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18. Following is the frequency distribution of yield of cane in tons per acre. Calculate the mean yield.

[tex]\[
\begin{tabular}{|c|c|}
\hline
Class-interval & Frequency \\
\hline
$35-40$ & 6 \\
$40-45$ & 7 \\
$45-50$ & 11 \\
$50-55$ & 25 \\
$55-60$ & 31 \\
$60-65$ & 41 \\
$65-70$ & 41 \\
$70-75$ & 14 \\
$75-80$ & 16 \\
$80-85$ & 8 \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnsweridn
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].
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