Measures of Location and Spread

3 The test marks of 14 students are displayed in a stem-and-leaf diagram, as shown below. \includegraphics[max width=\textwidth, alt={}, center]{e23cb28b-49e5-436a-942d-e6320029c634-2_234_261_1425_482} Key: 1 | 6 means 16 marks

1. Find the lower quartile.
2. Given that the median is 32 , find the values of \(w\) and \(x\).
3. Find the possible values of the upper quartile.
4. State one advantage of a stem-and-leaf diagram over a box-and-whisker plot.
5. State one advantage of a box-and-whisker plot over a stem-and-leaf diagram.

4 The weights in kilograms of packets of cereal were noted correct to 4 significant figures. The following stem-and-leaf diagram shows the data. Key: 748 | 5 represents 0.7485 kg .

1. On the grid, draw a box-and-whisker plot to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{556a1cc2-47ef-4ef7-a8f6-42850c303531-05_814_1604_1336_299}
2. Name a distribution that might be a suitable model for the weights of this type of cereal packet. Justify your answer.

1. The stem and leaf diagram shows the number of deliveries made by Pat each day for 24 days

\begin{table}[h] \end{table} where \(a\), \(b\) and \(c\) are positive integers with \(a < b < c\) An outlier is defined as any value greater than \(1.5 \times\) interquartile range above the upper quartile. Given that there is only one outlier for these data,

1. show that \(c = 9\) The number of deliveries made by Pat each day is represented by \(d\) The data in the stem and leaf diagram are coded using $$x = d - 125$$ and the following summary statistics are obtained $$\sum x = - 96 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 1306$$
2. Find the mean number of deliveries.
3. Find the standard deviation of the number of deliveries. One of these 24 days is selected at random. The random variable \(D\) represents the number of deliveries made by Pat on this day. The random variable \(X = D - 125\)
4. Find \(\mathrm { P } ( D > 118 \mid X < 0 )\)

1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

1 A computer can generate random numbers which are either 0 or 2 . On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers.

8 In the 2001 census, the household size (the number of people living in each household) was recorded. The percentages of households of different sizes were then calculated. The table shows the percentages for two wards, Withington and Old Moat, in Manchester.

1. Calculate the median and interquartile range of the household size for Withington.
2. Making an appropriate assumption for the last class, which should be stated, calculate the mean and standard deviation of the household size for Withington. Give your answers to an appropriate degree of accuracy. The corresponding results for Old Moat are as follows.

   Median	Interquartile range	Mean	Standard deviation
   2	2	2.4	1.5

3. State one advantage of using the median rather than the mean as a measure of the average household size.
4. By comparing the values for Withington with those for Old Moat, explain briefly why the interquartile range may be less suitable than the standard deviation as a measure of the variation in household size.
5. For one of the above wards, the value of Spearman's rank correlation coefficient between household size and percentage is - 1 . Without any calculation, state which ward this is. Explain your answer.

The results of 14 observations of a random variable \(V\) are summarised by $$n = 14, \quad \sum v = 3752, \quad \sum v^2 = 1007448.$$ Calculate unbiased estimates of E\((V)\) and Var\((V)\). [4]

2 A researcher is investigating the lengths, in kilometres, of the journeys to work of the employees at a certain firm. She takes a random sample of 10 employees.

1. State what is meant by 'random' in this context. The results of her sample are as follows. $$\begin{array} { l l l l l l l l l l } 1.5 & 2.0 & 3.6 & 5.9 & 4.8 & 8.7 & 3.5 & 2.9 & 4.1 & 3.0 \end{array}$$
2. Find unbiased estimates of the population mean and variance.
3. State what is meant by 'population' in this context.

