2.02a Interpret single variable data: tables and diagrams

6 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm . To try to reduce absence, the company decides to introduce 'flexi-time' and allow employees to work their seven hours each day at any time between 7 am and 9 pm . For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table. Use a paired sample \(t\)-test to test, at the \(10 \%\) significance level, whether the population mean number of hours of absence has decreased, following the introduction of flexi-time.

3 Two machines, \(A\) and \(B\), produce metal rods of a certain type. The lengths, in metres, of 19 rods produced by machine \(A\) and 19 rods produced by machine \(B\) are shown in the following back-to-back stem-and-leaf diagram. \begin{table}[h] \end{table}

1. Find the median and the interquartile range for machine \(A\).
   It is given that for machine \(B\) the median is 0.232 m , the lower quartile is 0.224 m and the upper quartile is 0.243 m .
2. Draw box-and-whisker plots for \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{a3b3ebd1-db9e-4552-9abe-bfdeba786d02-05_812_1205_616_511}
3. Hence make two comparisons between the lengths of the rods produced by machine \(A\) and those produced by machine \(B\).

6 The annual salaries, in thousands of dollars, for 11 employees at each of two companies \(A\) and \(B\) are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with company \(A\) on the left-hand side of the diagram.
2. Find the median and the interquartile range of the salaries of the employees in company \(A\). [3]
   A new employee joins company \(B\). The mean salary of the 12 employees is now \(\\) 38500$.
3. Find the salary of the new employee.

7 The heights, in cm, of the 11 basketball players in each of two clubs, the Amazons and the Giants, are shown below.

1. State an advantage of using a stem-and-leaf diagram compared to a box-and-whisker plot to illustrate this information.
2. Represent the data by drawing a back-to-back stem-and-leaf diagram with Amazons on the left-hand side of the diagram.
3. Find the interquartile range of the heights of the players in the Amazons.
   Four new players join the Amazons. The mean height of the 15 players in the Amazons in now 191.2 cm . The heights of three of the new players are \(180 \mathrm {~cm} , 185 \mathrm {~cm}\) and 190 cm .
4. Find the height of the fourth new player.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 The heights in cm of 160 sunflower plants were measured. The results are summarised on the following cumulative frequency curve. \includegraphics[max width=\textwidth, alt={}, center]{b72bd3eb-2c10-4d01-ab18-74d6fb812f27-02_1783_1424_404_356}

1. Use the graph to estimate the number of plants with heights less than 100 cm .
2. Use the graph to estimate the 65th percentile of the distribution.
3. Use the graph to estimate the interquartile range of the heights of these plants.

1 The time taken, \(t\) minutes, to complete a puzzle was recorded for each of 150 students. These times are summarised in the table.

1. Draw a cumulative frequency graph to illustrate the data.

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2. Use your graph to estimate the 20th percentile of the data.

4 The times taken, in seconds, by 15 members of each of two swimming clubs, the Penguins and the Dolphins, to swim 50 metres are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Penguins on the left-hand side. \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_2720_33_141_20} The diagram shows a box-and-whisker plot representing the times for the Penguins.
2. On the same diagram, draw a box-and-whisker plot to represent the times for the Dolphins. \includegraphics[max width=\textwidth, alt={}, center]{9b21cc0f-b043-4251-8aa9-cb1e5c2fb5d0-09_719_1219_424_424}
3. Hence state one difference between the distributions of the times for the Penguins and the Dolphins.

7 Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{f7c0e35d-1889-4e5b-b094-f467052a66cf-10_1593_1296_790_466}
2. 40\% of these fish have a length of \(d \mathrm {~cm}\) or more. Use your graph to estimate the value of \(d\).
   The mean length of these 150 fish is 15.295 cm .
3. Calculate an estimate for the variance of the lengths of the fish.
   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 A driver records the distance travelled in each of 150 journeys. These distances, correct to the nearest km , are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{3f05dc2a-b466-40bc-9f5f-0fd2bff120c8-06_1593_1397_852_415}
2. For 30\% of these journeys the distance travelled is \(d \mathrm {~km}\) or more. Use your graph to estimate the value of \(d\).
3. Calculate an estimate of the mean distance travelled for the 150 journeys.

