Data representation

2 The table summarises the diameters, \(d\) millimetres, of a random sample of 60 new cricket balls to be used in junior cricket.

2 The table summarises the diameters, \(d\) millimetres, of a random sample of 60 new cricket balls to be used in junior cricket.

A quality control department checks the lifetimes of batteries produced by a company. The lifetimes, \(x\) minutes, for a random sample of 80 'Superstrength' batteries are shown in the table below.

Lifetime	\(160 \leq x < 165\)	\(165 \leq x < 168\)	\(168 \leq x < 170\)	\(170 \leq x < 172\)	\(172 \leq x < 175\)	\(175 \leq x < 180\)
Frequency	5	14	20	21	16	4

1. Estimate the proportion of these batteries which have a lifetime of at least 174.0 minutes. [2]
2. Use the data in the table to estimate
   [3]

The data in the table on the previous page are represented in the following histogram, Fig 15. \includegraphics{figure_15} A quality control manager models the data by a Normal distribution with the mean and standard deviation you calculated in part (b).

1. Comment briefly on whether the histogram supports this choice of model. [2]
2. 1. Use this model to estimate the probability that a randomly selected battery will have a lifetime of more than 174.0 minutes.
   2. Compare your answer with your answer to part (a). [3]

The company also manufactures 'Ultrapower' batteries, which are stated to have a mean lifetime of 210 minutes.

1. A random sample of 8 Ultrapower batteries is selected. The mean lifetime of these batteries is 207.3 minutes. Carry out a hypothesis test at the 5% level to investigate whether the mean lifetime is as high as stated. You should use the following hypotheses \(\text{H}_0 : \mu = 210\), \(\text{H}_1 : \mu < 210\), where \(\mu\) represents the population mean for Ultrapower batteries. You should assume that the population is Normally distributed with standard deviation 3.4. [5]

The table below shows the frequencies for a set of data from a continuous variable \(X\) A histogram is drawn to represent this data. Find the frequency density of the bar in the histogram representing the class \(24 < x \leq 42\) Circle your answer. [1 mark] 2 \qquad 18 \qquad 36 \qquad 70

1.

1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.

   Age group	0 to 4
   Frequency	2143

   Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.

5 At a certain resort the number of hours of sunshine, measured to the nearest hour, was recorded on each of 21 days. The results are summarised in the table. The diagram shows part of a histogram to illustrate the data. The scale on the frequency density axis is 2 cm to 1 unit. \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-3_944_1778_699_148}

1. (a) Calculate the frequency density of the \(1 - 3\) class.
   (b) Fred wishes to draw the block for the 10 - 15 class on the same diagram. Calculate the height, in centimetres, of this block.
2. A cumulative frequency graph is to be drawn. Write down the coordinates of the first two points that should be plotted. You are not asked to draw the graph.
3. (a) Calculate estimates of the mean and standard deviation of the number of hours of sunshine.
   (b) Explain why your answers are only estimates.

2 The table summarises the lengths in centimetres of 104 dragonflies.

1. State which class contains the upper quartile.
2. Draw a histogram, on graph paper, to represent the data.

5 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

1. Illustrate these data by means of a histogram.
2. Identify the type of skewness of the distribution.

The probability distribution for \(X\), the lifetime of a light bulb, in hours, is given below. a) Suppose that a random sample of 40 light bulbs is tested, and a histogram is drawn of their lifetimes. Calculate the expected height of the bar for the interval \(262 \leqslant x < 265\).
b) Now suppose that the last two intervals are changed to \(265 \leqslant x < 268\) and \(268 \leqslant x < 300\). Explain why it is not possible to tell what will happen to the expected heights of the last two bars.
c) Celyn collects a different random sample of 40 light bulbs to test. She draws a histogram of their lifetimes and finds that it is different to the histogram referred to in part (a). Should Celyn be concerned that the two histograms are different?

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.

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.

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-04_1335_1319_404_413} Use the curve to estimate

1. the interquartile range of the marks,
2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
5. What does this information suggest about the estimate of the interquartile range found in part (i)? \section*{June 2005}

1 The number of passengers getting off the 11.45 am train at a railway station on each of 35 days is summarised as follows.

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.

4 The following are the house prices in thousands of dollars, arranged in ascending order, for 51 houses from a certain area.

1. Draw a box-and-whisker plot to represent the data. An expensive house is defined as a house which has a price that is more than 1.5 times the interquartile range above the upper quartile.
2. For the above data, give the prices of the expensive houses.
3. Give one disadvantage of using a box-and-whisker plot rather than a stem-and-leaf diagram to represent this set of data.