7. A random sample of 8 apples is taken from an orchard and the weight, in grams, of each apple is measured. The results are given below. $$\begin{array} { l l l l l l l l } 143 & 131 & 165 & 122 & 137 & 155 & 148 & 151 \end{array}$$

1. Calculate unbiased estimates for the mean and the variance of the weights of apples. A population has an unknown mean \(\mu\) and an unknown variance \(\sigma ^ { 2 }\) A random sample represented by \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 8 }\) is taken from this population.
2. Explain why \(\sum _ { i = 1 } ^ { 8 } \left( X _ { i } - \mu \right) ^ { 2 }\) is not a statistic. Given that \(\mathrm { E } \left( S ^ { 2 } \right) = \sigma ^ { 2 }\), where \(S ^ { 2 }\) is an unbiased estimator of \(\sigma ^ { 2 }\) and the statistic $$Y = \frac { 1 } { 8 } \left( \sum _ { i = 1 } ^ { 8 } X _ { i } ^ { 2 } - 8 \bar { X } ^ { 2 } \right)$$
3. find \(\mathrm { E } ( Y )\) in terms of \(\sigma ^ { 2 }\)
4. Hence find the bias, in terms of \(\sigma ^ { 2 }\), when \(Y\) is used as an estimator of \(\sigma ^ { 2 }\)

1 For \(n\) values of the variable \(x\), it is given that \(\Sigma ( x - 100 ) = 216\) and \(\Sigma x = 2416\). Find the value of \(n\).

1 For \(n\) values of the variable \(x\), it is given that $$\Sigma ( x - 50 ) = 144 \quad \text { and } \quad \Sigma x = 944 .$$ Find the value of \(n\).

2 A summary of 40 values of \(x\) gives the following information: $$\Sigma ( x - k ) = 520 , \quad \Sigma ( x - k ) ^ { 2 } = 9640$$ where \(k\) is a constant.

1. Given that the mean of these 40 values of \(x\) is 34 , find the value of \(k\).
2. Find the variance of these 40 values of \(x\).

3 The mean and standard deviation of 20 values of \(x\) are 60 and 4 respectively.

1. Find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\).
   Another 10 values of \(x\) are such that their sum is 550 and the sum of their squares is 40500 .
2. Find the mean and standard deviation of all these 30 values of \(x\).

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.

4 Farfield Travel and Lacket Travel are two travel companies which arrange tours abroad. The numbers of holidays arranged in a certain week are recorded in the table below, together with the means and standard deviations of the prices.

1. Calculate the mean price of all 51 holidays.
2. The prices of individual holidays with Farfield Travel are denoted by \(\\) x _ { F }\( and the prices of individual holidays with Lacket Travel are denoted by \)\\( x _ { L }\). By first finding \(\Sigma x _ { F } ^ { 2 }\) and \(\Sigma x _ { L } ^ { 2 }\), find the standard deviation of the prices of all 51 holidays.

The masses, in kilograms, of 100 chickens on sale in a large supermarket were recorded as follows.

Mass (\(x\) kg)	\(1.6 \leq x < 1.8\)	\(1.8 \leq x < 2.0\)	\(2.0 \leq x < 2.2\)	\(2.2 \leq x < 2.4\)	\(2.4 \leq x < 2.6\)
Number of chickens	16	27	28	18	11

Calculate estimates of the mean and standard deviation of the masses of these chickens. [5]

The masses, in kilograms, of 100 chickens on sale in a large supermarket were recorded as follows.

Mass (\(x\) kg)	\(1.6 \leq x < 1.8\)	\(1.8 \leq x < 2.0\)	\(2.0 \leq x < 2.2\)	\(2.2 \leq x < 2.4\)	\(2.4 \leq x < 2.6\)
Number of chickens	16	27	28	18	11

Calculate estimates of the mean and standard deviation of the masses of these chickens. [5]

7 Vernon, a service engineer, is expected to carry out a boiler service in one hour.
One hour is subtracted from each of his actual times, and the resulting differences, \(x\) minutes, for a random sample of 100 boiler services are summarised in the table.

