Video Transcript
A botanist records the height in
centimeters of 120 tomato plants in an experiment testing different growing
conditions. By drawing a cumulative frequency
curve to represent the data, estimate the number of plants with a height less than
116 centimeters, given that the maximum height of a plant is 150 centimeters. Option (A) 11 plants. Option (B) 22 plants. Option (C) 30 plants. Option (D) 90 plants. Or is it option (E) 98 plants?
In this question, we are given a
grouped frequency table representing the heights of 120 tomato plants. And we want to estimate the number
of plants with heights below 116 centimeters by drawing a cumulative frequency
curve. To do this, we first need to find
the cumulative frequency in the table. That is, the running total of all
of the other frequencies. The first frequency is 12, so we
can start by adding this value onto the row for cumulative frequency. To find the next value, we need to
add 12 to the next frequency. We calculate that 21 plus 12 is
equal to 33 and add this value onto the table.
We can follow this process again to
find the next cumulative frequency. We calculate that 52 plus 33 is
equal to 85 and add this value onto our table. We follow this process two more
times to obtain the values of 115 and 120. We should always check to make sure
that the final cumulative frequency is equal to the total population. In this case, that is 120 since
there are 120 tomato plants.
We now want to use this data to
sketch a cumulative frequency curve. To do this, we need to plot
coordinates using the table. The 𝑥-coordinates are the upper
bounds of the classes and the corresponding 𝑦-coordinates are the cumulative
frequencies. We can add upper bounds onto the
table by noting that the upper bounds are not included in each class and using the
fact that the maximum height of a plant is 150 centimeters.
Using the first column of the
table, we see that the upper bound is 110 and the cumulative frequency is 12. So, we need to plot the point 110,
12. Using the second column of the
table, we see that the upper bound is 120 and the cumulative frequency is 33. So, we need to plot the point 120,
33. We can follow this same process for
the remaining three columns in the table. This gives us the points 130, 85;
140, 115; and 150, 120.
Before we sketch our cumulative
frequency graph, we can note that we are given a lower bound on the heights of the
plants. There are no plants with heights
less than 100 centimeters. If there are no plants with heights
less than 100 centimeters, then the cumulative frequency up to 100 is zero. So, we can include the point 100,
zero on our graph.
We are now ready to sketch the
cumulative frequency graph of the data. We will start by clearing some
space and taking note of the points on our graph. The 𝑥-axis will record the heights
of the plants. Since this starts at 100
centimeters, we do not need to include the values below this. We also only need to include values
up to 150 centimeters. The 𝑦-axis will measure the
cumulative frequency. This starts at zero and will end at
the total population size, in this case 120.
Now, we need to plot the
coordinates we found from our cumulative frequency table onto the graph to obtain
the following. The cumulative frequency graph of
the data is then approximated by connecting these points with a smooth curve as
shown. We can use this graph to
approximate the number of plants with a height less than 116 centimeters by noting
that this is the same as estimating the cumulative frequency up to 116
centimeters. We can draw a vertical line at 116
centimeters on our sketch to see that the approximate cumulative frequency is at
22. Of course, this is an approximation
and any value around this will be valid depending on the accuracy of the sketch.
Hence, we can estimate that 22
plants have height less than 116 centimeters, which is option (B).
