Find the actual lower and upper class limits and also the class marks of the classes :
1.1 - 2.0, 2.1 - 3.0 and 3.1 - 4.0.

Answer

To find the actual lower and upper class limits and the class marks for the given classes \(1.1 - 2.0\), \(2.1 - 3.0\), and \(3.1 - 4.0\), we will follow these steps:

### Step 1: Identify the Class Intervals
The given class intervals are:
- Class 1: \(1.1 - 2.0\)
- Class 2: \(2.1 - 3.0\)
- Class 3: \(3.1 - 4.0\)

### Step 2: Determine the Breaks in the Intervals
Notice that there is a break between:
- \(2.0\) and \(2.1\)
- \(3.0\) and \(3.1\)

This indicates that the data is not continuous, and we need to adjust the limits.

### Step 3: Calculate the Half of the Difference
The difference between the upper limit of one class and the lower limit of the next class is:
- From \(2.0\) to \(2.1\): \(2.1 - 2.0 = 0.1\)
- From \(3.0\) to \(3.1\): \(3.1 - 3.0 = 0.1\)

Half of this difference is:
\[ \frac{0.1}{2} = 0.05 \]

### Step 4: Adjust the Class Limits
Now we will adjust the lower and upper class limits:
- For Class 1:
- Actual lower limit = \(1.1 - 0.05 = 1.05\)
- Actual upper limit = \(2.0 + 0.05 = 2.05\)

- For Class 2:
- Actual lower limit = \(2.1 - 0.05 = 2.05\)
- Actual upper limit = \(3.0 + 0.05 = 3.05\)

- For Class 3:
- Actual lower limit = \(3.1 - 0.05 = 3.05\)
- Actual upper limit = \(4.0 + 0.05 = 4.05\)

### Step 5: Summarize the Actual Limits
- Actual lower limits: \(1.05, 2.05, 3.05\)
- Actual upper limits: \(2.05, 3.05, 4.05\)

### Step 6: Calculate the Class Marks
The class mark is calculated using the formula:
\[ \text{Class Mark} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]

- For Class 1:
\[ \text{Class Mark} = \frac{1.05 + 2.05}{2} = \frac{3.10}{2} = 1.55 \]

- For Class 2:
\[ \text{Class Mark} = \frac{2.05 + 3.05}{2} = \frac{5.10}{2} = 2.55 \]

- For Class 3:
\[ \text{Class Mark} = \frac{3.05 + 4.05}{2} = \frac{7.10}{2} = 3.55 \]

### Step 7: Summarize the Class Marks
- Class Marks: \(1.55, 2.55, 3.55\)

### Final Summary
- Actual lower limits: \(1.05, 2.05, 3.05\)
- Actual upper limits: \(2.05, 3.05, 4.05\)
- Class marks: \(1.55, 2.55, 3.55\)

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