20 The histogram gives some information about the weights, in grams, of some books.
75 books weigh less than 100 grams.
A book is chosen at random.
Find an estimate for the probability that this book weighs between 400 grams and 600 grams.
(4)
Mark scheme
Scheme
Marks
(frequency density =) 75 ÷ 100 (= 0.75) or any one correct value marked on the FD axis or uses their own linear FD scale to find the area of at least one of the third, fourth or fifth bars
OR eg 1 (large) box / 1 (large) square = 12.5 books oe or 5 (sml) squares = 2.5 books or 1 (sml) square / 1 box = 0.5 books
M1
eg using frequency densities 75 + (“2.25” × 100) + (“1” × 250) + (“2.5” × 50) + (“0.25” × 200) (= 75 + 225 + 250 + 125 + 50 = 725) OR eg using a measure of scale (eg large squares) (6 + 18 + 20 + 10 + 4) × “12.5” (=725) oe OR eg total no. of large squares = 6 + 18 + 20 + 10 + 4 (=58) or total no. of sml squares = 150 + 450 + 500 + 250 + 100 (=1450)
M1
eg using frequency densities (“1” × 50) + (“2.5” × 50) + (“0.25” × 100) (= 50 + 125 + 25 = 200) OR eg using a measure of scale (eg large squares) (4 + 10 + 2) × “12.5” (=200) oe OR eg no. of large squares = 4 + 10 + 2 (= 16) or no. of sml squares = 100 + 250 + 50 (= 400)
M1
Correct answer only scores full marks (unless from obviously incorrect working)
Answer: \(\dfrac{200}{725}\)
A1
(4)
(4 marks)
Notes
M1: for a correct calculation of frequency density or a correct value on the frequency density axis or use of their own linear frequency density scale to find the area of at least one of the third, fourth or fifth bars OR or a correct measure of scale
Implied by a correct frequency for the third (250), fourth (125) or fifth (50) bar
M1: for a method to work out the total number of books OR the total number of squares Allow one error in a frequency density value, class width value or number of squares but not an omission
Allow ft their own linear frequency density scale
M1: for a method to estimate the number of books between 400g and 600 g OR the total number of squares between 400 g and 600 g Allow one error in a frequency density value, class width value or number of squares but not an omission Allow ft their own linear frequency density scale
A1: oe eg \(\dfrac{8}{29}\) or any correct decimal or correct percentage rounded or truncated to 2sf. NB \(\dfrac{200}{725} = 0.27(58\ldots)\)
M3 for any complete method that relies only on counting squares, eg counting large squares: \(\dfrac{4 + 10 + 2}{6 + 18 + 20 + 10 + 4}\) Allow one error in number of squares in each of numerator and denominator but do not allow any omissions
Must use a consistent method for the award of more than one mark
Correct answer scores full marks (unless from obvious incorrect working)
Answer: \(\dfrac{11}{63}\)
A1
(3)
(3 marks)
Notes
M1: for finding the area of at least 2 bars
either using freq density × mins
or
use of counting small squares or cm2
Values may be seen on the diagram
22 or 220 or 8.8 implies M1
M1: for method to find total number of people (allow one error or omission) or total number of small squares/ cm2 for method used (allow one error or omission)
A1: oe eg \(\dfrac{22}{126}\) or \(\dfrac{220}{1260}\) or 0.174(60…) or 0.175 or 17.4(60…)% or 17.5%
or 22 out of 126
If \(\dfrac{22}{126}\) is seen in the workings and 22 is on the answer line, award M2A0
18 The histogram shows information about the times some students took to complete a puzzle.
Work out an estimate for the fraction of these students who took between 20 minutes and 60 minutes to complete the puzzle.
(4)
Mark scheme
Scheme
Marks
eg two from 1.5 × 10 (= 15) or 0.8 × 15 (= 12) or 4 × 5 (= 20) or 2.5 × 30 (= 75) or 4.6 × 5 (= 23) or eg two from 15 or 12 or 20 or 75 or 23 or eg two from 150 or 120 or 200 or 750 or 230 or eg two from 6 or 4.8 or 8 or 30 or 9.2
Correct answer scores full marks (unless from obvious incorrect working)
Answer: \(\dfrac{99}{145}\)
A1
(4)
(4 marks)
Notes
M1: for at least two correct frequencies
or
for counting squares or blocks
M1: for a method to find the number of students in each time interval with an intention to add Allow one error or for a method to find the total number of squares or blocks oe with an intention to add Allow one error or for a correct method to find the frequency between 20 and 60 minutes eg 4 + 20 + 75 M2 for a value of 99 for 20 – 60 minute interval
M1: for a complete method
Allow the one error in the previous M mark to follow through