1. 1. Calculate estimates of the mean and the standard deviation of these differences.
      (4 marks)
   2. Hence deduce, in minutes, estimates of the mean and the standard deviation of Vernon's actual service times for this sample.
2. 1. Construct an approximate \(98 \%\) confidence interval for the mean time taken by Vernon to carry out a boiler service.
   2. Give a reason why this confidence interval is approximate rather than exact.
3. Vernon claims that, more often than not, a boiler service takes more than an hour and that, on average, a boiler service takes much longer than an hour. Comment, with a justification, on each of these claims.

Given that \(\sum x = 364\), \(\sum x^2 = 19412\), \(n = 10\), find \(\sigma\), the standard deviation of \(X\). Circle your answer. 24.8 \quad 44.1 \quad 616.2 \quad 1941.2 [1 mark]

Given that \(\sum x = 364\), \(\sum x^2 = 19412\), \(n = 10\), find \(\sigma\), the standard deviation of \(X\). Circle your answer. 24.8 \quad 44.1 \quad 616.2 \quad 1941.2 [1 mark]

5. Each member of a group of 27 people was timed when completing a puzzle.
The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-08_381_1557_504_264}

1. Find the range of the times.
2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
3. calculate the mean time taken to complete the puzzle,
4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
   When their times are included with the data of the other 27 people

1 Find the mean and variance of the following data. $$\begin{array} { l l l l l l l l l l } 5 & - 2 & 12 & 7 & - 3 & 2 & - 6 & 4 & 0 & 8 \end{array}$$

1 Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded below. $$\begin{array} { l l l l l l l l l l } 32 & 35 & 45 & 37 & 38 & 44 & 33 & 39 & 36 & 45 \end{array}$$ Find the mean and standard deviation of the lengths of these leaves.

4 The six faces of a fair die are numbered \(1,1,1,2,3,3\). The score for a throw of the die, denoted by the random variable \(W\), is the number on the top face after the die has landed.

1. Find the mean and standard deviation of \(W\).
2. The die is thrown twice and the random variable \(X\) is the sum of the two scores. Draw up a probability distribution table for \(X\).
3. The die is thrown \(n\) times. The random variable \(Y\) is the number of times that the score is 3 . Given that \(\mathrm { E } ( Y ) = 8\), find \(\operatorname { Var } ( Y )\).

4 The weights in kilograms of 11 bags of sugar and 7 bags of flour are as follows.
Sugar: 1.9611 .98312 .00812 .0141 .9681 .9941 .2 .0112 .0171 .9771 .9841 .989
Flour: \(\begin{array} { l l l l l l l } 1.945 & 1.962 & 1.949 & 1.977 & 1.964 & 1.941 & 1.953 \end{array}\)

1. Represent this information on a back-to-back stem-and-leaf diagram with sugar on the left-hand side.
2. Find the median and interquartile range of the weights of the bags of sugar.

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

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

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

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

4 The back-to-back stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\). Key: \(4 | 16 | 7\) means \(A = 0.164\) and \(B = 0.167\).

1. Find the median and the interquartile range for variable \(A\).
2. You are given that, for variable \(B\), the median is 0.171 , the upper quartile is 0.179 and the lower quartile is 0.164 . Draw box-and-whisker plots for \(A\) and \(B\) in a single diagram on graph paper.

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

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

6 The lengths of some insects of the same type from two countries, \(X\) and \(Y\), were measured. The stem-and-leaf diagram shows the results. Key: 5 | 81 | 3 means an insect from country \(X\) has length 0.815 cm and an insect from country \(Y\) has length 0.813 cm .

1. Find the median and interquartile range of the lengths of the insects from country \(X\).
2. The interquartile range of the lengths of the insects from country \(Y\) is 0.028 cm . Find the values of \(q\) and \(r\).
3. Represent the data by means of a pair of box-and-whisker plots in a single diagram on graph paper.
4. Compare the lengths of the insects from the two countries.

