Statistics, Probability and Data Depth Worksheet with Answers

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13+ · Mathematics

CE.M30 Statistics, Probability and Data Depth

ISEB COMMON ENTRANCE ISEB CE 13+ Mathematics · Calculator allowed · about 100 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

The number of goals scored by a school hockey team in their last 9 matches: 2, 4, 2, 5, 3, 2, 1, 4, 2

(a)Write down the mode of the number of goals scored.(1)

(b)Work out the range of the number of goals scored.(1)

(Total for Question 1 is 2 marks)

2

A jar contains 9 red beads, 6 blue beads and 5 green beads, and nothing else (20 beads in total). A bead is taken from the jar at random.

(a)Write down the probability that the bead is red.(1)

(b)Find the probability that the bead is not green.(1)

(Total for Question 2 is 2 marks)

3

A fair spinner has 5 equal sections: two are coloured red, and one each is coloured blue, green and yellow. The spinner is spun once.

(a)Write down the probability that the spinner lands on red.(1)

(b)Find the probability that the spinner does not land on red.(1)

(Total for Question 3 is 2 marks)

4

The table shows the number of pets owned by 24 pupils in a class.
Number of pets (x): 0, 1, 2, 3
Frequency (f): 9, 8, 5, 2

(a)Write down the modal number of pets.(1)

(b)Find the number of pupils who own at least 1 pet.(1)

(Total for Question 4 is 2 marks)

5

The resting heart rates, in beats per minute (bpm), of 8 athletes are recorded below:
58, 62, 55, 60, 58, 64, 61, 58

(a)Calculate the mean heart rate.(2)

(b)Find the median heart rate.(2)

(Total for Question 5 is 4 marks)

6

The table shows the number of siblings of 30 pupils in a year group.
Number of siblings (x): 0, 1, 2, 3, 4
Frequency (f): 6, 10, 7, 4, 3

(a)Write down the modal number of siblings.(1)

(b)Find the number of pupils who have at least 2 siblings.(1)

(c)Calculate the mean number of siblings per pupil.(3)

(Total for Question 6 is 5 marks)

7

The bar chart shows the number of pupils who chose each drink at the school cafe on Monday. In total, 60 pupils were surveyed.

(a)Write down the number of pupils who chose Juice.(1)

(b)Given that 60 pupils were surveyed in total, calculate the number of pupils who chose Milkshake.(2)

(c)A pupil is chosen at random from the 60 pupils surveyed. Find the probability that this pupil chose Water, giving your answer as a fraction in its simplest form.(2)

(Total for Question 7 is 5 marks)

8

120 pupils were asked whether they attend the Chess club. Some of the results are shown in the two-way table below (numbers of pupils).
Chess No Chess Total
Year 8 22 ? 50
Year 9 ? 38 70
Total ? ? 120

(a)Find the number of Year 8 pupils who do not attend the Chess club.(1)

(b)Find the number of Year 9 pupils who attend the Chess club.(1)

(c)A pupil is chosen at random from all 120 pupils. Find the probability that the pupil is in Year 9 and does not attend the Chess club, giving your answer as a fraction in its simplest form.(2)

(Total for Question 8 is 4 marks)

9

180 pupils were asked how they usually travel to school. The results are shown in the table and drawn as a pie chart.
Walk: 63 pupils
Car: 45 pupils
Bus: (not given)
Other: 18 pupils

(a)Calculate the angle of the pie-chart sector that represents pupils who travel by Bus.(2)

(b)The Walk sector of the pie chart has an angle of 126 degrees. Use this angle to find the number of pupils who walk to school.(2)

(Total for Question 9 is 4 marks)

10

The graph shows the temperature, in degrees Celsius, recorded at a weather station every 2 hours from 06:00 to 18:00.

(a)Write down the temperature at 12:00.(1)

(b)Find the difference between the highest and lowest temperatures shown on the graph.(2)

(c)Describe how the temperature changes between 14:00 and 18:00.(1)

(Total for Question 10 is 4 marks)

11

A market trader plots a scatter graph of the daily temperature (degrees Celsius) against the number of cold drinks sold, for 12 days. The points show a pattern rising from bottom left to top right, and a line of best fit has been drawn through the points (10, 20) and (30, 100).

