2.02i Select/critique data presentation

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.

14 The pre-release material contains medical data for 103 women and 97 men.
The boxplot represents the weights in kg of 101 of the women from the pre-release material. \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-09_421_1232_735_244}

1. Use your knowledge of the pre-release material to give a reason why the weights of all 103 women were not included in the diagram.
2. Determine the range of values in which any outliers lie.
3. Use your knowledge of the pre-release material to explain whether these outliers should be removed from any further analysis of the data.
4. The median weight of men in the sample was found to be 79.9 kg . Explain what may be inferred by comparing the median weight of men with the median weight of women. Further analysis of the weights of both men and women is carried out. The table shows some of the results.

   	mean	standard deviation
   men	82.69 kg	19.98 kg
   women	72.5 kg	19.95 kg

5. Use the information in the table to make two inferences about the distribution of the weights of men compared with the distribution of the weights of women.

2. The number of caravans on Seaview caravan site on each night in August last year is summarised in the following stem and leaf diagram.

1. Find the three quartiles of these data. During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38,45 and 52 respectively.
2. On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
3. Compare and contrast these two box plots.
4. Give an interpretation to the upper quartiles of these two distributions.

6. Three swimmers Alan, Diane and Gopal record the number of lengths of the swimming pool they swim during each practice session over several weeks. The stem and leaf diagram below shows the results for Alan.

1. Find the three quartiles for Alan's results. The table below summarises the results for Diane and Gopal.

   	Diane	Gopal
   Smallest value	35	25
   Lower quartile	37	34
   Median	42	42
   Upper quartile	53	50
   Largest value	65	57

2. Using the same scale and on the same sheet of graph paper draw box plots to represent the data for Alan, Diane and Gopal.
3. Compare and contrast the three box plots.

6. The labelling on bags of garden compost indicates that the bags weigh 20 kg . The weights of a random sample of 50 bags are summarised in the table below.

1. On graph paper, draw a histogram of these data.
2. Using the coding \(y = 10\) (weight in \(\mathrm { kg } - 14\) ), find an estimate for the mean and standard deviation of the weight of a bag of compost.
   [0pt] [Use \(\Sigma f y ^ { 2 } = 171\) 503.75]
3. Using linear interpolation, estimate the median. The company that produces the bags of compost wants to improve the accuracy of the labelling. The company decides to put the average weight in kg on each bag.
4. Write down which of these averages you would recommend the company to use. Give a reason for your answer.

4. The attendance at college of a group of 18 students was recorded for a 4-week period. The number of students actually attending each of 16 classes are shown below.

1. 1. Calculate the mean and the standard deviation of the number of students attending these classes.
   2. Express the mean as a percentage of the 18 students in the group. In the same 4-week period, the attendance of a different group of 20, students is shown below.

      20	16	18	19
      15	14	14	15
      18	15	16	17
      16	18	15	14

2. Construct a back-to-back stem and leaf diagram to represent the attendance in both groups.
3. Find the mode, median and inter-quartile range for each group of students. The mean percentage attendance and standard deviation for the second group of students are 81.25 and 1.82 respectively.
4. Compare and contrast the attendance of these 2 groups of students.

2. The following table summarises the distances, to the nearest km , that 134 examiners travelled to attend a meeting in London.

1. Give a reason to justify the use of a histogram to represent these data.
2. Calculate the frequency densities needed to draw a histogram for these data.
   (DO NOT DRAW THE HISTOGRAM)
3. Use interpolation to estimate the median \(Q _ { 2 }\), the lower quartile \(Q _ { 1 }\), and the upper quartile \(Q _ { 3 }\) of these data. The mid-point of each class is represented by \(x\) and the corresponding frequency by \(f\). Calculations then give the following values $$\Sigma f _ { x } = 8379.5 \quad \text { and } \quad \Sigma f _ { x ^ { 2 } } = 557489.75$$
4. Calculate an estimate of the mean and an estimate of the standard deviation for these data. One coefficient of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
5. Evaluate this coefficient and comment on the skewness of these data.
6. Give another justification of your comment in part (e).

4. Aeroplanes fly from City \(A\) to City \(B\). Over a long period of time the number of minutes delay in take-off from City \(A\) was recorded. The minimum delay was 5 minutes and the maximum delay was 63 minutes. A quarter of all delays were at most 12 minutes, half were at most 17 minutes and \(75 \%\) were at most 28 minutes. Only one of the delays was longer than 45 minutes. 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.