6 The heights to the nearest metre of 134 office buildings in a certain city are summarised in the table below.

1. Draw a histogram on graph paper to illustrate the data.
2. Calculate estimates of the mean and standard deviation of these heights.

5 The lengths, \(t\) minutes, of 242 phone calls made by a family over a period of 1 week are summarised in the frequency table below.

1. Find the value of \(a\).
2. Calculate an estimate of the mean length of these phone calls.
3. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{a813e127-d116-411c-88ec-2443fdbc9391-07_2002_1513_486_356}

5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.

1. Draw a histogram to represent this information. \includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
2. Calculate an estimate of the mean time taken by these 200 players.
3. Find the greatest possible value of the interquartile range of these times.

The box plot below shows summary data for the number of minutes late that buses arrived at a rural bus stop. \includegraphics{figure_12} Identify which term best describes the distribution of this data. Circle your answer. [1 mark] negatively skewed \quad\quad normal \quad\quad positively skewed \quad\quad symmetrical

The box plot below shows summary data for the number of minutes late that buses arrived at a rural bus stop. \includegraphics{figure_12} Identify which term best describes the distribution of this data. Circle your answer. [1 mark] negatively skewed \quad\quad normal \quad\quad positively skewed \quad\quad symmetrical

3 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{b6d84f99-ee39-49c7-a5e8-05838efeef5a-2_804_1372_483_436} Data sourced from the 2001 Census, www.sta is \href{http://ics.gov.uk}{ics.gov.uk}

1. State the type of skewness shown by the distribution.
2. Use the histogram to estimate the number of people aged under 25.
3. The table below shows the cumulative frequency distribution.

   Age	20	30	40	50	65	100
   Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

   (A) Use the histogram to find the value of \(a\).
   (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.

   Age ( \(x\) years)	\(0 \leqslant x < 20\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 65\)	\(65 \leqslant x < 100\)
   Frequency (thousands)	1120	650	770	590	680	610

4. Illustrate these data by means of a histogram.
5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
   The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.

1. The histogram summarises the heights of 256 seedlings two weeks after they were planted. \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-06_1242_1810_287_132}
   1. Use linear interpolation to estimate the median height of the seedlings.
      (4)
   Chris decides to model the frequency density for these 256 seedlings by a curve with equation $$y = k x ( 8 - x ) \quad 0 \leqslant x \leqslant 8$$ where \(k\) is a constant.
2. Find the value of \(k\) Using this model,
3. write down the median height of the seedlings.

1. The table shows the time, to the nearest minute, spent waiting for a taxi by each of 80 people one Sunday afternoon.

1. Write down the upper class boundary for the \(2 - 4\) minute interval. A histogram is drawn to represent these data. The height of the tallest bar is 6 cm .
2. Calculate the height of the second tallest bar.
3. Estimate the number of people with a waiting time between 3.5 minutes and 7 minutes.
4. Use linear interpolation to estimate the median, the lower quartile and the upper quartile of the waiting times.
5. Describe the skewness of these data, giving a reason for your answer.

6 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.

1. Estimate the number of boys who have heights of at least 155 cm .
2. Calculate an estimate of the median height of the 100 boys.
3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{76283206-687f-45d6-9204-952d60843cf1-3_865_1349_1297_349}
4. How many more girls than boys had heights exceeding 160 cm ?
5. Calculate an estimate of the mean height of the 100 girls.

1 A study of the ages of car drivers in a certain country produced the results shown in the table. \begin{table}[h] \end{table} Illustrate these results diagrammatically.

1 Ashfaq and Kuljit have done a school statistics project on the prices of a particular model of headphones for MP3 players. Ashfaq collected prices from 21 shops. Kuljit used the internet to collect prices from 163 websites.

1. Name a suitable statistical diagram for Ashfaq to represent his data, together with a reason for choosing this particular diagram.
2. Name a suitable statistical diagram for Kuljit to represent her data, together with a reason for choosing this particular diagram.

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 number of minutes of recorded music on a sample of 100 CDs is summarised below.

1. Illustrate the data by means of a histogram.
2. Identify two features of the distribution.

1 The number of minutes of recorded music on a sample of 100 CDs is summarised below.

1. Illustrate the data by means of a histogram.
2. Identify two features of the distribution.

5 The number of people a football stadium can hold is called the 'capacity'. The capacities of 130 football stadiums in the UK, to the nearest thousand, are summarised in the table.

