2.02a Interpret single variable data: tables and diagrams

2. \begin{figure}[h] \end{figure} The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 An outlier is defined as a value
more than \(1.5 \times\) IQR below \(Q _ { 1 }\) or
more than \(1.5 \times\) IQR above \(Q _ { 3 }\) The three lowest air temperatures in the data set are \(7.6 ^ { \circ } \mathrm { C } , 8.1 ^ { \circ } \mathrm { C }\) and \(9.1 ^ { \circ } \mathrm { C }\) The highest air temperature in the data set is \(32.5 ^ { \circ } \mathrm { C }\)

1. Complete the box plot in Figure 1 showing clearly any outliers.
2. Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come. Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, \(x ^ { \circ } \mathrm { C }\), for Beijing in 2015 $$n = 184 \quad \sum x = 4153.6 \quad \mathrm {~S} _ { x x } = 4952.906$$
3. Show that, to 3 significant figures, the standard deviation is \(5.19 ^ { \circ } \mathrm { C }\) Simon decides to model the air temperatures with the random variable $$T \sim \mathrm {~N} \left( 22.6,5.19 ^ { 2 } \right)$$
4. Using Simon's model, calculate the 10th to 90th interpercentile range. Simon wants to model another variable from the large data set for Beijing using a normal distribution.
5. State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer. \includegraphics[max width=\textwidth, alt={}, center]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-09_473_1813_2161_127}
   (Total for Question 2 is 11 marks)

10 The table shows extracts from the "Method of travel by LA" tabs for 2001 and 2011 in the large data set.

1. In one of these four LAs a new tram system was opened in 2004. Suggest, with a reason taken from the data, which LA this could have been.
2. Julian suggests that the figures for "Bus, minibus or coach" for LA1 show that some new bus routes were probably introduced in this LA between 2001 and 2011. Use data from the table to comment on this suggestion.
3. In one of these four LAs a congestion charge on vehicles was introduced in 2003. Suggest, with a reason taken from the data, which LA this could have been.

9 A researcher is studying changes in behaviour in travelling to work by people who live outside London, between 2001 and 2011. He chooses the 15 Local Authorities (LAs) outside London with the largest decreases in the percentage of people driving to work, and arranges these in descending order. The table shows the changes in percentages from 2001 to 2011 in various travel categories, for these Local Authorities.

1. Explain why these LAs are not necessarily the 15 LAs with the largest decreases in the percentage of people driving to work.
2. The researcher wants to talk to those LAs outside London which have been most successful in encouraging people to change to cycling or walking to work.
   Suggest four LAs that he should talk to and why.
3. The researcher claims that Waverley is the LA outside London which has had the largest increase in the number of people working mainly at or from home.
   Does the data support his claim? Explain your answer.
4. Which two categories have replaced driving to work for the highest percentages of workers in these LAs? Support your answer with evidence from the table.
5. The researcher suggested that there would be strong correlation between the decrease in the percentage driving to work and the increase in percentage working mainly at or from home. Without calculation, use data from the table to comment briefly on this suggestion.

10 The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK. The first two digits of the Geography code give the type of each of the LAs:
06: Unitary authority
07: Non-metropolitan district
08: Metropolitan borough

1. In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram?
2. Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. Now assume the following.

5 Ali collected data from a random sample of 200 workers and recorded the number of days they each worked from home in the second week of September 2019. These data are shown in Fig. 5.1. \begin{table}[h] \end{table}

1. Represent the data by a suitable diagram.
2. Calculate
   Ali then collected data from a different random sample of 200 workers for the same week in September 2019. The mean number of days worked from home for this sample was 1.94 and the standard deviation was 1.75.
3. Explain whether there is any evidence to suggest that one or both of the samples must be flawed. Fig. 5.2 shows a cumulative frequency diagram for the ages of the workers in the first sample who worked from home on at least one day. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e0b502a8-c742-4d78-993c-8c0c7329ec9c-04_671_1362_1452_241} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Ali concludes that \(90 \%\) of the workers in this sample who worked from home on at least one day were under 60 years of age
4. Explain whether Ali's conclusion is correct.

1 A researcher collects data concerning the number of different social media platforms used by school pupils on a typical weekday. The frequency table for the data is shown below. The researcher uses software to represent the results in this diagram. \includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-04_961_1195_737_242}

1. Explain why this diagram is inappropriate.
2. Calculate the following for the number of social media platforms used:
   1. the mean,
   2. the standard deviation.