Explain briefly what you understand by

1. a sampling frame, [1]
2. a statistic. [2]

A large dental practice wishes to investigate the level of satisfaction of its patients.

1. Suggest a suitable sampling frame for the investigation. [1]
2. Identify the sampling units. [1]
3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]

A random sample \(X_1, X_2, \ldots, X_n\) is taken from a finite population. A statistic \(Y\) is based on this sample.

1. Explain what you understand by the statistic \(Y\). [2]
2. Give an example of a statistic. [1]
3. Explain what you understand by the sampling distribution of \(Y\). [2]

4 The marks, \(x\), of a random sample of 50 students in a test were summarised as follows. $$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$

1. Calculate unbiased estimates of the population mean and variance.
2. Each student's mark is scaled using the formula \(y = 1.5 x + 10\). Find estimates of the population mean and variance of the scaled marks, \(y\).

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.
      

4 The numbers of rides taken by two students, Fei and Graeme, at a fairground are shown in the following table.

1. The mean cost of Fei's rides is \(\\) 2.50\( and the standard deviation of the costs of Fei's rides is \)\\( 0\). Explain how you can tell that the roller coaster and the water slide each cost \(\\) 2.50\( per ride. [2]
2. The mean cost of Graeme's rides is \)\\( 3.76\). Find the standard deviation of the costs of Graeme's rides.

1 The stem and leaf diagram illustrates the weights in grams of 20 house sparrows. Key: \(\quad 27 \quad \mid \quad 7 \quad\) represents 27.7 grams

1. Find the median and interquartile range of the data.
2. Determine whether there are any outliers.

The table shows fuel economy figures in miles per gallon (mpg) for some new cars.

Car	A	B	C	D	E	F	G	H	I	J	K	L	M	N	O
Mpg	57	40	34	33	11	17	30	27	31	20	35	24	26	23	32

1. Find the median and quartiles for the mpg of these 15 cars. [2]
2. Use the values in part (a) to identify any cars for which the mpg is an outlier. [3]

2 A sprinter runs many 100 -metre trials, and the time, \(x\) seconds, for each is recorded. A sample of eight of these times is taken, as follows. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$

1. Calculate the sample mean, \(\bar { x }\), and sample standard deviation, \(s\), of these times.
2. Show that the time of 10.04 seconds may be regarded as an outlier.
3. Discuss briefly whether or not the time of 10.04 seconds should be discarded.

5 The pulse rates, in beats per minute, of a random sample of 15 small animals are shown in the following table.

1. Draw a stem-and-leaf diagram to represent the data.
2. Find the median and the quartiles.
3. On graph paper, using a scale of 2 cm to represent 10 beats per minute, draw a box-and-whisker plot of the data.

2 The numbers of people travelling on a certain bus at different times of the day are as follows.

1. Draw a stem-and-leaf diagram to illustrate the information given above.
2. Find the median, the lower quartile, the upper quartile and the interquartile range.
3. State, in this case, which of the median and mode is preferable as a measure of central tendency, and why.

4 Prices in dollars of 11 caravans in a showroom are as follows. \(\begin{array} { l l l l l l l l l l l } 16800 & 18500 & 17700 & 14300 & 15500 & 15300 & 16100 & 16800 & 17300 & 15400 & 16400 \end{array}\)

1. Represent these prices by a stem-and-leaf diagram.
2. Write down the lower quartile of the prices of the caravans in the showroom.
3. 3 different caravans in the showroom are chosen at random and their prices are noted. Find the probability that 2 of these prices are more than the median and 1 is less than the lower quartile.

2 The values, \(x\), in a particular set of data are summarised by $$\Sigma ( x - 25 ) = 133 , \quad \Sigma ( x - 25 ) ^ { 2 } = 3762 .$$ The mean, \(\bar { x }\), is 28.325 .

