2.02a Interpret single variable data: tables and diagrams

6

1. Give one advantage and one disadvantage of using a box-and-whisker plot to represent a set of data.
2. The times in minutes taken to run a marathon were recorded for a group of 13 marathon runners and were found to be as follows. $$\begin{array} { l l l l l l l l l l l l l } 180 & 275 & 235 & 242 & 311 & 194 & 246 & 229 & 238 & 768 & 332 & 227 & 228 \end{array}$$ State which of the mean, mode or median is most suitable as a measure of central tendency for these times. Explain why the other measures are less suitable.
3. Another group of 33 people ran the same marathon and their times in minutes were as follows.

   190	203	215	246	249	253	255	254	258	260	261
   263	267	269	274	276	280	288	283	287	294	300
   307	318	327	331	336	345	351	353	360	368	375

   1. On the grid below, draw a box-and-whisker plot to illustrate the times for these 33 people. \includegraphics[max width=\textwidth, alt={}, center]{f4d040a2-6a04-49ce-98ac-8ba5c515f905-09_611_1202_1270_555}
   2. Find the interquartile range of these times.

7 The times in minutes taken by 13 pupils at each of two schools in a cross-country race are recorded in the table below.

1. Draw a back-to-back stem-and-leaf diagram to illustrate these times with Thaters School on the left.
2. Find the interquartile range of the times for pupils at Thaters School.
   The times taken by pupils at Whitefay Park School are denoted by \(x\) minutes.
3. Find the value of \(\Sigma ( x - 60 ) ^ { 2 }\).
4. It is given that \(\Sigma ( x - 60 ) = 46\). Use this result, together with your answer to part (iii), to find the variance of \(x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A survey was made of the journey times of 63 people who cycle to work in a certain town. The results are summarised in the following cumulative frequency table.

1. State how many journey times were between 25 and 45 minutes.
2. Draw a histogram on graph paper to represent the data.
3. Calculate an estimate of the mean journey time.

5 The weights, in kg, of the 11 members of the Dolphins swimming team and the 11 members of the Sharks swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Dolphins on the left-hand side of the diagram and Sharks on the right-hand side.
2. Find the median and interquartile range for the Dolphins.

7 The weights in kilograms of two groups of 17-year-old males from country \(P\) and country \(Q\) are displayed in the following back-to-back stem-and-leaf diagram. In the third row of the diagram, ... \(4 | 7 | 1 \ldots\) denotes weights of 74 kg for a male in country \(P\) and 71 kg for a male in country \(Q\).

1. Find the median and quartile weights for country \(Q\).
2. You are given that the lower quartile, median and upper quartile for country \(P\) are 84,94 and 98 kg respectively. On a single diagram on graph paper, draw two box-and-whisker plots of the data.
3. Make two comments on the weights of the two groups.

7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.

1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
   \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).

2 The lengths of cars travelling on a car ferry are noted. The data are summarised in the following table.

1. Find the value of \(a\).
2. Draw a histogram on graph paper to represent the data.
3. Find the probability that a randomly chosen car on the ferry is less than 3.20 m in length.

1 The weights of 30 children in a class, to the nearest kilogram, were as follows. Construct a grouped frequency table for these data such that there are five equal class intervals with the first class having a lower boundary of 41.5 kg and the fifth class having an upper boundary of 61.5 kg .

5 The arrival times of 204 trains were noted and the number of minutes, \(t\), that each train was late was recorded. The results are summarised in the table.

1. Explain what \(- 2 \leqslant t < 0\) means about the arrival times of trains.
2. Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.

5 The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.

1. Draw a stem-and-leaf diagram to represent the data.
2. Find the median and the quartiles.
3. On graph paper, using a scale of 2 cm to represent 10 beats per minute, draw a box-and-whisker plot of the data.

4 A library has many identical shelves. All the shelves are full and the numbers of books on each shelf in a certain section are summarised by the following stem-and-leaf diagram. Key: 3 | 6 represents 36 books

1. Find the number of shelves in this section of the library.
2. Draw a box-and-whisker plot to represent the data. In another section all the shelves are full and the numbers of books on each shelf are summarised by the following stem-and-leaf diagram.

   2	12222334566679	\(( 13 )\)
   3	01112334456677788	\(( 15 )\)
   4	223357789

   Key: 3 | 6 represents 36 books
3. There are fewer books in this section than in the previous section. State one other difference between the books in this section and the books in the previous section.

6 The following table gives the marks, out of 75, in a pure mathematics examination taken by 234 students.

1. Draw a histogram on graph paper to represent these results.
2. Calculate estimates of the mean mark and the standard deviation.

4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table. A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The \(1 - 10\) rectangle has height 3 cm .

1. Calculate the value of \(x\) and the height of the 51-70 rectangle.
2. Calculate an estimate of the mean weight of the stones.

3 The table summarises the times that 112 people took to travel to work on a particular day.

1. State which time interval in the table contains the median and which time interval contains the upper quartile.
2. On graph paper, draw a histogram to represent the data.
3. Calculate an estimate of the mean time to travel to work.