3 A student conducts an investigation into the number of hours spent cooking per week by people who live in village A. The student represents the data in the cumulative frequency diagram below. \section*{Hours spent cooking per week by people who live in village A} \includegraphics[max width=\textwidth, alt={}, center]{ce94c1ea-ffe5-42d0-8f8a-43c47105d6bf-3_796_1494_918_233}

1. How many people were involved in the investigation?
2. Use the copy of the diagram in the Printed Answer Booklet to determine an estimate for the interquartile range. The student conducts a similar investigation into the number of hours spent cooking per week by 200 people who live in village B. The interquartile range is found to be 3.9 hours.
3. Explain whether the evidence suggests that the number of hours spent cooking by people who live in village B is more variable, equally variable or less variable than the number of hours spent cooking by people who live in village A .

2 A student measures the upper arm lengths of a sample of 97 women. The results are summarised in the frequency table in Fig. 2.1. \begin{table}[h] \end{table} The student constructs two cumulative frequency diagrams to represent the data using different class intervals. These are shown in Fig. 2.2 opposite One of these diagrams is correct and the other is incorrect.

1. State which diagram is incorrect, justifying your answer.
2. Use the correct diagram in Fig. 2.2 to find an estimate of the median. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-05_2256_1230_191_148} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}

3 A researcher is conducting an investigation into the number of portions of fruit adults consume each day. The researcher decides to ask 50 men and 50 women to complete a simple questionnaire.

1. State the type of sampling procedure the researcher is using.
2. Write down one disadvantage of this sampling procedure. The researcher represents the data in Fig. 3.1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Number of portions of fruit consumed by adults} \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-06_531_991_701_248}
   \end{figure} Fig. 3.1
3. Describe the shape of the distribution. The data are summarised in the frequency table in Fig. 3.2. \begin{table}[h]

   Number of portions of fruit	0	1	2	3	4	5
   Number of adults	18	34	26	11	7	4

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
4. For the data in Fig. 3.2, use your calculator to find
   Give your answers correct to 2 decimal places. A second researcher chooses a proportional stratified sample of 100 children from years 5 and 6 in a certain primary school. There are 220 children to choose from. In year 5 there are 125 children, of whom 81 are boys.
5. How many year 5 girls should be included in the sample? The second researcher found that the mean number of portions of fruit consumed per day by the children in this sample was 1.61 and the standard deviation was 0.53 .
6. Comment on the amount of fruit consumed per day by the children compared to the amount of fruit consumed per day by the adults.

2 Mia rolls a six-sided die 24 times and records the scores. She displays her results in a vertical line chart. This is shown in Fig. 2.1. \begin{figure}[h] \end{figure}

1. Describe the shape of the distribution. She repeats the experiment, but this time she rolls the die 50 times. Her results are displayed in Fig. 2.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Scores on a six-sided die} \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-03_476_1161_1617_242}
   \end{figure} Fig. 2.2 Her brother Kai rolls the same die 1000 times and displays his results in a similar diagram.
2. Assuming the die is fair, describe the distribution you would expect to see in Kai's diagram.

11 James is investigating the amount of time retired people spend each day using social media. He collects a sample by advertising in a local newspaper for people to complete an online survey.

1. State
   James processes his data in order to draw a histogram. His table of results is shown below.

   Time spent using social media in minutes	\(0 -\)	\(15 -\)	\(30 -\)	\(60 -\)	\(120 - 240\)
   Number of people per minute	12.2	14.0	8.4	7.3	3.1

2. Show that the size of the sample is 1455 .
3. Calculate an estimate of the probability that a retired person spends more than an hour per day using social media.

9 Fig. 9.1 shows box and whisker diagrams which summarise the birth rates per 1000 people for all the countries in three of the regions as given in the pre-release data set.
The diagrams were drawn as part of an investigation comparing birth rates in different regions of the world. Africa (Sub-Saharan) \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_104_991_557_730} East and South East Asia \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_109_757_744_671} Caribbean \includegraphics[max width=\textwidth, alt={}, center]{05376a51-e768-4b45-9c18-c98255a4bd70-08_99_369_982_730} \begin{figure}[h] \end{figure}

1. Discuss the distributions of birth rates in these regions of the world. Make three different statements. You should refer to both information from the box and whisker diagrams and your knowledge of the large data set.
2. The birth rates for all the countries in Australasia are shown below.

