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.
