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To determine the number of farmers in each category, we need to fill out the contingency table for the given data. Here's the detailed step-by-step solution:
### Step-by-Step Solution
1. Total Farmers: 126
2. Farmers Planted Carrots: 72
3. Farmers Planted Potatoes: 45
4. Farmers Planted Both Carrots and Potatoes: 38
5. Farmers Not Planted Either Carrots or Potatoes: 47
First, we need to calculate the number of farmers who planted only carrots, only potatoes, and either carrots or potatoes.
### Only Carrots:
Farmers who planted only carrots are those who planted carrots but not potatoes. We subtract the farmers who planted both from those who planted carrots:
[tex]\[ \text{Only Carrots} = \text{Farmers Planted Carrots} - \text{Farmers Planted Both} \][/tex]
[tex]\[ = 72 - 38 = 34 \][/tex]
### Only Potatoes:
Farmers who planted only potatoes are those who planted potatoes but not carrots. We subtract the farmers who planted both from those who planted potatoes:
[tex]\[ \text{Only Potatoes} = \text{Farmers Planted Potatoes} - \text{Farmers Planted Both} \][/tex]
[tex]\[ = 45 - 38 = 7 \][/tex]
### Either Carrots or Potatoes (or both):
To find the number of farmers who planted either carrots or potatoes or both, we subtract the farmers who did not plant either from the total number of farmers:
[tex]\[ \text{Planted Either} = \text{Total Farmers} - \text{Farmers Not Planted} \][/tex]
[tex]\[ = 126 - 47 = 79 \][/tex]
### Filling in the Table:
Using our calculated values, we can fill in the contingency table:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & Carrots & No Carrots & Total \\ \hline Potatoes & 38 & 7 & 45 \\ \hline No Potatoes & 34 & 47 & 81 \\ \hline Total & 72 & 54 & 126 \\ \hline \end{tabular} \][/tex]
### Explanation for Each Box:
- Carrots + Potatoes (Top-Left): 38 (farmers who planted both)
- Carrots + No Potatoes (Bottom-Left): 34 (only carrots)
- No Carrots + Potatoes (Top-Right): 7 (only potatoes)
- No Carrots + No Potatoes (Bottom-Right): 47 (neither)
Summarizing, the contingency table provides a clear breakdown of the survey results, allowing us to understand the planting habits of the farmers.
### Step-by-Step Solution
1. Total Farmers: 126
2. Farmers Planted Carrots: 72
3. Farmers Planted Potatoes: 45
4. Farmers Planted Both Carrots and Potatoes: 38
5. Farmers Not Planted Either Carrots or Potatoes: 47
First, we need to calculate the number of farmers who planted only carrots, only potatoes, and either carrots or potatoes.
### Only Carrots:
Farmers who planted only carrots are those who planted carrots but not potatoes. We subtract the farmers who planted both from those who planted carrots:
[tex]\[ \text{Only Carrots} = \text{Farmers Planted Carrots} - \text{Farmers Planted Both} \][/tex]
[tex]\[ = 72 - 38 = 34 \][/tex]
### Only Potatoes:
Farmers who planted only potatoes are those who planted potatoes but not carrots. We subtract the farmers who planted both from those who planted potatoes:
[tex]\[ \text{Only Potatoes} = \text{Farmers Planted Potatoes} - \text{Farmers Planted Both} \][/tex]
[tex]\[ = 45 - 38 = 7 \][/tex]
### Either Carrots or Potatoes (or both):
To find the number of farmers who planted either carrots or potatoes or both, we subtract the farmers who did not plant either from the total number of farmers:
[tex]\[ \text{Planted Either} = \text{Total Farmers} - \text{Farmers Not Planted} \][/tex]
[tex]\[ = 126 - 47 = 79 \][/tex]
### Filling in the Table:
Using our calculated values, we can fill in the contingency table:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & Carrots & No Carrots & Total \\ \hline Potatoes & 38 & 7 & 45 \\ \hline No Potatoes & 34 & 47 & 81 \\ \hline Total & 72 & 54 & 126 \\ \hline \end{tabular} \][/tex]
### Explanation for Each Box:
- Carrots + Potatoes (Top-Left): 38 (farmers who planted both)
- Carrots + No Potatoes (Bottom-Left): 34 (only carrots)
- No Carrots + Potatoes (Top-Right): 7 (only potatoes)
- No Carrots + No Potatoes (Bottom-Right): 47 (neither)
Summarizing, the contingency table provides a clear breakdown of the survey results, allowing us to understand the planting habits of the farmers.
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