4 A random sample of 25 people recorded the number of glasses of water they drank in a particular week. The results are shown below.

1. Draw a stem-and-leaf diagram to represent the data.
2. On graph paper draw a box-and-whisker plot to represent the data.

5 The weights, in kilograms, of the 15 rugby players in each of two teams, \(A\) and \(B\), are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with team \(A\) on the lefthand side of the diagram and team \(B\) on the right-hand side.
2. Find the interquartile range of the weights of the players in team \(A\).
3. A new player joins team \(B\) as a substitute. The mean weight of the 16 players in team \(B\) is now 93.9 kg . Find the weight of the new player.

7 The masses, in grams, of components made in factory \(A\) and components made in factory \(B\) are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the masses of components made in the two factories.
2. Find the median and the interquartile range for the masses of components made in factory \(B\).
3. Make two comparisons between the masses of components made in factory \(A\) and the masses of those made in factory \(B\).

5 The tables summarise the heights, \(h \mathrm {~cm}\), of 60 girls and 60 boys.

1. On graph paper, using the same set of axes, draw two cumulative frequency graphs to illustrate the data.
2. On a school trip the students have to enter a cave which is 165 cm high. Use your graph to estimate the percentage of the girls who will be unable to stand upright.
   [0pt]
3. The students are asked to compare the heights of the girls and the boys. State one advantage of using a pair of box-and-whisker plots instead of the cumulative frequency graphs to do this. [1]

2 The time taken by a car to accelerate from 0 to 30 metres per second was measured correct to the nearest second. The results from 48 cars are summarised in the following table.

1. On the grid, draw a cumulative frequency graph to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{ee1e5987-315b-48eb-8dba-b9d4d34c87c9-03_1207_1406_897_411}
2. 35 of these cars accelerated from 0 to 30 metres per second in a time more than \(t\) seconds. Estimate the value of \(t\).

2 The circumferences, \(c \mathrm {~cm}\), of some trees in a wood were measured. The results are summarised in the table.

1. On the grid, draw a cumulative frequency graph to represent the information. \includegraphics[max width=\textwidth, alt={}, center]{9c23b94b-e573-4e13-be90-e63a0daf18e5-03_1401_1404_854_413}
2. Estimate the percentage of trees which have a circumference larger than 75 cm .

5 The number of Olympic medals won in the 2012 Olympic Games by the top 27 countries is shown below.

1. Draw a stem-and-leaf diagram to illustrate the data.
2. Find the median and quartiles and draw a box-and-whisker plot on the grid. \includegraphics[max width=\textwidth, alt={}, center]{4c2afa86-960c-473e-970c-ed16c8434fec-07_1006_1406_1007_411}

6 The daily rainfall, \(x \mathrm {~mm}\), in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequency table.

1. On the grid, draw a cumulative frequency graph to illustrate the data.
2. On 100 of the days, the rainfall was \(k \mathrm {~mm}\) or more. Use your graph to estimate the value of \(k\).
3. Calculate estimates of the mean and standard deviation of the daily rainfall in this village.

2 The following back-to-back stem-and-leaf diagram shows the reaction times in seconds in an experiment involving two groups of people, \(A\) and \(B\). Key: 5 | 22 | 6 means a reaction time of 0.225 seconds for \(A\) and 0.226 seconds for \(B\)

1. Find the median and the interquartile range for group \(A\).
   The median value for group \(B\) is 0.235 seconds, the lower quartile is 0.217 seconds and the upper quartile is 0.245 seconds.
2. Draw box-and-whisker plots for groups \(A\) and \(B\) on the grid. \includegraphics[max width=\textwidth, alt={}, center]{62812433-baee-490a-bad4-b6b0f917c234-03_805_1495_1729_365}

7 The heights, in cm, of the 11 members of the Anvils athletics team and the 11 members of the Brecons swimming team are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Anvils on the left-hand side of the diagram and Brecons on the right-hand side.
2. Find the median and the interquartile range for the heights of the Anvils.
   The heights of the 11 members of the Anvils are denoted by \(x \mathrm {~cm}\). It is given that \(\Sigma x = 1923\) and \(\Sigma x ^ { 2 } = 337221\). The Anvils are joined by 3 new members whose heights are \(166 \mathrm {~cm} , 172 \mathrm {~cm}\) and 182 cm .
3. Find the standard deviation of the heights of all 14 members of the Anvils.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Ransha measured the lengths, in centimetres, of 160 palm leaves. His results are illustrated in the cumulative frequency graph below. \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-08_1090_1424_404_356}

1. Estimate how many leaves have a length between 14 and 24 centimetres.
2. \(10 \%\) of the leaves have a length of \(L\) centimetres or more. Estimate the value of \(L\).
3. Estimate the median and the interquartile range of the lengths.
   Sharim measured the lengths, in centimetres, of 160 palm leaves of a different type. He drew a box-and-whisker plot for the data, as shown on the grid below. \includegraphics[max width=\textwidth, alt={}, center]{7ea494c0-5e1a-4da9-a189-30128654fa1d-09_540_1287_1181_424}
4. Compare the central tendency and the spread of the two sets of data.