1. On graph paper, draw a histogram to represent this information. Use a scale of 2 cm for a capacity of 10000 on the horizontal axis.
2. Calculate an estimate of the mean capacity of these 130 stadiums.
3. Find which class in the table contains the median and which contains the lower quartile.

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.

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.

   \multirow{2}{*}{}

   \multirow{3}{*}}{

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

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.

   \multirow{2}{*}{}

   \multirow{3}{*}}{

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

The masses, \(x\) grams, of 800 apples are summarised in the histogram. \includegraphics{figure_6}

1. On the frequency density axis, 1 cm represents \(a\) units. Find the value of \(a\). [3]
2. Find an estimate of the median mass of the apples. [4]

8

1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.

   Age group	0 to 4
   Frequency	2143

   Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.

5. In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below. A histogram was drawn and the group ( \(8 - 10\) ) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.

1. Calculate the width and height of the rectangle representing the group (16-25) hours.
2. Use linear interpolation to estimate the median and interquartile range.
3. Estimate the mean and standard deviation of the number of hours spent shopping.
4. State, giving a reason, the skewness of these data.
5. State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.

The manager of a company noted the times spent in 80 meetings. The results were as follows. Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range. [6]

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 \(t\).

Each year the total number of hours, \(x\), of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.

\(x\)	\(30 \leqslant x < 60\)	\(60 \leqslant x < 90\)	\(90 \leqslant x < 110\)	\(110 \leqslant x < 140\)	\(140 \leqslant x < 180\)	\(180 \leqslant x \leqslant 240\)
Number of years	4	8	14	25	7	2

1. Draw a cumulative frequency graph to illustrate the data. [3]
2. Use your graph to estimate the 70th percentile of the data. [2]
3. Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years. [3]

The box plot below summarises the CO\(_2\) emissions, in g/km, for cars in the Large Data Set from the London and North West regions. \includegraphics{figure_12}

1. Using the box plot, give one comparison of central tendency and one comparison of spread for the two regions. [2 marks]
2. Jaspal, an environmental researcher, used all of the data in the Large Data Set to produce a statistical comparison of the CO\(_2\) and CO emissions in regions of England. Using your knowledge of the Large Data Set, give two reasons why his conclusions may be invalid. [2 marks]

The box plot below summarises the CO\(_2\) emissions, in g/km, for cars in the Large Data Set from the London and North West regions. \includegraphics{figure_12}

1. Using the box plot, give one comparison of central tendency and one comparison of spread for the two regions. [2 marks]
2. Jaspal, an environmental researcher, used all of the data in the Large Data Set to produce a statistical comparison of the CO\(_2\) and CO emissions in regions of England. Using your knowledge of the Large Data Set, give two reasons why his conclusions may be invalid. [2 marks]

The back-to-back stem and leaf diagram shows the journey times, to the nearest minute, of the commuter services into a big city provided by the trains of two operating companies. Key: \(4|3|6\) means 34 minutes for Company \(A\) and 36 minutes for Company \(B\).

1. Write down the numbers needed to complete the diagram. [1 mark]
2. Find the median and the quartiles for each company. [6 marks]
3. On graph paper, draw box plots for the two companies. Show your scale. [6 marks]
4. Use your plots to compare the two sets of data briefly. [2 marks]
5. Describe the skewness of each company's distribution of times. [2 marks]

1 The times taken, in minutes, by 80 people to complete a crossword puzzle are summarised by the box and whisker plot below. \includegraphics[max width=\textwidth, alt={}, center]{acb05873-e441-4b95-9732-6ebd5ae79fa6-2_147_848_507_612}

1. Write down the range and the interquartile range of the times.
2. Determine whether any of the times can be regarded as outliers.
3. Describe the shape of the distribution of the times.

1 The times taken, in minutes, by 80 people to complete a crossword puzzle are summarised by the box and whisker plot below. \includegraphics[max width=\textwidth, alt={}, center]{acb05873-e441-4b95-9732-6ebd5ae79fa6-2_147_848_507_612}

1. Write down the range and the interquartile range of the times.
2. Determine whether any of the times can be regarded as outliers.
3. Describe the shape of the distribution of the times.

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.

