Mick measured the length, [tex]$x$[/tex], of each of the insects he found underneath a rock. He recorded the lengths in the table below. Calculate an estimate of the mean length of the insects he found. Give your answer in millimeters (mm).
| Length (mm) | Frequency |
| :---------- | :-------- |
| [tex]$0\ \textless \ x \leq 10$[/tex] | 9 |
| [tex]$10\ \textless \ x \leq 20$[/tex] | 6 |
| [tex]$20\ \textless \ x \leq 30$[/tex] | 5 |
Answer (2)
Calculate the midpoint of each interval: 5, 15, 25.
Multiply each midpoint by its frequency: 45, 90, 125.
Sum the products and divide by the total frequency: 20 45 + 90 + 125 = 20 260 .
The estimated mean length is: 13 mm.
Explanation
Understand the problem We are given a table with the lengths of insects grouped into intervals and their corresponding frequencies. Our goal is to estimate the mean length of these insects. To do this, we'll find the midpoint of each interval, multiply it by the frequency of that interval, sum these products, and then divide by the total frequency.
Calculate the midpoints of the intervals First, we need to find the midpoint of each length interval:
For the interval 0 < x ≤ 10 , the midpoint is 2 0 + 10 = 5 .
For the interval 10 < x ≤ 20 , the midpoint is 2 10 + 20 = 15 .
For the interval 20 < x ≤ 30 , the midpoint is 2 20 + 30 = 25 .
Multiply midpoints by frequencies Next, we multiply each midpoint by its corresponding frequency:
For the interval 0 < x ≤ 10 : 5 × 9 = 45
For the interval 10 < x ≤ 20 : 15 × 6 = 90
For the interval 20 < x ≤ 30 : 25 × 5 = 125
Sum the products Now, we sum these products: 45 + 90 + 125 = 260 .
Calculate the total frequency The total frequency is the sum of the frequencies: 9 + 6 + 5 = 20 .
Estimate the mean length Finally, we divide the sum of the products by the total frequency to estimate the mean length: 20 260 = 13 . Therefore, the estimated mean length of the insects is 13 mm.
Examples
Estimating the mean length of insects can be useful in ecological studies to understand the size distribution of insect populations in a particular habitat. This information can be used to assess the health of the ecosystem, study the effects of environmental changes on insect populations, or compare insect sizes across different locations. For example, if you are studying the impact of pollution on insect growth, you might measure the lengths of insects in polluted and unpolluted areas and compare their estimated mean lengths.
The estimated mean length of the insects Mick found is 13 mm, calculated by finding the midpoints of each length interval, multiplying by the frequency, and dividing the sum of products by the total frequency.
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