Answer :
Alright! Let's address each part of the question step-by-step.
### 1.1 Draw a histogram to represent the results of the survey.
A histogram is a graphical representation of data where the data is grouped into ranges (bins) and plotted as bars. The height of each bar indicates the number of data points in that range.
Here’s how you should plot the histogram:
1. Draw the x-axis and label it as "Fat Content (%)".
2. Draw the y-axis and label it as "Number of Cows".
3. Create bars for each fat content range where the height corresponds to the number of cows in each category.
For the given data:
- [tex]\(2.6-3.0\%\)[/tex]: 11 cows
- [tex]\(3.1-3.5\%\)[/tex]: 66 cows
- [tex]\(3.6-4.0\%\)[/tex]: 93 cows
- [tex]\(4.1-4.5\%\)[/tex]: 61 cows
- [tex]\(4.6-5.0\%\)[/tex]: 15 cows
Your histogram should have bars with heights corresponding to these numbers.
### 1.2 Calculate the percentage of the farmer's cows that produce low-fat milk. Show ALL your working.
Low-fat milk is defined as milk with fat content of [tex]\(3\%\)[/tex] or less. This corresponds to the [tex]\(2.6-3.0 \%\)[/tex] fat content range.
To find the percentage of cows that produce low-fat milk:
1. Calculate the total number of cows:
[tex]\[ \text{Total cows} = 11 + 66 + 93 + 61 + 15 = 246 \][/tex]
2. Identify the number of cows that produce low-fat milk (in the [tex]\(2.6-3.0 \%\)[/tex] range):
[tex]\[ \text{Low-fat cows} = 11 \][/tex]
3. Calculate the percentage of low-fat cows:
[tex]\[ \text{Percentage of low-fat cows} = \left(\frac{\text{Low-fat cows}}{\text{Total cows}}\right) \times 100 = \left(\frac{11}{246}\right) \times 100 \approx 4.47\% \][/tex]
So, approximately [tex]\(4.47\%\)[/tex] of the farmer’s cows produce low-fat milk.
### 1.3 State the TYPE of variation that occurs in the cows, based on the evidence in the table.
The type of variation observed in the fat content of the cows' milk is continuous variation.
### 1.4 Give an explanation for your answer in QUESTION 1.3
Continuous variation is characterized by a range of small differences in a shared trait that is influenced by multiple factors. In this case, the fat content in milk isn’t limited to a few discrete categories but instead can take any value within a range (from [tex]\(2.6\%\)[/tex] to [tex]\(5.0\%\)[/tex]).
This is indicative of continuous variation because:
- The trait (fat content) shows a smooth transition from one value to another without distinct jumps.
- The histograms' bars reflect a spread of data points across a continuum rather than clustered into distinct groups (which would suggest discontinuous variation).
- Traits with continuous variation, such as milk fat content, are often influenced by multiple genes and the environment, leading to a wide spectrum of possible values.
### 1.1 Draw a histogram to represent the results of the survey.
A histogram is a graphical representation of data where the data is grouped into ranges (bins) and plotted as bars. The height of each bar indicates the number of data points in that range.
Here’s how you should plot the histogram:
1. Draw the x-axis and label it as "Fat Content (%)".
2. Draw the y-axis and label it as "Number of Cows".
3. Create bars for each fat content range where the height corresponds to the number of cows in each category.
For the given data:
- [tex]\(2.6-3.0\%\)[/tex]: 11 cows
- [tex]\(3.1-3.5\%\)[/tex]: 66 cows
- [tex]\(3.6-4.0\%\)[/tex]: 93 cows
- [tex]\(4.1-4.5\%\)[/tex]: 61 cows
- [tex]\(4.6-5.0\%\)[/tex]: 15 cows
Your histogram should have bars with heights corresponding to these numbers.
### 1.2 Calculate the percentage of the farmer's cows that produce low-fat milk. Show ALL your working.
Low-fat milk is defined as milk with fat content of [tex]\(3\%\)[/tex] or less. This corresponds to the [tex]\(2.6-3.0 \%\)[/tex] fat content range.
To find the percentage of cows that produce low-fat milk:
1. Calculate the total number of cows:
[tex]\[ \text{Total cows} = 11 + 66 + 93 + 61 + 15 = 246 \][/tex]
2. Identify the number of cows that produce low-fat milk (in the [tex]\(2.6-3.0 \%\)[/tex] range):
[tex]\[ \text{Low-fat cows} = 11 \][/tex]
3. Calculate the percentage of low-fat cows:
[tex]\[ \text{Percentage of low-fat cows} = \left(\frac{\text{Low-fat cows}}{\text{Total cows}}\right) \times 100 = \left(\frac{11}{246}\right) \times 100 \approx 4.47\% \][/tex]
So, approximately [tex]\(4.47\%\)[/tex] of the farmer’s cows produce low-fat milk.
### 1.3 State the TYPE of variation that occurs in the cows, based on the evidence in the table.
The type of variation observed in the fat content of the cows' milk is continuous variation.
### 1.4 Give an explanation for your answer in QUESTION 1.3
Continuous variation is characterized by a range of small differences in a shared trait that is influenced by multiple factors. In this case, the fat content in milk isn’t limited to a few discrete categories but instead can take any value within a range (from [tex]\(2.6\%\)[/tex] to [tex]\(5.0\%\)[/tex]).
This is indicative of continuous variation because:
- The trait (fat content) shows a smooth transition from one value to another without distinct jumps.
- The histograms' bars reflect a spread of data points across a continuum rather than clustered into distinct groups (which would suggest discontinuous variation).
- Traits with continuous variation, such as milk fat content, are often influenced by multiple genes and the environment, leading to a wide spectrum of possible values.
