Compare or interpret cumulative frequency graphs

5 The numbers of births, in thousands, to mothers of different ages in England and Wales, in 1991 and 2001 are illustrated by the cumulative frequency curves. Cumulative frequency (000's)
\includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-3_949_1338_461_479}

1. In which of these two years were there more births? How many more births were there in this year?
2. The following quantities were estimated from the diagram.

   Year	M edian age (years)	Interquartile range (years)	Proportion of mothers giving birth aged below 25	Proportion of mothers giving birth aged 35 or above
   1991	27.5	7.3	\(33 \%\)	\(9 \%\)
   2001				\(18 \%\)

   (a) Find the values missing from the table.
   (b) Did the women who gave birth in 2001 tend to be younger or older or about the same age as the women who gave birth in 1991? Using the table and your values from part (a), give two reasons for your answer.

6
\includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-3_803_1180_1018_413} The diagram shows the cumulative frequency graphs for the marks scored by the candidates in an examination. The 2000 candidates each took two papers; the upper curve shows the distribution of marks on paper 1 and the lower curve shows the distribution on paper 2. The maximum mark on each paper was 100.

1. Use the diagram to estimate the median mark for each of paper 1 and paper 2.
2. State with a reason which of the two papers you think was the easier one.
3. To achieve grade A on paper 1 candidates had to score 66 marks out of 100. What mark on paper 2 gives equal proportions of candidates achieving grade A on the two papers? What is this proportion?
4. The candidates' marks for the two papers could also be illustrated by means of a pair of box-and whisker plots. Give two brief comments comparing the usefulness of cumulative frequency graphs and box-and-whisker plots for representing the data.

8 Ali conducted an investigation into the distances ridden by those members of a cycling club who rode at least 120 km in a training week. She grouped all the distances into intervals of length 10 km and then constructed a cumulative frequency diagram, which is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{57007d39-abb0-475e-9ed8-03021fa1273b-06_1086_1627_587_233}

1. Explain whether the data Ali used is a sample or a population. The club is taking part in a competition. Eight team members and one reserve are to be selected. The club captain decides that the team members should be those cyclists who rode the furthest during the training week, and that the reserve should be the cyclist who rode the next furthest.
2. Use the graph to estimate the shortest distance cycled by a team member. The captain's best friend rode 156 km in the training week and was selected as reserve. Ali complained that this was unjustifiable.
3. Explain whether there is sufficient evidence in the diagram to support Ali's complaint.

12 Paul drew a cumulative frequency graph showing information about the numbers of people in various age-groups in a certain region X. He forgot to include the scale on the cumulative frequency axis, as shown below.
\includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-09_758_936_408_561}

1. Find an estimate of the median age of the population of region X .
2. Find an estimate of the proportion of people aged over 60 in region X . Sonika drew similar cumulative graphs for another two regions, Y and Z , but she included the scales on the cumulative frequency axes, as shown below.
   \includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-10_748_935_358_116}
   \includegraphics[max width=\textwidth, alt={}, center]{166bcf11-c812-4077-91c8-916b093cbbd0-10_746_940_358_1011}
3. Find an age group, of width 20 years, in which region Z has approximately 3 times as many people as region Y.
4. State one advantage and one disadvantage of using Sonika's two diagrams to compare the populations in Regions Y and Z.
5. Without calculation state, with a reason, which of regions Y or Z has the greater proportion of people aged under 40.

1. 200 candidates took each of two examination papers. The diagram shows the cumulative frequency graphs for their marks.
\includegraphics[max width=\textwidth, alt={}, center]{5b55a372-2cc8-454e-a10a-cabdc9801421-04_1091_1484_429_285}

1. State, with a reason, which of the two papers was the easier one.
2. The minimum mark for grade A , the top grade, on Paper 1 was 10 marks lower than the minimum mark for grade A on Paper 2. Given that 25 candidates gained grade A in Paper 1, find the number of candidates who gained grade A in Paper 2.
3. The mean and standard deviation of the marks on Paper 1 were 36.5 and 28.2 respectively. Later, a marking error was discovered and it was decided to add 1 mark to each of the 200 marks on Paper 1. State the mean and standard deviation of the new marks on Paper 1.
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