   Country	Birth rate per 1000
   Australia	12.19
   New Zealand	13.4
   Papua New Guinea	24.89

   1. Explain why the calculation below is not a correct method for finding the birth rate per 1000 for Australasia as a whole. $$\frac { 12.19 + 13.4 + 24.89 } { 3 } \approx 16.83$$
   2. Without doing any calculations, explain whether the birth rate per 1000 for Australasia as a whole is higher or lower than 16.83 . The scatter diagram in Fig. 9.2 shows birth rate per 1000 and physicians/ 1000 population for all the countries in the pre-release data set. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-09_898_1698_386_274} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
      \end{figure}
3. Describe the correlation in the scatter diagram.
4. Discuss briefly whether the scatter diagram shows that high birth rates would be reduced by increasing the number of physicians in a country.

9 At the end of each school term at North End College all the science classes in year 10 are given a test. The marks out of 100 achieved by members of set 1 are shown in Fig. 9. \begin{table}[h] \end{table} Key \(5 \quad\) 2 represents a mark of 52

1. Describe the shape of the distribution.
2. The teacher for set 1 claimed that a typical student in his class achieved a mark of 95. How did he justify this statement?
3. Another teacher said that the average mark in set 1 is 76 . How did she justify this statement? Benson's mark in the test is 35 . If the mark achieved by any student is an outlier in the lower tail of the distribution, the student is moved down to set 2 .
4. Determine whether Benson is moved down to set 2 .

14 The pre-release material includes data on unemployment rates in different countries. A sample from this material has been taken. All the countries in the sample are in Europe. The data have been grouped and are shown in Fig 14.1. \begin{table}[h] \end{table} A cumulative frequency curve has been generated for the sample data using a spreadsheet. This is shown in Fig. 14.2. \begin{figure}[h] \end{figure} Hodge used Fig. 14.2 to estimate the median unemployment rate in Europe. He obtained the answer 5.0. The correct value for this sample is 6.9.

1. (A) There is a systematic error in the diagram.
   The scatter diagram shown in Fig. 14.3 shows the unemployment rate and life expectancy at birth for the 47 countries in the sample for which this information is available. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Scatter diagram to show life expectancy at birth against unemployment rate} \includegraphics[alt={},max width=\textwidth]{d8ff9511-aff7-45ea-ba55-e6667e8ba760-09_627_1281_456_367}
   \end{figure} Fig. 14.3 The product moment correlation coefficient for the 47 items in the sample is - 0.2607 .
   The \(p\)-value associated with \(r = - 0.2607\) and \(n = 47\) is 0.0383 .
2. Does this information suggest that there is an association between unemployment rate and life expectancy at birth in countries in Europe? Hodge uses the spreadsheet tools to obtain the equation of a line of best fit for this data.
3. The unemployment rate in Kosovo is 35.3 , but there is no data available on life expectancy. Is it reasonable to use Hodge's line of best fit to estimate life expectancy at birth in Kosovo?

8 A garden centre stocks coniferous hedging plants. These are displayed in 10 rows, each of 120 plants. An employee collects a sample of the heights of these plants by recording the height of each plant on the front row of the display.

1. Explain whether the data collected by the employee is a simple random sample. The data are shown in the cumulative frequency curve below. \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-06_1376_1344_680_233} The owner states that at least \(75 \%\) of the plants are between 40 cm and 80 cm tall.
2. Show that the data collected by the employee supports this statement.
3. Explain whether all samples of 120 plants would necessarily support the owner's statement.

18 Riley is investigating the daily water consumption, in litres, of his household.
He records the amount used for a random sample of 120 days from the previous twelve-month period. The daily water consumption, in litres, is denoted by \(x\). Summary statistics for Riley's sample are given below. \(\sum \mathrm { x } = 31164.7 \sum \mathrm { x } ^ { 2 } = 8101050.91 \mathrm { n } = 120\)

1. Calculate the sample mean giving your answer correct to \(\mathbf { 3 }\) significant figures. Riley displays the data in a histogram. \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-13_832_1383_934_242}
2. Find the number of days on which between 255 and 260 litres were used.
3. Give two reasons why a Normal distribution may be an appropriate model for the daily consumption of water. Riley uses the sample mean and the sample variance, both correct to \(\mathbf { 3 }\) significant figures, as parameters of a Normal distribution to model the daily consumption of water.
4. Use Riley's model to calculate the probability that on a randomly chosen day the household uses less than 255 litres of water.
5. Calculate the probability that the household uses less than 255 litres of water on at least 5 days out of a random sample of 28 days. The company which supplies the water makes charges relating to water consumption which are shown in the table below.