1. Find the standard deviation of \(x\).
2. Find \(\Sigma x ^ { 2 }\).

1 The length of time, \(t\) minutes, taken to do the crossword in a certain newspaper was observed on 12 occasions. The results are summarised below. $$\Sigma ( t - 35 ) = - 15 \quad \Sigma ( t - 35 ) ^ { 2 } = 82.23$$ Calculate the mean and standard deviation of these times taken to do the crossword.

3. Data relating to the lifetimes (to the nearest hour) of a random sample of 200 light bulbs from the production line of a manufacturer were summarised in a group frequency table. The mid-point of each group in the table was represented by \(x\) and the corresponding frequency for that group by \(f\). The data were then coded using \(y = \frac { ( x - 755.0 ) } { 2.5 }\) and summarised as follows: $$\Sigma f y = - 467 , \Sigma f y ^ { 2 } = 9179 .$$

1. Calculate estimates of the mean and the standard deviation of the lifetimes of this sample of bulbs.
   (9 marks)
   An estimate of the interquartile range for these data was 27.7 hours.
2. Explain, giving a reason, whether you would recommend the manufacturer to use the interquartile range or the standard deviation to represent the spread of lifetimes of the bulbs from this production line.
   (2 marks)

2 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.

1. Write down the median and mode of these data.
2. Draw a vertical line diagram for these data.
3. State the type of skewness of the distribution.

2 The numbers of absentees per day from Mrs Smith's reception class over a period of 50 days are summarised below.

1. Illustrate these data by means of a vertical line chart.
2. Calculate the mean and root mean square deviation of these data.
3. There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.

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.

There are 10 numbers in a list. The first 9 numbers have mean 6 and variance 2. The 10th number is 3. Find the mean and variance of all 10 numbers. [6]

1. Keith records the amount of rainfall, in mm , at his school, each day for a week. The results are given below.
   0.0
   0.5
   1.8
   2.8
   2.3
   5.6
   9.4

Jenny then records the amount of rainfall, \(x \mathrm {~mm}\), at the school each day for the following 21 days. The results for the 21 days are summarised below. $$\sum x = 84.6$$

1. Calculate the mean amount of rainfall during the whole 28 days. Keith realises that he has transposed two of his figures. The number 9.4 should have been 4.9 and the number 0.5 should have been 5.0 Keith corrects these figures.
2. State, giving your reason, the effect this will have on the mean.

3 A sample of 36 data values, \(x\), gave \(\Sigma ( x - 45 ) = - 148\) and \(\Sigma ( x - 45 ) ^ { 2 } = 3089\).

1. Find the mean and standard deviation of the 36 values.
2. One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.

7 A random sample of 97 people who own mobile phones was used to collect data on the amount of time they spent per day on their phones. The results are displayed in the table below.

1. Calculate estimates of the mean and standard deviation of the time spent per day on these mobile phones.
2. On graph paper, draw a fully labelled histogram to represent the data.

1 Each of a group of 10 boys estimates the length of a piece of string. The estimates, in centimetres, are as follows. $$\begin{array} { l l l l l l l l l l } 37 & 40 & 45 & 38 & 36 & 38 & 42 & 38 & 40 & 39 \end{array}$$

1. Find the mode.
2. Find the median and the interquartile range.

1 Twelve values of \(x\) are shown below. Find the mean and standard deviation of \(( x - 1760 )\). Hence find the mean and standard deviation of \(x\). [4]

Which one of the following is not a measure of spread? Circle your answer. [1 mark] median \(\qquad\) range \(\qquad\) standard deviation \(\qquad\) variance

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

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

1 The total annual emissions of carbon dioxide, \(x\) tonnes per person, for 13 European countries are given below. $$\begin{array} { c c c c c c c c c c c c c } 6.2 & 6.7 & 6.8 & 8.1 & 8.1 & 8.5 & 8.6 & 9.0 & 9.9 & 10.1 & 11.0 & 11.8 & 22.8 \end{array}$$

1. Find the mean, median and midrange of these data.
2. Comment on how useful each of these is as a measure of central tendency for these data, giving a brief reason for each of your answers.