3 At a summer camp an arithmetic test is taken by 250 children. The times taken, to the nearest minute, to complete the test were recorded. The results are summarised in the table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{c1bc5ac2-6b0e-48c7-92e9-9b8b56b57d90-05_1000_1198_785_516}
2. State which class interval contains the median.
3. Given that an estimate of the mean time is 61.05 minutes, state what feature of the distribution accounts for the median and the mean being different.

6 The times, \(t\) minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{033ceb76-8fd4-4a89-ab05-5e20039d1c8d-08_1689_1195_744_516}
2. \(24 \%\) of the students take \(k\) minutes or longer to complete the challenge. Use your graph to estimate the value of \(k\).
3. Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.

5 The following table gives the weekly snowfall, in centimetres, for 11 weeks in 2018 at two ski resorts, Dados and Linva.

1. Represent the information in a back-to-back stem-and-leaf diagram.
2. Find the median and the interquartile range for the weekly snowfall in Dados.
3. The median, lower quartile and upper quartile of the weekly snowfall for Linva are 12, 9 and 32 cm respectively. Use this information and your answers to part (b) to compare the central tendency and the spread of the weekly snowfall in Dados and Linva.

6 The weights, in kg, of 15 rugby players in the Rebels club and 15 soccer players in the Sharks club are shown below.

1. Represent the data by drawing a back-to-back stem-and-leaf diagram with Rebels on the left-hand side of the diagram.
2. Find the median and the interquartile range for the Rebels.
   A box-and-whisker plot for the Sharks is shown below. \includegraphics[max width=\textwidth, alt={}, center]{a2709c37-6e81-4873-8f38-94cb9f3c3252-09_533_1246_388_445}
3. On the same diagram, draw a box-and-whisker plot for the Rebels.
4. Make one comparison between the weights of the players in the Rebels club and the weights of the players in the Sharks club.

2 Lakeview and Riverside are two schools. The pupils at both schools took part in a competition to see how far they could throw a ball. The distances thrown, to the nearest metre, by 11 pupils from each school are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with Lakeview on the left-hand side.
2. Find the interquartile range of the distances thrown by the 11 pupils at Lakeview school.

3 The Lions and the Tigers are two basketball clubs. The heights, in cm, of the 11 players in each of their first team squads are given in the table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Lions on the left.
2. Find the median and the interquartile range of the heights of the Lions first team squad.
   It is given that for the Tigers, the lower quartile is 183 cm , the median is 190 cm and the upper quartile is 195 cm .
3. Make two comparisons between the heights of the players in the Lions first team squad and the heights of the players in the Tigers first team squad.

3 The times, \(t\) minutes, taken to complete a walking challenge by 250 members of a club are summarised in the table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{1eb957f4-5088-4991-aa8a-f895d55d2bcf-04_1395_1298_705_466}
2. Use your graph to estimate the 60th percentile of the data.
   It is given that an estimate for the mean time taken to complete the challenge by these 250 members is 34.4 minutes.
3. Calculate an estimate for the standard deviation of the times taken to complete the challenge by these 250 members.

1 \includegraphics[max width=\textwidth, alt={}, center]{e8c2b51e-d788-4917-829e-1b056a24f520-03_1372_1194_260_479} The times taken by 120 children to complete a particular puzzle are represented in the cumulative frequency graph.

1. Use the graph to estimate the interquartile range of the data.
   35\% of the children took longer than \(T\) seconds to complete the puzzle.
2. Use the graph to estimate the value of \(T\).

4 The heights, in cm, of the 11 players in each of two teams, the Aces and the Jets, are shown in the following table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information with the Aces on the left-hand side of the diagram.
2. Find the median and the interquartile range of the heights of the players in the Aces.
3. Give one comment comparing the spread of the heights of the Aces with the spread of the heights of the Jets.

