Answer :
To estimate the mean length of the insects Kylie measured, we can use the midpoint of each length interval and then calculate the weighted average. Here's a detailed step-by-step solution:
### Step 1: Identify the intervals and their frequencies
Given the intervals and their respective frequencies:
[tex]\[ \begin{array}{|c|c|} \hline \text{Length } ( \text{mm} ) & \text{Frequency} \\ \hline 0 < x \leq 10 & 5 \\ \hline 10 < x \leq 20 & 8 \\ \hline 20 < x \leq 30 & 7 \\ \hline \end{array} \][/tex]
### Step 2: Calculate the midpoint of each interval
For each interval, the midpoint is calculated as follows:
[tex]\[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
- For [tex]\( 0 < x \leq 10 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{0 + 10}{2} = 5.0 \][/tex]
- For [tex]\( 10 < x \leq 20 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{10 + 20}{2} = 15.0 \][/tex]
- For [tex]\( 20 < x \leq 30 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{20 + 30}{2} = 25.0 \][/tex]
The midpoints are [tex]\( 5.0, 15.0, \)[/tex] and [tex]\( 25.0 \)[/tex] mm, respectively.
### Step 3: Calculate the total number of insects
Sum the frequencies to find the total number of insects:
[tex]\[ \text{Total Frequency} = 5 + 8 + 7 = 20 \][/tex]
### Step 4: Calculate the weighted sum of midpoints
Multiply each midpoint by its corresponding frequency and sum these values:
[tex]\[ \begin{aligned} \text{Weighted Sum} &= (5.0 \times 5) + (15.0 \times 8) + (25.0 \times 7) \\ &= 25.0 + 120.0 + 175.0 \\ &= 320.0 \end{aligned} \][/tex]
### Step 5: Calculate the mean length
Divide the weighted sum of midpoints by the total number of insects to find the mean length:
[tex]\[ \text{Mean Length} = \frac{320.0}{20} = 16.0 \, \text{mm} \][/tex]
### Final Answer
The estimated mean length of the insects Kylie found is [tex]\( \boxed{16.0} \)[/tex] mm.
### Step 1: Identify the intervals and their frequencies
Given the intervals and their respective frequencies:
[tex]\[ \begin{array}{|c|c|} \hline \text{Length } ( \text{mm} ) & \text{Frequency} \\ \hline 0 < x \leq 10 & 5 \\ \hline 10 < x \leq 20 & 8 \\ \hline 20 < x \leq 30 & 7 \\ \hline \end{array} \][/tex]
### Step 2: Calculate the midpoint of each interval
For each interval, the midpoint is calculated as follows:
[tex]\[ \text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2} \][/tex]
- For [tex]\( 0 < x \leq 10 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{0 + 10}{2} = 5.0 \][/tex]
- For [tex]\( 10 < x \leq 20 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{10 + 20}{2} = 15.0 \][/tex]
- For [tex]\( 20 < x \leq 30 \)[/tex]:
[tex]\[ \text{Midpoint} = \frac{20 + 30}{2} = 25.0 \][/tex]
The midpoints are [tex]\( 5.0, 15.0, \)[/tex] and [tex]\( 25.0 \)[/tex] mm, respectively.
### Step 3: Calculate the total number of insects
Sum the frequencies to find the total number of insects:
[tex]\[ \text{Total Frequency} = 5 + 8 + 7 = 20 \][/tex]
### Step 4: Calculate the weighted sum of midpoints
Multiply each midpoint by its corresponding frequency and sum these values:
[tex]\[ \begin{aligned} \text{Weighted Sum} &= (5.0 \times 5) + (15.0 \times 8) + (25.0 \times 7) \\ &= 25.0 + 120.0 + 175.0 \\ &= 320.0 \end{aligned} \][/tex]
### Step 5: Calculate the mean length
Divide the weighted sum of midpoints by the total number of insects to find the mean length:
[tex]\[ \text{Mean Length} = \frac{320.0}{20} = 16.0 \, \text{mm} \][/tex]
### Final Answer
The estimated mean length of the insects Kylie found is [tex]\( \boxed{16.0} \)[/tex] mm.