A reporter is writing an article on the CO₂ emissions from vehicles using the Large Data Set. The reporter claims that the Large Data Set shows that the CO₂ emissions from all vehicles in the UK have declined every year from 2002 to 2016. Using your knowledge of the Large Data Set, give two reasons why this claim is invalid. [2 marks]

1. Jiang is studying the variable Daily Mean Pressure from the large data set.

He drew the following box and whisker plot for these data for one of the months for one location using a linear scale but

• he failed to label all the values on the scale
• he gave an incorrect value for the median \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_248_1264_573_402}

Daily Mean Pressure (hPa)
Using your knowledge of the large data set, suggest a suitable value for

1. the median,
2. the range.
   (You are not expected to have memorised values from the large data set. The question is simply looking for sensible answers.)
   1. Jiang is studying the variable Daily Mean Pressure from the large data set.
   He drew the following box and whisker plot for these data for one of the months for one nong asing at
   • he gave an incorrect value for the median "
   Using your knowledge of the large data set, suggest a suitable value for
3. the median,
   " \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_42_31_1213_1304}
   "

13 The four pie charts illustrate the numbers of employees using different methods of travel in four Local Authorities in 2011. \includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-10_1131_1077_347_242}

1. State, with reasons, which of the four Local Authorities is most likely to be a rural area with many hills.
2. Explain why pie charts are more suitable for answering part (a) than bar charts showing the same data.
3. Two of the Local Authorities represent urban areas.
   1. State with a reason which two Local Authorities are likely to be urban.
   2. One urban Local Authority introduced a Park-and-Ride service in 2006. Users of this service drive to the edge of the urban area and then use buses to take them into the centre of the area. A student claims that a comparison of the corresponding pie charts for 2001 (not shown) and 2011 would enable them to identify which Local Authority this was. State with a reason whether you agree with the student.

4 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.

1. Draw a histogram on graph paper to illustrate these data.
2. Calculate an estimate of the median distance.

4 The frequency table below shows the distance travelled by 1200 visitors to a particular UK tourist destination in August 2008.

1. Draw a histogram on graph paper to illustrate these data.
2. Calculate an estimate of the median distance.

5. The values of daily sales, to the nearest \(\pounds\), taken at a newsagents last year are summarised in the table below.

1. Draw a histogram to represent these data.
2. Use interpolation to estimate the median and inter-quartile range of daily sales.
3. Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
4. State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
   (2)

Kareem bought some tomatoes. He recorded the mass of each tomato and displayed the results in a histogram, which is shown below. \includegraphics{figure_7} Determine how many tomatoes Kareem bought. [2]

4 The following histogram summarises the times, in minutes, taken by 190 people to complete a race. \includegraphics[max width=\textwidth, alt={}, center]{df246a50-157b-49f7-bba0-f9b86960b8b9-2_1210_1125_1251_513}

1. Show that 75 people took between 200 and 250 minutes to complete the race.
2. Calculate estimates of the mean and standard deviation of the times of the 190 people.
3. Explain why your answers to part (ii) are estimates.

2 The histogram shows the amount of money, in pounds, spent by the customers at a supermarket on a particular day. \includegraphics[max width=\textwidth, alt={}, center]{c7cb0f6b-7b6b-4c52-8287-7efc6bd70247-2_985_1473_470_379}

1. Express the data in the form of a grouped frequency table.
2. Use your table to estimate the total amount of money spent by customers on that day.

1. The students in a class were each asked to write down how many CDs they owned. The student with the least number of CDs had 14 and all but one of the others owned 60 or fewer. The remaining student owned 65 . The quartiles for the class were 30,34 and 42 respectively.

Outliers are defined to be any values outside the limits of \(1.5 \left( Q _ { 3 } - Q _ { 1 } \right)\) below the lower quartile or above the upper quartile. On graph paper draw a box plot to represent these data, indicating clearly any outliers.
(7 marks)

2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

1. On the insert, draw a cumulative frequency diagram to illustrate the data.
2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
3. State the type of skewness of the distribution of the data.

1 Some adults and some children each tried to estimate, without using a watch, the number of seconds that had elapsed in a fixed time-interval. Their estimates are shown below.

1. Draw a back-to-back stem-and-leaf diagram to represent the data.
2. Make two comparisons between the estimates of the adults and the children.

1 A business analyst collects data about the distribution of hourly wages, in \(\pounds\), of shop-floor workers at a factory. These data are illustrated in the box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{56f1bd5c-4b45-4e36-a324-e7e0edbb5bdd-1_206_1420_505_397}

1. Name the type of skewness of the distribution.
2. Find the interquartile range and hence show that there are no outliers at the lower end of the distribution, but there is at least one outlier at the upper end.
3. Suggest possible reasons why this may be the case.

1. A random sample is to be taken from the A-level results obtained by the final-year students in a Sixth Form College. Suggest
   1. suitable sampling units,
   2. a suitable sampling frame.
   3. Would it be advisable simply to use the results of all those doing A-level Maths?
   Explain your answer.

