Video: Pack 1 • Paper 2 • Question 17

Pack 1 • Paper 2 • Question 17

05:34

Video Transcript

The table shows information about the lengths of 62 giant South African earthworms. There are 10 earthworms between 100 and 150 centimeters, 15 earthworms between 150 and 200 centimeters, 20 earthworms between 200 and 275 centimeters, 12 earthworms between 275 and 350 centimeters, and finally five earthworms between 350 and 400 centimeters. Part a) On the grid provided, draw a histogram for this data. Part b) Work out an estimate for the percentage of earthworms that are between 250 and 350 centimeters long.

In order to draw a histogram, we firstly need to calculate the frequency density. The frequency density is calculated by dividing the frequency by the class width. The class width of the first row in our table is 50, as 150 minus 100 equals 50. In the same way, we can calculate the other class widths. They are 50, 75, 75, and 50.

We noticed that some of these values are different. This means that the bars in our histogram will have different widths. To work out the frequency density for the top row in our table, we now divide 10 by 50, the frequency by the class width. 10 divided by 50 is 0.2.

The frequency density for the second row is 15 divided by 50. This is equal to 0.3. Our next frequency density is 20 divided by 75. This gives us 0.26 recurring. In the same way, 12 divided by 75 is 0.16. And five divided by 50 is 0.1.

We now need to draw a histogram using the length of the worms and the frequency density. The longest length of worm is 400 centimeters. And the highest number in our frequency density column is 0.3. We can, therefore, label the 𝑥-axis from zero to 400 and the 𝑦-axis from zero to 0.3.

Our first bar has width 50, from 100 to 150, and has a height of 0.2. The second bar also has a width of 50, from 150 to 200 centimeters, and has a frequency density or height of 0.3. The third and fourth bars have widths of 75, from 200 to 275 and from 275 to 350. They have heights of 0.26 recurring and 0.16, respectively. The final bar has width 50, from 350 to 400 centimeters, and a height or frequency density of 0.1. This completes the histogram for the data in the table.

The second part of our question asked us to estimate the percentage of earthworms that are between 250 and 350 centimeters long. The earthworms between 250 and 350 centimeters long as shown by the shaded area on the diagram. We can split this into two parts: firstly part 𝐴, those earthworms between 250 and 275 centimeters long, and part 𝐵, the earthworms between 275 and 350 centimeters long.

The total number of earthworms can be calculated by working out the area of each of these blocks. Section 𝐴 has an area of 25 multiplied by 0.26 recurring. And section 𝐵 has an area of 75 multiplied by 0.16. This is equal to 18.6 recurring. Therefore, we can estimate that there are 18.6 recurring earthworms between 250 and 350 centimeters long.

To calculate the percentage of earthworms in the range 250 to 350 centimeters, we need to divide 18.6 recurring by 62 and multiply the answer by 100. We divide by 62 as the total number of earthworms was 62. Typing this into our calculator gives us an answer of 30.1 percent to one decimal place. This means that our estimate for the number of earthworms that were between 250 and 350 centimeters long is 30.1 percent.

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