4 The weights, \(x \mathrm {~kg}\), of 120 students in a sports college are recorded. The results are summarised in the following table.

1. Draw a cumulative frequency graph to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{82c36c11-878c-47d1-a07f-fbf8b2a22d97-06_1390_1389_660_418}
2. It is found that \(35 \%\) of the students weigh more than \(W \mathrm {~kg}\). Use your graph to estimate the value of \(W\).
3. Calculate estimates for the mean and standard deviation of the weights of the 120 students. [6]

3 The time taken, in minutes, to walk to school was recorded for 200 pupils at a certain school. These times are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-04_1217_1509_705_278}
2. Use your graph to estimate the median and the interquartile range of the data. \includegraphics[max width=\textwidth, alt={}, center]{ad3a6a8a-23fe-415a-b2f4-7c49136ccc6c-05_2723_35_101_20}
3. Calculate an estimate for the mean value of the times taken by the 200 pupils to walk to school.

6 Teams of 15 runners took part in a charity run last Saturday. The times taken, in minutes, to complete the course by the runners from the Falcons and the runners from the Kites are shown in the table.

1. Draw a back-to-back stem-and-leaf diagram to represent this information, with the Falcons on the left-hand side.
2. Find the median and the interquartile range of the times for the Falcons.
   Let \(x\) and \(y\) denote the times, in minutes, of a runner from the Falcons and a runner from the Kites respectively. It is given that $$\sum x = 792 , \quad \sum x ^ { 2 } = 43504 , \quad \sum y = 783 , \quad \sum y ^ { 2 } = 42223 .$$
3. Find the mean and the standard deviation of the times taken by all 30 runners from the two teams.

4 On a certain day, the heights of 150 sunflower plants grown by children at a local school are measured, correct to the nearest cm . These heights are summarised in the following table.

1. Draw a cumulative frequency graph to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-06_1600_1301_760_383}
2. Use your graph to estimate the 30th percentile of the heights of the sunflower plants. \includegraphics[max width=\textwidth, alt={}, center]{915661eb-2544-4293-af72-608fedb43d70-07_2723_35_101_20}
3. Calculate estimates for the mean and the standard deviation of the heights of the 150 sunflower plants.

1

1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Sales of Superclene Toothpaste} \includegraphics[alt={},max width=\textwidth]{df20f053-8d67-428d-bb19-9447049deed5-2_725_1073_347_497}
   \end{figure} The diagram represents the sales of Superclene toothpaste over the last few years. Give a reason why it is misleading.
2. The following data represent the daily ticket sales at a small theatre during three weeks. $$52,73,34,85,62,79,89,50,45,83,84,91,85,84,87,44,86,41,35,73,86 \text {. }$$
   1. Construct a stem-and-leaf diagram to illustrate the data.
   2. Use your diagram to find the median of the data.

2 In a recent survey, 640 people were asked about the length of time each week that they spent watching television. The median time was found to be 20 hours, and the lower and upper quartiles were 15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours, and the greatest amount was 60 hours.

1. On graph paper, show these results using a fully labelled cumulative frequency graph.
2. Use your graph to estimate how many people watched more than 50 hours of television each week.

4 The following back-to-back stem-and-leaf diagram shows the cholesterol count for a group of 45 people who exercise daily and for another group of 63 who do not exercise. The figures in brackets show the number of people corresponding to each set of leaves. Key: 2 | 8 | 1 represents a cholesterol count of 8.2 in the group who exercise and 8.1 in the group who do not exercise.

1. Give one useful feature of a stem-and-leaf diagram.
2. Find the median and the quartiles of the cholesterol count for the group who do not exercise. You are given that the lower quartile, median and upper quartile of the cholesterol count for the group who exercise are 4.25, 5.3 and 6.6 respectively.
3. On a single diagram on graph paper, draw two box-and-whisker plots to illustrate the data.