Video: Pack 4 • Paper 2 • Question 7

Pack 4 • Paper 2 • Question 7

06:30

Video Transcript

The residents in a block of flats recorded how much each person spent on food in a particular week. The table below shows the cumulative frequencies for the data recorded. Part a) Using the grid, plot a cumulative frequency graph for the information above.

To begin this question, let us remind ourselves of the definition of cumulative frequency. The cumulative frequency can be thought of as the running total of the frequencies up to a given amount. The question provides a table with amount spent in a week for our first column and cumulative frequency for our second column.

Given our definition, we can begin to interpret this data. The first row of the table tells us that four people spent 10 pounds or less on food in this particular week. Taking a closer look at the format of the first column, we know this because the amount of food spent in a week represented by 𝑥 is greater than or equal to zero and less than or equal to 10.

Now, we can see this pattern repeated down the first column, where 𝑥 is greater than or equal to zero and less than or equal to some amount. To continue our interpretation, the second row of the table tells us that 11 people spent 20 pounds or less on food in this particular week. 19 people spent 30 pounds or less and so on.

Now, let’s get to plotting the graph. For questions of this type, cumulative frequency will always be plotted along the 𝑦-axis. And here, we’ll be using amount spent in a week for our 𝑥-axis. When plotting the individual points, it’s important to remember they will be taking our 𝑥-coordinate as the top of the given ranges. So the 𝑥-coordinate of our first point will be 10 and the 𝑦-coordinate will be four.

Looking at the 𝑥-axis, we can see that we have 10 squares to represent 10 pounds for the amount spent in a week. We can, therefore, say that each square represents one pound. Looking at the 𝑦-axis, we can see the we have five squares to represent 10 people for our cumulative frequency. We can, therefore, say that each square represents two people.

We can now carefully plot our first point at the 𝑥-coordinate of 10 and the 𝑦-coordinate of four. Our next point is plotted at the 𝑥-coordinate of 20 and the 𝑦-coordinate of 11. Be careful with the 𝑦-coordinate of this point since the 𝑦-coordinate of 11 comes in between two of the grid lines.

We can now continue by plotting all of the data given in the table on our grid. To complete our cumulative frequency graph, we need to join up all of our points. Now, the best way to do this is by drawing a smooth curve that goes through each of them.

However, for most instances of this type of question, you’ll also be awarded a correct answer by joining each point to its neighbours by a straight line segment instead. As a final point here, it is worth noting that since our 𝑥 range starts at zero pounds, we have started our graph at the origin or zero, zero. We’ve now completed our cumulative frequency graph and therefore completed part a of the question.

Part b) Find the median amount of money spent on food during this particular week.

To begin, we can remind ourselves that the median is the middle value for a given set of data. For this question, we can see that the highest number for cumulative frequency that we have is 80. This tells us that we’ve been given data about 80 people in total. To find the median amount of money spent, we need to find the median person of these 80 people when ordering by amount spent.

For a cumulative frequency graph, we can simply do this by finding the total frequency divided by two. So 80 over two equals the 40th person. We now moved to the graph. And the curve we drew during part a of the question comes into action.

To find an estimate for the amount of money that the 40th person spent, we first draw a horizontal line from 40 on the 𝑦-axis to meet our cumulative frequency curve. We then draw a vertical line downwards to the 𝑥-axis. And we read off that we have a value of 43.

Since this is our estimate for the amount of money that the 40th person spent, this is our answer to part b. And we can say that the median amount of money spent on food during this particular week is 43 pounds. We now move on to part c.

A local supermarket wanted to give a discount to a quarter of the residents. They gave it to the residents who spent the most in this particular week. Find an estimate for the minimum amount spent during this given week to be eligible for the discount.

Once again, to begin this part of the question, let’s draw our attention to the fact that we have data on 80 people or 80 residents in total. Now, the local supermarket wants to give a discount to a quarter of the residents. So they want to give a discount to 80 over four, which is 20 residents. So the supermarket wants to give a discount to the 20 residents who spent the most in this particular week.

If we suppose that resident one spent the least and resident 80 spent the most in this particular week, then we should be able to see that the range of residents eligible for the discount goes from resident 60 to resident 80 since 80 minus 20 is equal to 60. Here, resident 60 lies at the lower bound of this range. And therefore, of all the residents eligible for the discount during this given week, they will have the minimum spent.

To answer this part of the question, we, therefore, need to find an estimate for the spent of the 60th resident. We can now use the same technique that we used to answer part b of this question.

We first draw a horizontal line from 60 on the 𝑦-axis to meet our cumulative frequency curve. We then draw a vertical line downwards to meet the 𝑥-axis and read off a value of 52. We, therefore, have an estimate for the 60th resident. And we have answered part c of the question since we can say that our estimate for the minimum amount spent during this given week to be eligible for the discount is 52 pounds.

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