1. On the graph paper opposite draw a box plot to represent these data.
2. Comment on the distribution of delays. Justify your answer.
3. Suggest how the distribution might be interpreted by a passenger who frequently flies from City \(A\) to City \(B\). \includegraphics[max width=\textwidth, alt={}, center]{9698650f-ef85-468d-a703-1b40df7f9d02-07_1190_1487_278_223}

1. (a) Describe the main features and uses of a box plot.
   

Children from schools \(A\) and \(B\) took part in a fun run for charity. The times, to the nearest minute, taken by the children from school \(A\) are summarised in Figure 1. \begin{figure}[h] \end{figure} (b) (i) Write down the time by which \(75 \%\) of the children in school \(A\) had completed the run.
(ii) State the name given to this value.
(c) Explain what you understand by the two crosses ( X ) on Figure 1.
For school \(B\) the least time taken by any of the children was 25 minutes and the longest time was 55 minutes. The three quartiles were 30,37 and 50 respectively.
(d) Draw a box plot to represent the data from school \(B\). \includegraphics[max width=\textwidth, alt={}, center]{c8bade79-a39a-4055-bfae-928f5338fdfc-03_798_1196_580_372}
(e) Compare and contrast these two box plots.

4. The following table summarises the times, \(t\) minutes to the nearest minute, recorded for a group of students to complete an exam. $$\text { [You may use } \sum \mathrm { f } t ^ { 2 } = 134281.25 \text { ] }$$

1. Estimate the mean and standard deviation of these data.
2. Use linear interpolation to estimate the value of the median.
3. Show that the estimated value of the lower quartile is 18.6 to 3 significant figures.
4. Estimate the interquartile range of this distribution.
5. Give a reason why the mean and standard deviation are not the most appropriate summary statistics to use with these data. The person timing the exam made an error and each student actually took 5 minutes less than the times recorded above. The table below summarises the actual times.

   Time (minutes) \(t\)	\(6 - 15\)	\(16 - 20\)	\(21 - 25\)	\(26 - 30\)	\(31 - 40\)	\(41 - 55\)
   Number of students f	62	88	16	13	11	10

6. Without further calculations, explain the effect this would have on each of the estimates found in parts (a), (b), (c) and (d).

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.

1. A random sample of 35 homeowners was taken from each of the villages Greenslax and Penville and their ages were recorded. The results are summarised in the back-to-back stem and leaf diagram below.

Key: 7 | 3 | 1 means 37 years for Greenslax and 31 years for Penville
Some of the quartiles for these two distributions are given in the table below.

1. Find the value of \(a\) and the value of \(b\). An outlier is a value that falls either $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 } \\ & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
2. On the graph paper opposite draw a box plot to represent the data from Penville. Show clearly any outliers.
3. State the skewness of each distribution. Justify your answers. \includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-03_930_1237_1800_367}

6. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below. A histogram was drawn to represent these data. The \(30 - 60\) group was represented by a bar of width 1.5 cm and height 1 cm .

1. Find the width and the height of the \(70 - 80\) group.
2. Use linear interpolation to estimate the median of this distribution. Given that \(x\) denotes the midpoint of each group in the table and $$\sum f x = 6460 \quad \sum f x ^ { 2 } = 529400$$
3. calculate an estimate for
   1. the mean,
   2. the standard deviation,
      for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
4. Evaluate this coefficient and comment on the skewness of these data.

2. A botany student counted the number of daisies in each of 42 randomly chosen areas of 1 m by 1 m in a large field. The results are summarised in the following stem and leaf diagram.

1. Write down the modal value of these data.
2. Find the median and the quartiles of these data.
3. On graph paper and showing your scale clearly, draw a box plot to represent these data.
4. Comment on the skewness of this distribution. The student moved to another field and collected similar data from that field.
5. Comment on how the student might summarise both sets of raw data before drawing box plots.
   (1 mark)

6. A travel agent sells holidays from his shop. The price, in \(\pounds\), of 15 holidays sold on a particular day are shown below. For these data, find

