S1: Representation and summary of dataEdexcel International A Level Maths: Topic test

20 questions, 54 marks

Edexcel International A Level Maths

S1: Representation and summary of data topic test

Total 54 marks

Name

Class

Date

1
The masses, in kg, of seven parcels sent by a courier are 2.4, 3.1, 3.1, 3.8, 4.5, 5.2 and 9.0.
(a)
Find the median mass of the parcels.
[1 mark]
A $3.8$ kg
B $3.1$ kg
C $4.4$ kg
D $4.5$ kg
(b)
A value is defined to be an outlier if it is more than $1.5\times$ the interquartile range above the upper quartile. Which of the masses is an outlier?
[1 mark]
A $2.4$ kg
B $5.2$ kg
C $9.0$ kg
D $4.5$ kg
(c)
Calculate the mean mass and use it, with the median, to describe the skewness of the data.
[2 marks]
Total for question 1: 4 marks
2
The ages, $x$ years, of 10 members of a choir are coded using $y=\dfrac{x-30}{5}$ . It is given that $\sum y=14$ and $\sum y^2=70$ .
(a)
Find the mean age of the 10 members.
[1 mark]
A $7$ years
B $37$ years
C $1.4$ years
D $35$ years
(b)
One year later every member is exactly one year older. What happens to the mean and the standard deviation of the ages?
[1 mark]
ABoth increase by 1 year.
BThe mean is unchanged and the standard deviation increases by 1 year.
CThe mean increases by 1 year and the standard deviation increases by 5 years.
DThe mean increases by 1 year and the standard deviation is unchanged.
(c)
Find the variance of the ages $x$ .
[2 marks]
Total for question 2: 4 marks
3
The times taken by 50 runners to complete a 5 km race are summarised in a grouped frequency table: 15 to under 20 minutes, 6 runners; 20 to under 25 minutes, 14 runners; 25 to under 30 minutes, 18 runners; 30 to under 40 minutes, 8 runners; 40 to under 60 minutes, 4 runners.
(a)
Estimate the mean time taken.
[3 marks]
(b)
Use linear interpolation to estimate the interquartile range of the times.
[4 marks]
Total for question 3: 7 marks
4
The lengths of stay, in days, of 11 patients on a surgical ward were 2, 3, 3, 4, 5, 5, 6, 8, 9, 12 and 25. The lengths of stay of 120 patients on a medical ward are summarised in a grouped table: 0 to under 5 days, 18 patients; 5 to under 10 days, 30 patients; 10 to under 15 days, 36 patients; 15 to under 25 days, 24 patients; 25 to under 45 days, 12 patients.
(a)
For the surgical ward:
(i) find the mean length of stay; [1]
(ii) find the standard deviation, given that $\sum x^2=1038$ ; [2]
(iii) given that a value is an outlier if it is more than $1.5\times$ the interquartile range above the upper quartile, show that 25 is an outlier, and state, with a reason, the skewness of the data. [3]
[6 marks]
(b)
For the medical ward, a histogram is drawn with frequency density on the vertical axis.
(i) Find the frequency density of the class 15 to under 25 days. [1]
(ii) The bar for the class 10 to under 15 days is 9 cm tall. Find the height of the bar for the class 25 to under 45 days. [2]
(iii) Estimate the mean length of stay on the medical ward. [3]
[6 marks]
Total for question 4: 12 marks
5
The distances travelled to work by employees at two companies, A and B, are summarised as follows. Company A: shortest 12 km, lower quartile 20 km, median 26 km, upper quartile 31 km, longest 45 km. Company B: shortest 8 km, lower quartile 22 km, median 27 km, upper quartile 40 km, longest 60 km.
(a)
Which company has the greater interquartile range of distances?
[1 mark]
ACompany B, with an interquartile range of 18 km.
BCompany A, with an interquartile range of 11 km.
CBoth companies have the same interquartile range.
DIt cannot be found from the summary.
(b)
Which of the following describes the skewness of the distances for Company B?
[1 mark]
ANegative skew.
BNo skew (symmetrical).
CPositive skew.
DIt cannot be decided from the summary.
(c)
A value is an outlier if it is greater than $Q_3+1.5\times(Q_3-Q_1)$ . Show that the longest distance for Company B is not an outlier.
[2 marks]
Total for question 5: 4 marks
6
In a quiz, 60 players each answered five questions. The numbers of players who answered 0, 1, 2, 3, 4 and 5 questions correctly were 3, 7, 12, 20, 13 and 5 respectively.
(a)
Find the mean number of correct answers.
[1 mark]
A $3$
B $2.8$
C $2.5$
D $168$
(b)
Find the standard deviation of the number of correct answers.
[1 mark]
A $1.63$
B $9.47$
C $3.08$
D $1.28$
(c)
Find the interquartile range of the number of correct answers.
[2 marks]
Total for question 6: 4 marks
7
The masses, $x$ grams, of 30 chocolate bars from a production line are summarised by $\sum x=1506$ and $\sum x^2=75\,735$ .
(a)
Find the mean and the standard deviation of the masses.
[3 marks]
(b)
A bar of mass 56 g is judged to be an outlier and is removed. Find the mean and the variance of the masses of the remaining 29 bars.
[4 marks]
Total for question 7: 7 marks
8
A company has two shops, P and Q. Over 30 days the daily takings, in pounds, are summarised as follows. Shop P: lower quartile 380, median 420, upper quartile 460, mean 424, standard deviation 58. Shop Q: lower quartile 330, median 370, upper quartile 470, mean 412, standard deviation 96, highest takings 700. The company also timed 40 customers queuing at shop P. The times $t$ minutes are: $0\le t<5$ , 6 customers; $5\le t<10$ , $a$ customers; $10\le t<20$ , 14 customers; $20\le t<40$ , $b$ customers. The estimated mean queuing time is 12.75 minutes.
(a)
(i) Compare the typical daily takings of the two shops. [2]
(ii) Compare the variability of the daily takings of the two shops. [2]
(iii) A value is an outlier if it is more than $1.5\times$ the interquartile range above the upper quartile. Show that the highest takings of shop Q is an outlier. [2]
[6 marks]
(b)
(i) Write down an equation connecting $a$ and $b$ . [1]
(ii) Use the mean to form a second equation, and find $a$ and $b$ . [3]
(iii) Use linear interpolation to estimate the median queuing time. [2]
[6 marks]
Total for question 8: 12 marks

End of questions

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).