3 Jagdeesh measured the lengths, \(x\) minutes, of 60 randomly chosen lectures. His results are summarised below.

1. Calculate unbiased estimates of the population mean and variance.
2. Calculate a \(98 \%\) confidence interval for the population mean.

2 The amounts of money, \(x\) dollars, that 24 people had in their pockets are summarised by \(\Sigma ( x - 36 ) = - 60\) and \(\Sigma ( x - 36 ) ^ { 2 } = 227.76\). Find \(\Sigma x\) and \(\Sigma x ^ { 2 }\).

3 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 The masses in kilograms of 50 children having a medical check-up were recorded correct to the nearest kilogram. The results are shown in the table.

1. Find which class interval contains the lower quartile.
2. On the grid, draw a histogram to illustrate the data in the table. \includegraphics[max width=\textwidth, alt={}, center]{dd75fa20-fead-48d6-aff4-c5e733769f9f-02_1397_1397_1187_415}

1 Levels of nitrogen dioxide in the atmosphere are being monitored at the side of a road in a busy city centre. A sample of 18 measurements taken (in suitable units) is as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l } 83 & 44 & 95 & 92 & 98 & 63 & 69 & 76 & 19 & 91 & 70 & 91 & 74 & 65 & 62 & 70 & 95 & 108 \end{array}$$

1. Find the mean and standard deviation of the sample.
2. Hence identify, with justification, any possible outliers.

1 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$

1. Construct a sorted stem and leaf diagram to represent these data.
2. State the type of skewness suggested by your stem and leaf diagram.
3. For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.

A random sample of ten CO₂ emissions was selected from the Large Data Set. The emissions in grams per kilogram were: 13 \quad 45 \quad 45 \quad 0 \quad 49 \quad 77 \quad 49 \quad 49 \quad 49 \quad 78

1. Find the standard deviation of the sample. [1 mark]
2. An environmentalist calculated the average CO₂ emissions for cars in the Large Data Set registered in 2002 and in 2016. The averages are listed below.

   Year of registration	2002	2016
   Average CO₂ emission	171.2	120.4

   The environmentalist claims that the average CO₂ emissions for 2002 and 2016 combined is 145.8 Determine whether this claim is correct. Fully justify your answer. [2 marks]

The survey described in the article was based on a small sample. State one conclusion which is unlikely to be influenced by the size of the sample. [1]

Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.

1. State an advantage of using a stratified sample rather than a simple random sample. [1]
2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK. [1]

4 The heights, \(x \mathrm {~cm}\), of a group of 28 people were measured. The mean height was found to be 172.6 cm and the standard deviation was found to be 4.58 cm . A person whose height was 161.8 cm left the group.

1. Find the mean height of the remaining group of 27 people.
2. Find \(\Sigma x ^ { 2 }\) for the original group of 28 people. Hence find the standard deviation of the heights of the remaining group of 27 people.

1 Twelve tourists were asked to estimate the height, in metres, of a new building. Their estimates were as follows. $$\begin{array} { l l l l l l l l l l l l } 50 & 45 & 62 & 30 & 40 & 55 & 110 & 38 & 52 & 60 & 55 & 40 \end{array}$$

1. Find the median and the interquartile range for the data.
2. Give a disadvantage of using the mean as a measure of the central tendency in this case.

Test scores, \(X\), have mean 54 and variance 144. The scores are scaled using the formula \(Y = a + bX\), where \(a\) and \(b\) are constants and \(b > 0\). The scaled scores, \(Y\), have mean 50 and variance 100. Find the values of \(a\) and \(b\). [4]

6 A sample of 5 randomly selected values of a variable \(X\) is as follows: $$\begin{array} { l l l l l } 1 & 2 & 6 & 1 & a \end{array}$$ where \(a > 0\).
Given that an unbiased estimate of the variance of \(X\) calculated from this sample is \(\frac { 11 } { 2 }\), find the value of \(a\).

