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