2.02g Calculate mean and standard deviation

1. Stav is studying the large data set for September 2015

He codes the variable Daily Mean Pressure, \(x\), using the formula \(y = x - 1010\) The data for all 30 days from Hurn are summarised by $$\sum y = 214 \quad \sum y ^ { 2 } = 5912$$

1. State the units of the variable \(x\)
2. Find the mean Daily Mean Pressure for these 30 days.
3. Find the standard deviation of Daily Mean Pressure for these 30 days. Stav knows that, in the UK, winds circulate
   The table gives the Daily Mean Pressure for 3 locations from the large data set on 26/09/2015

   Location	Heathrow	Hurn	Leuchars
   Daily Mean Pressure	1029	1028	1028
   Cardinal Wind Direction

   The Cardinal Wind Directions for these 3 locations on 26/09/2015 were, in random order, $$\begin{array} { l l l } W & N E & E \end{array}$$ You may assume that these 3 locations were under a single region of pressure.
4. Using your knowledge of the large data set, place each of these Cardinal Wind Directions in the correct location in the table.
   Give a reason for your answer. \section*{Question 3 continued.}

9 The table shows information about the number of days absent last year by students in class 2A at a certain school.

1. Calculate an estimate of the mean for these data.
2. Find the median of these data. The headteacher is writing a report on the numbers of absences at her school. She wishes to include a figure for the average number of absences in class 2A. A governor suggests that she should quote the mean. The class teacher suggests that she should quote the median, because it is lower than the mean.
3. Give another reason for using the median rather than the mean for the average number of absences in class 2A.

8 A random sample of 10 students from a college was chosen. They were asked how much time, \(x\) hours, they spent studying, and how much money, \(\pounds y\), they earned, in a typical week during term time. The results are shown in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{c4cc2cd8-46bf-448f-b223-92378984bfde-5_544_741_555_242}

1. Comment on the relationship shown by the diagram between hours spent studying and money earned, during term time, by these 10 students. The coordinates of the points in the diagram are \(( 18,23 ) , ( 20,21 ) , ( 23,20 ) , ( 25,19 ) , ( 25,21 )\), \(( 27,18 ) , ( 32,16 ) , ( 38,17 ) , ( 40,16 )\) and \(( 41,23 )\).
2. Find the mean and standard deviation of the number of hours spent per week studying during term time by these 10 students.

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 .

10 The pre-release material contains information about the birth rate per 1000 people in different countries of the world. These countries have been classified into different regions. The table shows some data for three of these regions: the mean and standard deviation (sd) of the birth rate per 1000, and the number of countries for which data was used, n. \section*{Birth rate per 1000 by region}

1. Use the information in the table to compare and contrast the birth rate per 1000 in Africa with the birth rate per 1000 in Europe.
2. The birth rate per 1000 in Mauritius, which is in Africa, is recorded as 9.86. Use the information in the table to show that this value is an outlier.
3. Use your knowledge of the pre-release material to explain whether the value for Mauritius should be discarded.
4. The pre-release material identifies 27 countries in Oceania. Suggest a reason why only 21 values were used to calculate the mean and standard deviation.

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.

7 A farmer has 200 apple trees. She is investigating the masses of the crops of apples from individual trees. She decides to select a sample of these trees and find the mass of the crop for each tree.

1. Explain how she can select a random sample of 10 different trees from the 200 trees. The masses of the crops from the 10 trees, measured in kg, are recorded as follows. \(\begin{array} { l l l l l l l l l l } 23.5 & 27.4 & 26.2 & 29.0 & 25.1 & 27.4 & 26.2 & 28.3 & 38.1 & 24.9 \end{array}\)
2. For these data find

4 A survey of the number of cars per household in a certain village generated the data in Fig. 4. \begin{table}[h] \end{table}

1. Calculate the mean number of cars per household.
2. Calculate the standard deviation of the number of cars per household.

3 Fig. 3 shows the time Lorraine spent in hours, \(t\), answering e-mails during the working day. The data were collected over a number of months. \begin{table}[h] \end{table}

1. Calculate an estimate of the mean time per day that Lorraine spent answering e-mails over this period.
2. Explain why your answer to part (a) is an estimate. When Lorraine accepted her job, she was told that the mean time per day spent answering e-mails would not be more than 3 hours.
3. Determine whether, according to the data in Fig. 3, it is possible that the mean time per day Lorraine spends answering e-mails is in fact more than 3 hours.

14 The pre-release material includes data concerning crude death rates in different countries of the world. Fig. 14.1 shows some information concerning crude death rates in countries in Europe and in Africa. \begin{table}[h] \end{table}

1. Use your knowledge of the large data set to suggest a reason why the statistics in Fig. 14.1 refer to only 48 of the 51 European countries.
2. Use the information in Fig. 14.1 to show that there are no outliers in either data set. The crude death rate in Libya is recorded as 3.58 and the population of Libya is recorded as 6411776.
3. Calculate an estimate of the number of deaths in Libya in a year. The median age in Germany is 46.5 and the crude death rate is 11.42. The median age in Cyprus is 36.1 and the crude death rate is 6.62 .
4. Explain why a country like Germany, with a higher median age than Cyprus, might also be expected to have a higher crude death rate than Cyprus. Fig. 14.2 shows a scatter diagram of median age against crude death rate for countries in Africa and Fig. 14.3 shows a scatter diagram of median age against crude death rate for countries in Europe. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-10_678_1221_1975_248} \captionsetup{labelformat=empty} \caption{Fig. 14.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{95eb3bcc-6d3c-4f7e-9b27-5e046ab57ec5-11_588_1248_223_228} \captionsetup{labelformat=empty} \caption{Fig. 14.3}
   \end{figure} The rank correlation coefficient for the data shown in Fig. 14.2 is - 0.281206 .
   The rank correlation coefficient for the data shown in Fig. 14.3 is 0.335215 .
5. Compare and contrast what may be inferred about the relationship between median age and crude death rate in countries in Africa and in countries in Europe.

