Alright, let's work through these questions step-by-step.
### Given Data
We have the following information from the survey on the number of eggs in bird nests:
| Number of eggs | 2 | 3 | 4 | 5 | 6 | 7 |
|----------------|---|---|---|---|---|---|
| Number of nests| 4 | 6 | 24 | 5 | 12 | 4 |
### a) (i) Modal Number of Eggs per Nest
The mode is the value that appears most frequently in a data set.
1. We look at the "Number of nests" row to determine which "Number of eggs" has the highest frequency.
2. The highest frequency is 24 nests with 4 eggs.
So, the modal number of eggs per nest is 4.
### a) (ii) Median Number of Eggs per Nest
The median is the middle value of the ordered data set. If the data set has an even number of observations, the median is the average of the two middle values.
1. List each egg count by its corresponding number of nests:
- 2 eggs × 4 nests = 8 entries
- 3 eggs × 6 nests = 18 entries
- 4 eggs × 24 nests = 96 entries
- 5 eggs × 5 nests = 25 entries
- 6 eggs × 12 nests = 72 entries
- 7 eggs × 4 nests = 28 entries
2. Combine these into a single list and sort it:
```
[2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6,
6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7]
```
3. Count the total number of nests: [tex]\( 4 + 6 + 24 + 5 + 12 + 4 = 55 \)[/tex]. Thus, there are 55 entries in the list above.
4. Since 55 is odd, the median is the middle value, which is the 28th entry in the sorted list. The 28th entry is 4.
So, the median number of eggs per nest is 4.
### 3) Average Number of Eggs per Nest
The average number of eggs per nest is calculated by dividing the total number of eggs by the total number of nests.
1. Calculate the total number of eggs:
- [tex]\( 2 \times 4 = 8 \)[/tex]
- [tex]\( 3 \times 6 = 18 \)[/tex]
- [tex]\( 4 \times 24 = 96 \)[/tex]
- [tex]\( 5 \times 5 = 25 \)[/tex]
- [tex]\( 6 \times 12 = 72 \)[/tex]
- [tex]\( 7 \times 4 = 28 \)[/tex]
2. Total number of eggs is [tex]\( 8 + 18 + 96 + 25 + 72 + 28 = 247 \)[/tex].
3. Total number of nests is [tex]\( 4 + 6 + 24 + 5 + 12 + 4 = 55 \)[/tex].
4. Calculate the average:
[tex]\[
\text{Average number of eggs per nest} = \frac{\text{Total number of eggs}}{\text{Total number of nests}} = \frac{247}{55} \approx 4.5
\][/tex]
So, the average number of eggs per nest is 4.5 (rounded to one decimal place).
### Summary:
- Modal number of eggs per nest: 4
- Median number of eggs per nest: 4
- Average number of eggs per nest: 4.5 (rounded to 1 decimal place)
- Total number of nests: 55