1. the mean and the standard deviation,
2. the median and the inter-quartile range. An outlier is an observation that falls either more than \(1.5 \times\) (inter-quartile range) above the upper quartile or more than \(1.5 \times\) (inter-quartile range) below the lower quartile.
3. Determine if any of the prices are outliers. The travel agent also sells holidays from a website on the Internet. On the same day, he recorded the price, \(\pounds x\), of each of 20 holidays sold on the website. The cheapest holiday sold was \(\pounds 98\), the most expensive was \(\pounds 2400\) and the quartiles of these data were \(\pounds 305 , \pounds 1379\) and \(\pounds 1805\). There were no outliers.
4. On graph paper, and using the same scale, draw box plots for the holidays sold in the shop and the holidays sold on the website.
5. Compare and contrast sales from the shop and sales from the website. \section*{END}

1 The number of matches in each of a sample of 85 boxes is summarised in the table.

1. For these data:
   1. state the modal value;
   2. determine values for the median and the interquartile range.
2. Given that, on investigation, the 2 extreme values in the above table are 227 and 271 :
   1. calculate the range;
   2. calculate estimates of the mean and the standard deviation.
3. For the numbers of matches in the 85 boxes, suggest, with a reason, the most appropriate measure of spread.

2 A small chapel was open to visitors for 55 days during the summer of 2015. The table summarises the daily numbers of visitors.

1. For these data:
   1. state the modal value;
   2. find values for the median and the interquartile range.
2. Name one measure of average and one measure of spread that cannot be calculated exactly from the data in the table.
   [0pt] [2 marks]
3. Reference to the raw data revealed that the 3 unknown exact values in the table were 13,37 and 58. Making use of this additional information, together with the data in the table, calculate the value of each of the two measures that you named in part (b).
   [0pt] [3 marks]

5. For a project, a student asked 40 people to draw two straight lines with what they thought was an angle of \(75 ^ { \circ }\) between them, using just a ruler and a pencil. She then measured the size of the angles that had been drawn and her data are summarised in this stem and leaf diagram.

1. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
2. determine if there are any outliers in these data,
3. draw a box plot representing these data on graph paper,
4. describe the skewness of the distribution and suggest a reason for it.

10 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.
4. Comment on the availability of public transport in Local Authority B as suggested by the table.

10 The table shows the age structure of usual residents of 18 Local Authorities (LAs) in the North West region of the UK in 2011. \section*{Percentage of residents}

1. Without reference to any other columns, explain how you would use only the columns for the age ranges 0 to 17 and 18 to 24 to decide whether an LA might be one of the following.
   1. An LA that includes a university
   2. An LA that attracts young couples to live
   3. An LA that attracts retired people to live
2. Using your answers to part (a), identify the following.
   1. Four LAs that might include a university
   2. Three LAs that might be attractive to retired people
3. Explain why your answer to part (b)(ii), based only on the columns for the age ranges 0 to 17 and 18 to 24, may not be reliable.
4. The lower quartile, median and upper quartile of the percentages in the column "Age 65 and over" are \(14.56 \% , 15.99 \%\) and \(16.76 \%\) respectively. Use this information to comment on your answers to part (b)(ii) and part (c). In a magazine article, a councillor plans to describe a typical LA in the North West region. He wants to quote the average percentage of residents aged 65 or over.
5. The mean of the percentages in the column "Age 65 and over" is \(16.90 \%\). Use this information, and the information given in part (d), to explain whether the median or the mean better represents the data in the column "Age 65 and over".

15
The number of hours of sunshine and the daily maximum temperature were recorded over a 9-day period in June at an English seaside town. A scatter diagram representing the recorded data is shown below. \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-20_872_1511_488_264} One of the points on the scatter diagram is an error. 15

1. 1. Write down the letter that identifies this point.
      15
      1. (ii) Suggest one possible action that could be taken to deal with this error.
         15
   2. It is claimed that the scatter diagram proves that longer hours of sunshine cause
      higher maximum daily temperatures. Comment on the validity of this claim.
      [0pt] [1 mark]

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

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?

The following stem and leaf diagram shows the aptitude scores \(x\) obtained by all the applicants for a particular job.

1. Write down the modal aptitude score. [1]
2. Find the three quartiles for these data. [3]

Outliers can be defined to be outside the limits \(Q_1 - 1.0(Q_3 - Q_1)\) and \(Q_3 + 1.0(Q_3 - Q_1)\).

1. On a graph paper, draw a box plot to represent these data. [7]

For these data, \(\Sigma x = 3363\) and \(\Sigma x^2 = 238305\).

1. Calculate, to 2 decimal places, the mean and the standard deviation for these data. [3]
2. Use two different methods to show that these data are negatively skewed. [4]