   Standing charge per day in pence	7.8
   Charge per litre in pence	0.18

6. Adapt Riley's model for daily water consumption to model the daily charges for water consumption. \section*{END OF QUESTION PAPER}

3 The histogram shows the amount spent on electricity in pounds in a sample of households in March 2023. \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-04_542_1276_1133_244}

1. Describe the shape of the distribution. A total of 16 households each spent between \(\pounds 60\) and \(\pounds 65\) on electricity.
2. Determine how many households were in the sample altogether.

4 Fig. 4 shows a cumulative frequency diagram for the time spent revising mathematics by year 11 students at a certain school during a week in the summer term. \begin{figure}[h] \end{figure}

1. Use the diagram to estimate the median time spent revising mathematics by these students. [1] A teacher comments that \(90 \%\) of the students spent less than an hour revising mathematics during this week.
2. Determine whether the information in the diagram supports this comment.

10 Ben has an interest in birdwatching. For many years he has identified, at the start of the year, 32 days on which he will spend an hour counting the number of birds he sees in his garden. He divides the year into four using the Meteorological Office definition of seasons. Each year he uses stratified sampling to identify the 32 days on which he will count the birds in his garden, drawn equally from the four seasons. Ben's data for 2019 are shown in the stem and leaf diagram in Fig. 10.1. \begin{table}[h] \end{table}

1. Suggest a reason why Ben chose to use stratified sampling instead of simple random sampling.
2. Describe the shape of the distribution.
3. Explain why the mode is not a useful measure of central tendency in this case.
4. For Ben's sample, determine
   Ben found a boxplot for the sample of size 32 he collected using stratified sampling in 2015. The boxplot is shown in Fig. 10.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c9d14a4d-a1c8-42ad-9c0b-42cef6b3612f-06_483_1163_1982_242} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{figure} In 2016 Ben replaced his hedge with a garden fence.
   Ben now believes that
   Jane says she can tell that the data for 2015 is definitely uniformly distributed by looking at the boxplot.
5. Explain why Jane is wrong.

1 Triangle ABC is shown in Fig. 1. \begin{figure}[h] \end{figure} Find the perimeter of triangle ABC .

2. Two youth clubs, Eastyou and Westyou, decided to raise money for charity by running a 5 km race. All the members of the youth clubs took part and the time, in minutes, taken for each member to run the 5 km was recorded. The times for the Westyou members are summarised in Figure 1. \begin{figure}[h] \end{figure}

1. Write down the time that is exceeded by \(75 \%\) of Westyou members. The times for the Eastyou members are summarised by the stem and leaf diagram below.

   Stem	Leaf
   2	0	2	3	4											\(( 4 )\)
   2	5	6	8	8	8	9	9
   3	0	0	0	0	0	1	1	1	2	2	2	2	3	4	\(( 14 )\)
   3	5	5	5	7	9										\(( 5 )\)

   Key: 2|0 means 20 minutes
2. Find the value of the median and interquartile range for the Eastyou members. An outlier is a value that falls either
3. On the grid on page 7, draw a box plot to represent the times of the Eastyou members.
4. State the skewness of each distribution. Give reasons for your answers. $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 } \\ & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
   \includegraphics[max width=\textwidth, alt={}]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-06_2255_50_314_1976}
   \includegraphics[max width=\textwidth, alt={}, center]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-07_406_1390_2224_262} Turn over for a spare grid if you need to redraw your box plot. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Only use this grid if you need to redraw your box plot.} \includegraphics[alt={},max width=\textwidth]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-09_401_1399_2261_258}
   \end{figure}

1. A random sample of 100 carrots is taken from a farm and their lengths, \(L \mathrm {~cm}\), recorded. The data are summarised in the following table.

A histogram is drawn to represent these data.
The bar representing the class \(5 \leqslant L < 8\) is 1.5 cm wide and 1 cm high.

