" class width that makes the classes easy to interpret and ensures that each class contains its lower class limit as a multiple of the class width, we can follow this approach:

The class width $w$ can be approximated by: $w = \frac{\text{Range}}{\text{Number of classes}}$

Given that there should be 10 classes: $w \approx \frac{93}{10} = 9.3$

We round this to a "nice" number, typically a value that is easy to work with (e.g., 5, 10, 15, etc.). The best "nice" number closest to 9.3 is 10.

Step 3: Determine the First Lower Class Limit

The first lower class limit should be a multiple of the class width. Since the smallest data value is 183, the first lower class limit should be the multiple of 10 that is less than or equal to 183. The closest multiple of 10 is 180.

Summary:

• Optimal class width = 10
• First lower class limit = 180

Would you like more details on constructing the entire grouped frequency distribution table?

Here are five questions you might consider next:

1. How do you calculate the upper class limits for each class?
2. How would you determine the class boundaries in this grouped frequency distribution?
3. What steps are involved in tallying the frequencies for each class?
4. How can you represent this data visually using a histogram?
5. How does changing the class width affect the grouped frequency distribution?

Tip: When determining class widths, always ensure that the width is "nice" to make the data easier to interpret and use.