The histogram below shows the heights, in cm, of male A-level students at a particular school. \includegraphics{figure_12} Which class interval contains the median height? Circle your answer. [1 mark] \([155, 160)\) \quad \([160, 170)\) \quad \([170, 180)\) \quad \([180, 190]\)

6 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{05b96db3-93c7-4921-a1c6-c7b2f8952a8f-4_1264_1553_486_296}

1. Use the diagram to estimate the median and interquartile range of the data.
2. Use your answers to part (i) to estimate the number of outliers in the sample.
3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
5. Find the probability that
   (A) exactly 2 of these 17 babies require special care,
   (B) more than 2 of the 17 babies require special care.
6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.

2. A video rental shop needs to find out whether or not videos have been rewound when they are returned; it will do this by taking a sample of returned videos

1. State one advantage and one disadvantage of taking a sample.
2. Suggest a suitable sampling frame.
3. Describe the sampling units.
4. Criticise the sampling method of looking at just one particular shelf of videos.

1. (a) Briefly describe the difference between a census and a sample survey.
   (b) Illustrate the difference by considering the case of a village council which has to decide whether or not to build a new village hall.

Given that the council decides to use a sample survey,
(c) suggest suitable sampling units.

1 A residents' association has 654 members, numbered from 1 to 654 . The secretary wishes to send a questionnaire to a random sample of members. In order to choose the members for the sample she uses a table of random numbers. The first line in the table is as follows. $$\begin{array} { l l l l l l } 1096 & 4357 & 3765 & 0431 & 0928 & 9264 \end{array}$$ The numbers of the first two members in the sample are 109 and 643. Find the numbers of the next three members in the sample.

\includegraphics{figure_3} The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies. [6]

6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.

1. State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
2. Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
3. Calculate estimates of the mean and variance of the times taken by these athletes.

1. \begin{figure}[h] \end{figure} The histogram in Figure 1 shows the times taken to complete a crossword by a random sample of students. The number of students who completed the crossword in more than 15 minutes is 78
Estimate the percentage of students who took less than 11 minutes to complete the crossword.

7. The table shows information, derived from the 2011 UK census, about the percentage of employees who used various methods of travel to work in four Local Authorities. One of the Local Authorities is a London borough and two are metropolitan boroughs, not in London.

1. Which one of the Local Authorities is a London borough? Give a reason for your answer.
2. Which two of the Local Authorities are metropolitan boroughs outside London? In each case give a reason for your answer.
3. Describe one difference between the public transport available in the two metropolitan boroughs, as suggested by the table.
   [0pt]

1 The distance of a student's home from college, correct to the nearest kilometre, was recorded for each of 55 students. The distances are summarised in the following table. Dominic is asked to draw a histogram to illustrate the data. Dominic's diagram is shown below. \includegraphics[max width=\textwidth, alt={}, center]{d6836b62-75e7-410e-ab1e-83c391b85948-2_1225_1303_628_422} Give two reasons why this is not a correct histogram.

\includegraphics{figure_3} In an open-plan office there are 88 computers. The times taken by these 88 computers to access a particular web page are represented in the cumulative frequency diagram.

1. On graph paper draw a box-and-whisker plot to summarise this information. [4]

An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.

1. Show that there are no outliers. [2]

Doug has a list of times taken by competitors in a 'fun run'. He has grouped the data and calculated the frequency densities in order to draw a histogram to represent the information. Some of the data are presented in Fig. 2.

Time in minutes	\(15-\)	\(20-\)	\(25-\)	\(35-\)	\(45-60\)
Number of runners	12	23	59	71
Frequency density	2.4		5.9	7.1	1.4

Fig. 2

1. Write down the missing values in the copy of Fig. 2 in the Printed Answer Booklet. [2]
2. Doug labels the horizontal axis on the histogram 'time in minutes' and the vertical axis 'number of minutes per runner'. State which one of these labels is incorrect and write down a correct version. [1]

6. The number of bags of potato crisps sold per day in a bar was recorded over a two-week period. The results are shown below. $$20,15,10,30,33,40,5,11,13,20,25,42,31,17$$

1. Calculate the mean of these data.
2. Draw a stem and leaf diagram to represent these data.
3. Find the median and the quartiles of these data. An outlier is an observation that falls either \(1.5 \times\) (interquartile range) above the upper quartile or \(1.5 \times\) (interquartile range) below the lower quartile.
4. Determine whether or not any items of data are outliers.
5. On graph paper draw a box plot to represent these data. Show your scale clearly.
6. Comment on the skewness of the distribution of bags of crisps sold per day. Justify your answer.