1. Find the width and height of the bar representing the class \(15 \leqslant L < 20\)
2. Use linear interpolation to estimate the median length of these carrots.
3. Estimate
   1. the mean length of these carrots,
   2. the standard deviation of the lengths of these carrots. A supermarket will only buy carrots with length between 9 cm and 22 cm .
4. Estimate the proportion of carrots from the farm that the supermarket will buy. Any carrots that the supermarket does not buy are sold as animal feed. The farm makes a profit of 2.2 pence on each carrot sold to the supermarket, a profit of 0.8 pence on each carrot longer than 22 cm and a loss of 1.2 pence on each carrot shorter than 9 cm .
5. Find an estimate of the mean profit per carrot made by the farm.

1. The company Seafield requires contractors to record the number of hours they work each week. A random sample of 38 weeks is taken and the number of hours worked per week by contractor Kiana is summarised in the stem and leaf diagram below.

Key : 3|2 means 32 The quartiles for this distribution are summarised in the table below.

1. Find the values of \(w , x\) and \(y\) Kiana is looking for outliers in the data. She decides to classify as outliers any observations greater than $$Q _ { 3 } + 1.0 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
2. Showing your working clearly, identify any outliers that Kiana finds.
3. Draw a box plot for these data in the space provided on the grid opposite.
4. Use the formula $$\text { skewness } = \frac { \left( Q _ { 3 } - Q _ { 2 } \right) - \left( Q _ { 2 } - Q _ { 1 } \right) } { \left( Q _ { 3 } - Q _ { 1 } \right) }$$ to find the skewness of these data. Give your answer to 2 significant figures. Kiana's new employer, Landacre, wishes to know the average number of hours per week she worked during her employment at Seafield to help calculate the cost of employing her.
5. Explain why Landacre might prefer to know Kiana's mean, rather than median, number of hours worked per week. Turn over for a spare grid if you need to redraw your box plot.

1. Gill buys a bag of logs to use in her stove. The lengths, \(l \mathrm {~cm}\), of the 88 logs in the bag are summarised in the table below.

A histogram is drawn to represent these data.
The bar representing logs with length \(27 < l \leqslant 30\) has a width of 1.5 cm and a height of 4 cm .

1. Calculate the width and height of the bar representing log lengths of \(20 < l \leqslant 25\)
2. Use linear interpolation to estimate the median of \(l\) The maximum length of log Gill can use in her stove is 26 cm .
   Gill estimates, using linear interpolation, that \(x\) logs from the bag will fit into her stove.
3. Show that \(x = 62\) Gill randomly selects 4 logs from the bag.
4. Using \(x = 62\), find the probability that all 4 logs will fit into her stove. The weights, \(W\) grams, of the logs in the bag are coded using \(y = 0.5 w - 255\) and summarised by $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
5. Calculate
   1. the mean of \(W\)
   2. the variance of \(W\)

1. A researcher is investigating the growth of two types of tree, Birch and Maple. The height, to the nearest cm, a seedling grows in one year is recorded for 35 Birch trees and 32 Maple trees. The results are summarised in the back-to-back stem and leaf diagram below.

Key: 5 | 6 | 3 means 65 cm for a Birch tree and 63 cm for a Maple tree
The median height that these Maple trees grow in one year is 45 cm .

1. Find the value of \(\boldsymbol { k }\), used in the stem and leaf diagram.
2. Find the lower quartile and the upper quartile of the height grown in one year for these Birch trees. The researcher defines an outlier as an observation that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or less than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$
3. Show that there is only one outlier amongst the Birch trees. The grid on page 3 shows a box plot for the heights that the Maple trees grow in one year.
4. On the same grid draw a box plot for the heights that the Birch trees grow in one year.
5. Comment on any difference in the distributions of the growth of these Birch trees and the growth of these Maple trees.
   State the values of any statistics you have used to support your comment. The researcher realises he has missed out 4 pieces of data for the Maple trees. The heights each seedling grows in one year, to the nearest cm, in ascending order, for these 4 Maple trees are \(27 \mathrm {~cm} , a \mathrm {~cm} , 48 \mathrm {~cm} , 2 a \mathrm {~cm}\). Given that there is no change to the box plot for the Maple trees given on page 3
6. find the range of possible values for \(a\) Show your working clearly.
   \includegraphics[max width=\textwidth, alt={}]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-03_1243_1659_1464_210}
   Only use this grid if you need to redraw your answer for part (d) \includegraphics[max width=\textwidth, alt={}, center]{ee0c7c12-84f3-479c-b36a-3357f8529a1c-05_1154_1643_1503_217}
   (Total for Question 1 is 13 marks)