To calculate the mean yield from the given frequency distribution data, we need to follow a systematic approach. Here are the steps involved:
### Step 1: Identify the Class Intervals and Frequencies
The class intervals and their corresponding frequencies are as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Class-Interval} & \text{Frequency} \\
\hline
35-40 & 6 \\
40-45 & 7 \\
45-50 & 11 \\
50-55 & 25 \\
55-60 & 31 \\
60-65 & 41 \\
65-70 & 41 \\
70-75 & 14 \\
75-80 & 16 \\
80-85 & 8 \\
\hline
\end{array}
\][/tex]
### Step 2: Determine the Midpoints of Each Class Interval
The midpoint (or class mark) of each interval can be calculated using the formula:
[tex]\[
\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}
\][/tex]
Calculating the midpoints for each class interval:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Class-Interval} & \text{Midpoint (x)} \\
\hline
35-40 & 37.5 \\
40-45 & 42.5 \\
45-50 & 47.5 \\
50-55 & 52.5 \\
55-60 & 57.5 \\
60-65 & 62.5 \\
65-70 & 67.5 \\
70-75 & 72.5 \\
75-80 & 77.5 \\
80-85 & 82.5 \\
\hline
\end{array}
\][/tex]
### Step 3: Calculate [tex]\(\sum f_i \cdot x_i\)[/tex]
Multiply each midpoint by its corresponding frequency and sum the results:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Class-Interval} & \text{Frequency (f)_i} & \text{Midpoint (x)_i} & f_i \cdot x_i \\
\hline
35-40 & 6 & 37.5 & 225.0 \\
40-45 & 7 & 42.5 & 297.5 \\
45-50 & 11 & 47.5 & 522.5 \\
50-55 & 25 & 52.5 & 1312.5 \\
55-60 & 31 & 57.5 & 1782.5 \\
60-65 & 41 & 62.5 & 2562.5 \\
65-70 & 41 & 67.5 & 2767.5 \\
70-75 & 14 & 72.5 & 1015.0 \\
75-80 & 16 & 77.5 & 1240.0 \\
80-85 & 8 & 82.5 & 660.0 \\
\hline
& \sum f_i = 200 & & \sum f_i \cdot x_i = 12385.0 \\
\hline
\end{array}
\][/tex]
### Step 4: Calculate the Mean Yield
The mean yield [tex]\(\bar{x}\)[/tex] is calculated using the formula:
[tex]\[
\bar{x} = \frac{\sum f_i \cdot x_i}{\sum f_i}
\][/tex]
Using the values obtained:
[tex]\[
\bar{x} = \frac{12385.0}{200} = 61.925
\][/tex]
### Conclusion
The mean yield of cane in tons per acre is [tex]\(\boxed{61.925}\)[/tex].
