Video Transcript
The following table shows the
amount of money customers put in a donation box. Note that no customer donated more
than 100 dollars. Which of the following is the
descending cumulative frequency graph? Is it option (A), (B), (C), (D), or
(E)?
We begin by recalling that the
descending cumulative frequency of a value 𝑥 indicates the frequency of values that
are greater than or equal to 𝑥. Before we can identify the correct
graph, we need to calculate the descending cumulative frequency values. We begin by adding an extra row to
our table. The first group in the table, 10
dash, indicates donations that are 10 dollars or greater up to, but not including,
20 dollars, since this is the lower bound of the next group. This means that the first
descending cumulative frequency will be for all donations that are 10 dollars or
greater. The total frequency for this
distribution is 65. And we know that the first
descending cumulative frequency must be equal to this, since all the donations were
at least 10 dollars.
To calculate the second descending
cumulative frequency value, those donations greater than or equal to 20 dollars, we
need to subtract the three customers in the first group from the total number of
customers. 65 minus three is equal to 62. So, our second descending
cumulative frequency is 62.
Next, we need to subtract the
number of customers who donated 20 dollars or more but less than 30 dollars from the
second descending cumulative frequency. 62 minus seven is equal to 55. So, the third descending cumulative
frequency value is 55. Repeating this process, we obtain
further descending cumulative frequency values of 45, 33, 20, 11, four, and
four.
We commonly finish a descending
cumulative frequency with a value of zero. To do this, we consider the last
class to have the same class width as the others and define an additional class to
the distribution. Here, this would be 100 dash. We know that no customer donated
more than 100 dollars, so the frequency here is zero.
We are now in a position where we
can compare our descending cumulative frequency values with the five graphs. We have the 10 coordinates 10, 65;
20, 62; 30, 55; 40, 45; 50, 33; 60, 20; 70, 11; 80, four; 90, four; and 100,
zero. This corresponds to graph (E). Whilst there are several incorrect
points on some of the other graphs, we have highlighted one on each. The point 90, zero wrongly appears
in graphs (A), (B), and (C) and the point 100, four wrongly appears in graph
(D).
We can therefore conclude that the
correct graph is option (E). This is the descending cumulative
frequency graph for the information in the table.