1 Two cricket teams kept records of the number of runs scored by their teams in 8 matches. The scores are shown in the following table.

1. Find the mean and standard deviation of the scores for team \(A\). The mean and standard deviation for team \(B\) are 130.75 and 29.63 respectively.
2. State with a reason which team has the more consistent scores.

8 A club secretary wishes to survey a sample of members of his club. He uses all members present at a particular meeting as his sample.

1. Explain why this sample is likely to be biased. Later the secretary decides to choose a random sample of members.
   The club has 253 members and the secretary numbers the members from 1 to 253 . He then generates random 3-digit numbers on his calculator. The first six random numbers generated are 156, 965, 248, 156, 073 and 181. The secretary uses each number, where possible, as the number of a member in the sample.
2. Find possible numbers for the first four members in the sample.

2 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]{088972e9-bfcd-429c-9145-af274a4c0a58-2_163_857_436_642}

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.

4 The Mathematics and English A-level marks of 1400 pupils all taking the same examinations are shown in the cumulative frequency graphs below. Both examinations are marked out of 100 . \includegraphics[max width=\textwidth, alt={}, center]{be6c6525-a20c-42d0-8fef-1cd254baaa76-06_1682_1246_404_445} Use suitable data from these graphs to compare the central tendency and spread of the marks in Mathematics and English.

Alison selects 10 of her male friends. For each one she measures the distance between his eyes. The distances, measured in mm, are as follows: 51 57 58 59 61 64 64 65 67 68 The mean of these data is 61.4. The sample standard deviation is 5.232, correct to 3 decimal places. One of the friends decides he does not want his measurement to be used. Alison replaces his measurement with the measurement from another male friend. This increases the mean to 62.0 and reduces the standard deviation. Give a possible value for the measurement which has been removed and find the measurement which has replaced it. [3]

A student collected data on the number of text messages, \(t\), sent by 30 students in her year group in the previous week. Her results are summarised as follows: \(\Sigma t = 1039\), \(\Sigma t^2 = 65393\).

1. Calculate unbiased estimates of the mean and variance of the number of text messages sent by these students per week. [4]

Another student collected similar data for 20 different students and calculated unbiased estimates of the mean and variance of 32.0 and 963.4 respectively.

1. Calculate unbiased estimates of the mean and variance for the combined sample of 50 students. [6]

2. An estate agent recorded the price per square metre, \(p \pounds / \mathrm { m } ^ { 2 }\), for 7 two-bedroom houses. He then coded the data using the coding \(q = \frac { p - a } { b }\), where \(a\) and \(b\) are positive constants. His results are shown in the table below.

1. Find the value of \(a\) and the value of \(b\) The estate agent also recorded the distance, \(d \mathrm {~km}\), of each house from the nearest train station. The results are summarised below. $$\mathrm { S } _ { d d } = 1.02 \quad \mathrm {~S} _ { q q } = 8.22 \quad \mathrm {~S} _ { d q } = - 2.17$$
2. Calculate the product moment correlation coefficient between \(d\) and \(q\)
3. Write down the value of the product moment correlation coefficient between \(d\) and \(p\) The estate agent records the price and size of 2 additional two-bedroom houses, \(H\) and \(J\).

   House	Price \(( \pounds )\)	Size \(\left( \mathrm { m } ^ { 2 } \right)\)
   \(H\)	156400	85
   \(J\)	172900	95

4. Suggest which house is most likely to be closer to a train station. Justify your answer.

1 A summary of 20 values of \(x\) gives $$\Sigma ( x - 30 ) = 439 , \quad \Sigma ( x - 30 ) ^ { 2 } = 12405 .$$ A summary of another 25 values of \(x\) gives $$\sum ( x - 30 ) = 470 , \quad \sum ( x - 30 ) ^ { 2 } = 11346 .$$

1. Find the mean of all 45 values of \(x\).
2. Find the standard deviation of all 45 values of \(x\).