To find the mean of the given frequency distribution using the class midpoints, we will follow these steps:
1. Identify the class midpoints for each class.
2. Multiply each class midpoint by its corresponding frequency to get the weighted sums.
3. Sum all the frequencies to get the total frequency.
4. Sum the weighted sums.
5. Divide the sum of the weighted sums by the total frequency to find the mean.
6. Round the mean to two decimal places.
Let's work through these steps in detail.
### Step 1: Identify the Class Midpoints
The class midpoint for a class interval is calculated by taking the average of the lower and upper boundaries of the class.
[tex]\[
\text{Class Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}
\][/tex]
For each class:
- For the class [tex]\([60-69]\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{60 + 69}{2} = 64.5
\][/tex]
- For the class [tex]\([70-79]\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{70 + 79}{2} = 74.5
\][/tex]
- For the class [tex]\([80-89]\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{80 + 89}{2} = 84.5
\][/tex]
- For the class [tex]\([90-99]\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{90 + 99}{2} = 94.5
\][/tex]
So, the class midpoints are [tex]\(64.5, 74.5, 84.5, \text{ and } 94.5\)[/tex].
### Step 2: Multiply Each Midpoint by Its Corresponding Frequency
Next, we multiply each class midpoint by the frequency of its class to find the weighted sum for each class:
[tex]\[
\text{Weighted Sum for Each Class} = \text{Class Midpoint} \times \text{Frequency}
\][/tex]
For each class:
- Class [tex]\(60-69\)[/tex]:
[tex]\[
64.5 \times 3 = 193.5
\][/tex]
- Class [tex]\(70-79\)[/tex]:
[tex]\[
74.5 \times 12 = 894
\][/tex]
- Class [tex]\(80-89\)[/tex]:
[tex]\[
84.5 \times 7 = 591.5
\][/tex]
- Class [tex]\(90-99\)[/tex]:
[tex]\[
94.5 \times 2 = 189
\][/tex]
### Step 3: Sum All the Frequencies
[tex]\[
\text{Total Frequency} = 3 + 12 + 7 + 2 = 24
\][/tex]
### Step 4: Sum All the Weighted Sums
[tex]\[
\text{Total Weighted Sum} = 193.5 + 894 + 591.5 + 189 = 1868
\][/tex]
### Step 5: Divide the Total Weighted Sum by the Total Frequency
[tex]\[
\text{Mean} = \frac{\text{Total Weighted Sum}}{\text{Total Frequency}} = \frac{1868}{24} \approx 77.83
\][/tex]
### Step 6: Round the Mean to Two Decimal Places
The mean of the grouped frequency distribution, rounded to two decimal places, is:
[tex]\[
77.83
\][/tex]
Hence, the approximate mean for the given frequency distribution is [tex]\(77.83\)[/tex].