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}

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.

15 Bottles of Fizzipop nominally contain 330 ml of drink. A consumer affairs researcher collects a random sample of 55 bottles of Fizzipop and records the volume of drink in each bottle. Summary statistics for the researcher's sample are shown in the table.

1. 1. Calculate the mean volume of drink in a bottle of Fizzipop.
   2. Show that the standard deviation of the volume of drink in a bottle of Fizzipop is 3.78 ml . The researcher uses software to produce a histogram with equal class intervals, which is shown below. \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-10_533_759_1181_251}
2. Explain why the researcher decides that the Normal distribution is a suitable model for the volume of drink in a bottle of Fizzipop.
3. Use your answers to parts (a) and (b) to determine the expected number of bottles which contain less than 330 ml in a random sample of 100 bottles. In order to comply with new regulations, no more than 1\% of bottles of Fizzipop should contain less than 330 ml . The manufacturer decides to meet the new regulations by adjusting the manufacturing process so that the mean volume of drink in a bottle of Fizzipop is increased. The standard deviation is unaltered.
4. Determine the minimum mean volume of drink in a bottle of Fizzipop which should ensure that the new regulations are met. Give your answer to \(\mathbf { 3 }\) significant figures. The mean volume of drink in a bottle of Fizzipop is set to 340 ml . After several weeks the quality control manager suspects the mean volume may have reduced. She collects a random sample of 100 bottles of Fizzipop. The mean volume of drink in a bottle in the sample is found to be 339.37 ml .
5. Assuming the standard deviation is unaltered, conduct a hypothesis test at the \(5 \%\) level to determine whether there is any evidence to suggest that the mean volume of drink in a bottle of Fizzipop is less than 340 ml .

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.

4. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club. The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below. $$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$

1. Write down the value of \(a\) and the value of \(b\).
2. Calculate an estimate of the mean of \(v\).
3. Calculate an estimate of the standard deviation of \(v\).
4. Use linear interpolation to estimate the median of \(v\).
5. Hence describe the skewness of the distribution. Give a reason for your answer.
6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club.

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}

5. The weights, in grams, of a random sample of 48 broad beans are summarised in the table. (You may assume \(\sum \mathrm { fy } { } ^ { 2 } = 101.56\) ) A histogram was drawn to represent these data. The \(2.1 < x \leqslant 2.7\) class was represented by a bar of width 1.5 cm and height 1 cm .

1. Find the width and height of the \(0.9 < x \leqslant 1.1\) class.
2. Give a reason to justify the use of a histogram to represent these data.
3. Estimate the mean and the standard deviation of the weights of these broad beans.
4. Use linear interpolation to estimate the median of the weights of these broad beans. One of these broad beans is selected at random.
5. Estimate the probability that its weight lies between 1.1 grams and 1.6 grams. One of these broad beans having a recorded weight of 0.95 grams was incorrectly weighed. The correct weight is 1.4 grams.
6. State, giving a reason, the effect this would have on your answers to part (c). Do not carry out any further calculations.

1. The heights, \(x\) metres, of 40 children were recorded by a teacher. The results are summarised as follows

$$\sum x = 58 \quad \sum x ^ { 2 } = 84.829$$

1. Find the mean and the variance of the heights of these 40 children. The teacher decided that these statistics would be more useful in centimetres.
2. Find
   1. the mean of these heights in centimetres,
   2. the standard deviation of these heights in centimetres. Two more children join the group. Their heights are 130 cm and 160 cm .
3. 1. State, giving a reason, the mean height of the 42 children.
   2. Without recalculating the standard deviation, state, giving a reason, whether the standard deviation of the heights of the 42 children will be greater than, less than or the same as the standard deviation of the heights of the group of 40 children.
      

2. Chi wanted to summarise the scores of the 39 competitors in a village quiz. He started to produce the following stem and leaf diagram Key: 2|5 is a score of 25 \begin{table}[h] \end{table} He did not complete the stem and leaf diagram but instead produced the following box plot. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-04_357_1237_772_356} Chi defined an outlier as a value that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$ or
less than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\)

1. Find
   1. the interquartile range
   2. the range.
2. Describe, giving a reason, the skewness of the distribution of scores. Albert and Beth asked for their scores to be checked.
   Albert's score was changed from 25 to 37
   Beth's score was changed from 54 to 60
3. On the grid on page 5, draw an updated box plot. Show clearly any calculations that you used. Some of the competitors complained that the questions were biased towards the younger generation. The product moment correlation coefficient between the age of the competitors and their score in the quiz is - 0.187
4. State, giving a reason, whether or not the complaint is supported by this statistic. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-05_360_1242_2238_351} Turn over for a spare grid if you need to redraw your box plot. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-07_367_1246_2261_351}

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)