To calculate the geometric mean of the given data, we need to follow these steps:
### Step 1: Understand the Data
We have two columns of data: 'Yield' and 'No. of Farms' corresponding to each yield value. Here's the data:
| Yield | No. of Farms |
|-------|--------------|
| 7.5 | 5 |
| 13.0 | 8 |
| 18.5 | 10 |
| 20.5 | 14 |
| 22.0 | 6 |
| 23.0 | 7 |
| 24.0 | 3 |
| 25.0 | 4 |
| 26.0 | 2 |
| 28.0 | 1 |
### Step 2: Calculate the Total Number of Farms
First, sum up the numbers in the 'No. of Farms' column to find the total number of farms.
[tex]\[
5 + 8 + 10 + 14 + 6 + 7 + 3 + 4 + 2 + 1 = 60
\][/tex]
### Step 3: Calculate the Weighted Product
Next, calculate the product of each yield raised to the power of its corresponding frequency (number of farms).
[tex]\[
(7.5^5) \times (13.0^8) \times (18.5^{10}) \times (20.5^{14}) \times (22.0^6) \times (23.0^7) \times (24.0^3) \times (25.0^4) \times (26.0^2) \times (28.0^1)
\][/tex]
[tex]\[
\approx 8.303360589043404 \times 10^{75}
\][/tex]
### Step 4: Calculate the Geometric Mean
The formula for the geometric mean (GM) for a set of values [tex]\( x_1, x_2, \ldots, x_n \)[/tex] each with frequencies [tex]\( f_1, f_2, \ldots, f_n \)[/tex] is given by:
[tex]\[
GM = \left( \prod_{i=1}^{n} x_i^{f_i} \right)^{\frac{1}{\sum_{i=1}^{n} f_i}}
\][/tex]
Using the total number of farms and the weighted product calculated above:
[tex]\[
GM = \left( 8.303360589043404 \times 10^{75} \right)^{\frac{1}{60}}
\][/tex]
### Step 5: Obtain the Result
[tex]\[
GM \approx 18.421326425299853
\][/tex]
### Final Answer
The geometric mean of the given yields is approximately 18.4213.
