Let's analyze this problem step-by-step.

Given Data

The goal is to construct a Grouped Frequency Distribution Table (GFDT) with the following parameters:

1. 10 classes.
2. A "nice" class width (typically an integer that allows for convenient calculations).
3. Class limits that are multiples of the class width.

Solution Steps

1. Identify the Range of Data:

   • Calculate the minimum and maximum values in the data set.
   • Find the range by subtracting the minimum from the maximum.

2. Determine Optimal Class Width:

   • Divide the range by the number of desired classes (10 in this case).
   • Round this result to a convenient number (often the nearest multiple of 5 or 10) to get a "nice" class width.

3. Set the First Lower Class Limit:

   • The first lower class limit should be a number that is a multiple of the class width and ideally close to or below the minimum value of the dataset.

Applying to This Dataset

1. Find the Range

• The maximum value in the data set is $370$.
• The minimum value is $199$.
• Range = $370 - 199 = 171$.

2. Calculate Initial Class Width

• Initial class width (based on range and classes) = $171 \div 10 = 17.1$.
• A "nice" width close to this would be $20$, as specified in the question.

3. Set the First Lower Class Limit

• Since the class width is $20$, we need the first lower class limit to be a multiple of $20$ and below the minimum value $199$.
• $190$ is a suitable starting point, as it meets both criteria.

Thus, the provided answers:

• Optimal class width = 20
• 1st lower class limit = 190

These align with the solution steps and should be correct.

Would you like further details on how to construct the full GFDT or have any questions on these calculations?

Related Questions

1. How do you construct a full frequency distribution table from this dataset?
2. Why is it important for class limits to be multiples of the class width?
3. What are some benefits of using a "nice" class width?
4. How would the table change if you used a class width of 15 instead?
5. What other considerations are important when setting class limits?

Tip: In frequency distribution, choosing an appropriate class width helps in visualizing data patterns clearly and reduces overlapping.