Sure, let's proceed step-by-step to solve the question.
### 1.1 Draw a Histogram
To draw a histogram representing the results of the survey, we take the given fat content ranges and plot them against the number of cows in each range.
The table provides:
| Fat Content (%) | Number of Cows |
|-----------------|----------------|
| 2.8 - 3.0 | 11 |
| 3.1 - 3.5 | 66 |
| 3.6 - 4.0 | 63 |
| 4.1 - 4.5 | 81 |
| 4.6 - 5.0 | 15 |
We will not actually draw the histogram here, but the steps are as follows:
- X-axis: Fat Content (%) ranges.
- Y-axis: Number of Cows.
- Each range is represented as a bar, with the height of the bar corresponding to the number of cows in that range.
### 1.2 Calculate the Percentage of Cows Producing Low-Fat Milk
The fat content range for low-fat milk is 2.8 - 3.0%. According to the table, there are 11 cows in this range.
First, calculate the total number of cows:
[tex]\[
\text{Total number of cows} = 11 + 66 + 63 + 81 + 15 = 236
\][/tex]
Next, calculate the percentage of cows that produce low-fat milk:
[tex]\[
\text{Percentage of low-fat milk producing cows} = \left( \frac{11}{236} \right) \times 100
\][/tex]
[tex]\[
\text{Percentage} = \left( \frac{11}{236} \right) \times 100 \approx 4.66\%
\][/tex]
Thus, approximately 4.66% of the farmer's cows produce low-fat milk (with fat content 3% or less).
### 1.3 State the Type of Variation
The type of variation that occurs in the cows, based on the evidence in the table, is continuous variation.
### 1.4 Explanation for the Type of Variation
Continuous variation occurs when the characteristic can take any value within a range. In this case, the fat content of the milk can vary smoothly over a continuous range (from 2.8% to 5%). This means there are no distinct categories or groups, and the data can be represented on a continuous scale. Thus, the distribution of fat content in the cows' milk is an example of continuous variation.
