Answer :
Let's analyze the data step-by-step as provided in the table:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline \text{Number of eggs} & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Number of nests} & 4 & 6 & 24 & 50 & 12 & 4 \\ \hline \end{array} \][/tex]
### (i) Modal number of eggs per nest
The mode is the value that appears most frequently. To find the modal number of eggs per nest, look for the highest frequency in the "Number of nests" row.
We compare the frequencies:
- For 2 eggs, there are 4 nests.
- For 3 eggs, there are 6 nests.
- For 4 eggs, there are 24 nests.
- For 5 eggs, there are 50 nests.
- For 6 eggs, there are 12 nests.
- For 7 eggs, there are 4 nests.
The highest frequency is 50 nests, which corresponds to the number of eggs being 5.
So, the modal number of eggs per nest is [tex]\(5\)[/tex].
### (ii) Median number of eggs per nest
To find the median, we must consider the total dataset by expanding the counts. The median is the middle value of a dataset when it is ordered from smallest to largest.
First, let's list the total dataset:
- [tex]\(2\)[/tex] eggs appearing [tex]\(4\)[/tex] times, i.e., [tex]\( [2, 2, 2, 2] \)[/tex]
- [tex]\(3\)[/tex] eggs appearing [tex]\(6\)[/tex] times, i.e., [tex]\( [3, 3, 3, 3, 3, 3] \)[/tex]
- [tex]\(4\)[/tex] eggs appearing [tex]\(24\)[/tex] times, i.e., [tex]\( [4 \text{ repeated } 24 \text{ times}] \)[/tex]
- [tex]\(5\)[/tex] eggs appearing [tex]\(50\)[/tex] times, i.e., [tex]\( [5 \text{ repeated } 50 \text{ times}] \)[/tex]
- [tex]\(6\)[/tex] eggs appearing [tex]\(12\)[/tex] times, i.e., [tex]\( [6 \text{ repeated } 12 \text{ times}] \)[/tex]
- [tex]\(7\)[/tex] eggs appearing [tex]\(4\)[/tex] times, i.e., [tex]\( [7, 7, 7, 7] \)[/tex]
The total number of nests is [tex]\(4 + 6 + 24 + 50 + 12 + 4 = 100\)[/tex].
Since there are 100 observations, the median is the average of the 50th and 51st values once the data is sorted.
From the counts, the first [tex]\(4 + 6 + 24 = 34\)[/tex] values are 2s, 3s, and 4s, and the next 50 values are 5s. Therefore, the 50th and 51st values will both be [tex]\(5\)[/tex].
Hence, the median number of eggs per nest is [tex]\(5\)[/tex].
### (iii) Mean number of eggs per nest
To calculate the mean number of eggs per nest, we use the weighted mean formula. The mean is the total number of eggs divided by the total number of nests.
- Total number of eggs = [tex]\(2 \times 4 + 3 \times 6 + 4 \times 24 + 5 \times 50 + 6 \times 12 + 7 \times 4\)[/tex]
[tex]\[ = 8 + 18 + 96 + 250 + 72 + 28 = 472 \][/tex]
- Total number of nests = [tex]\(4 + 6 + 24 + 50 + 12 + 4 = 100\)[/tex]
The mean number of eggs per nest:
[tex]\[ \text{Mean} = \frac{\text{Total number of eggs}}{\text{Total number of nests}} = \frac{472}{100} = 4.72 \][/tex]
We round 4.72 to one decimal place, so the mean number of eggs per nest is [tex]\(4.7\)[/tex].
### Final Results
So, the answers are as follows:
1. Modal number of eggs per nest: [tex]\(5\)[/tex]
2. Median number of eggs per nest: [tex]\(5\)[/tex]
3. Mean number of eggs per nest (rounded to one decimal place): [tex]\(4.7\)[/tex]
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline \text{Number of eggs} & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text{Number of nests} & 4 & 6 & 24 & 50 & 12 & 4 \\ \hline \end{array} \][/tex]
### (i) Modal number of eggs per nest
The mode is the value that appears most frequently. To find the modal number of eggs per nest, look for the highest frequency in the "Number of nests" row.
We compare the frequencies:
- For 2 eggs, there are 4 nests.
- For 3 eggs, there are 6 nests.
- For 4 eggs, there are 24 nests.
- For 5 eggs, there are 50 nests.
- For 6 eggs, there are 12 nests.
- For 7 eggs, there are 4 nests.
The highest frequency is 50 nests, which corresponds to the number of eggs being 5.
So, the modal number of eggs per nest is [tex]\(5\)[/tex].
### (ii) Median number of eggs per nest
To find the median, we must consider the total dataset by expanding the counts. The median is the middle value of a dataset when it is ordered from smallest to largest.
First, let's list the total dataset:
- [tex]\(2\)[/tex] eggs appearing [tex]\(4\)[/tex] times, i.e., [tex]\( [2, 2, 2, 2] \)[/tex]
- [tex]\(3\)[/tex] eggs appearing [tex]\(6\)[/tex] times, i.e., [tex]\( [3, 3, 3, 3, 3, 3] \)[/tex]
- [tex]\(4\)[/tex] eggs appearing [tex]\(24\)[/tex] times, i.e., [tex]\( [4 \text{ repeated } 24 \text{ times}] \)[/tex]
- [tex]\(5\)[/tex] eggs appearing [tex]\(50\)[/tex] times, i.e., [tex]\( [5 \text{ repeated } 50 \text{ times}] \)[/tex]
- [tex]\(6\)[/tex] eggs appearing [tex]\(12\)[/tex] times, i.e., [tex]\( [6 \text{ repeated } 12 \text{ times}] \)[/tex]
- [tex]\(7\)[/tex] eggs appearing [tex]\(4\)[/tex] times, i.e., [tex]\( [7, 7, 7, 7] \)[/tex]
The total number of nests is [tex]\(4 + 6 + 24 + 50 + 12 + 4 = 100\)[/tex].
Since there are 100 observations, the median is the average of the 50th and 51st values once the data is sorted.
From the counts, the first [tex]\(4 + 6 + 24 = 34\)[/tex] values are 2s, 3s, and 4s, and the next 50 values are 5s. Therefore, the 50th and 51st values will both be [tex]\(5\)[/tex].
Hence, the median number of eggs per nest is [tex]\(5\)[/tex].
### (iii) Mean number of eggs per nest
To calculate the mean number of eggs per nest, we use the weighted mean formula. The mean is the total number of eggs divided by the total number of nests.
- Total number of eggs = [tex]\(2 \times 4 + 3 \times 6 + 4 \times 24 + 5 \times 50 + 6 \times 12 + 7 \times 4\)[/tex]
[tex]\[ = 8 + 18 + 96 + 250 + 72 + 28 = 472 \][/tex]
- Total number of nests = [tex]\(4 + 6 + 24 + 50 + 12 + 4 = 100\)[/tex]
The mean number of eggs per nest:
[tex]\[ \text{Mean} = \frac{\text{Total number of eggs}}{\text{Total number of nests}} = \frac{472}{100} = 4.72 \][/tex]
We round 4.72 to one decimal place, so the mean number of eggs per nest is [tex]\(4.7\)[/tex].
### Final Results
So, the answers are as follows:
1. Modal number of eggs per nest: [tex]\(5\)[/tex]
2. Median number of eggs per nest: [tex]\(5\)[/tex]
3. Mean number of eggs per nest (rounded to one decimal place): [tex]\(4.7\)[/tex]