(a)Describe the type of correlation shown by the scatter graph.(1)

(b)Interpret what this correlation means in the context of the data.(1)

(c)Using the line of best fit, estimate the number of cold drinks sold on a day with a temperature of 24 degrees Celsius.(2)

(d)Explain why it would not be sensible to use the line of best fit to estimate sales on a day with a temperature of 45 degrees Celsius.(1)

(Total for Question 11 is 5 marks)

12

A bag contains only red, blue, green and yellow marbles. A marble is chosen at random. P(red) = 0.2, P(blue) = 0.35, P(green) = 0.15.

(a)Find P(yellow).(1)

(b)There are 40 marbles in the bag. Work out the number of yellow marbles.(2)

(Total for Question 12 is 3 marks)

13

A fair four-sided spinner numbered 1, 2, 3, 4 is spun once, and a fair coin is tossed once.

(a)List all the possible outcomes, writing each as a pair (number, letter).(1)

(b)Find the probability of getting an even number and a Head.(2)

(c)Find the probability of getting a number greater than 2, a Tail, or both.(2)

(Total for Question 13 is 5 marks)

14

A drawing pin is dropped 60 times. It lands point-up 42 times.

(a)Find the relative frequency (estimated probability) of the pin landing point-up.(1)

(b)The pin is now dropped a further 150 times. Estimate the number of times it will land point-up.(2)

(Total for Question 14 is 3 marks)

15

The table shows the time, h hours, spent on homework in one week by 40 pupils.

Time (h hours)	Frequency
0 ≤ h < 2	5
2 ≤ h < 4	12
4 ≤ h < 6	14
6 ≤ h < 8	6
8 ≤ h < 10	3

(a)Write down the modal class.(1)

(b)Calculate an estimate for the mean time spent on homework.(3)

(c)Explain why your answer to part (b) is only an estimate.(1)

(Total for Question 15 is 5 marks)

16

The stem-and-leaf diagram shows the high-jump heights, in cm, cleared by 13 pupils in a school sports day.

Stem	Leaf
11	2 5 8
12	1 4 4 7 9
13	2 5 8
14	1 5

Key: 11 | 2 means a height of 112 cm

(a)Find the median height.(1)

(b)Find the range of the heights.(1)

(c)Find the number of pupils who jumped at least 130 cm.(1)

(Total for Question 16 is 3 marks)

17

Two sprinters, Osei and Ffion, each ran 5 races over 100 m. Their times, in seconds, were:
Osei: 13.2, 13.5, 13.1, 13.8, 13.4
Ffion: 13.6, 13.3, 13.5, 13.4, 13.2

(a)Calculate the mean time for each sprinter.(2)

(b)Calculate the range for each sprinter.(2)

(c)Using your answers to parts (a) and (b), determine which sprinter was more consistent, giving a reason.(1)

(Total for Question 17 is 5 marks)

18

In a survey of 45 pupils, 25 attend the Art club, 18 attend the Music club, and 8 attend both clubs. You may find it helpful to draw a Venn diagram.

(a)Find the number of pupils who attend neither club.(2)

(b)Find the probability that a pupil chosen at random from the 45 pupils attends the Art club only, giving your answer as a fraction in its simplest form.(2)

(Total for Question 18 is 4 marks)

19

A bag contains counters that are either red or blue. On any single pick, the probability of picking a red counter at random is 0.4. A counter is picked at random, its colour is noted, and it is replaced before a second counter is picked at random. You may find it helpful to draw a tree diagram.

(a)Find the probability that both counters picked are red.(2)

(b)Find the probability that exactly one of the two counters picked is red.(3)

(c)Find the probability that at least one of the two counters picked is red.(2)

(Total for Question 19 is 7 marks)

20

A pencil case contains 4 blue pens and 6 black pens, and nothing else (10 pens in total). Two pens are taken at random, one after the other, without replacement.

(a)Find the probability that both pens taken are blue.(2)

(b)Find the probability that the two pens taken are the same colour.(3)

(Total for Question 20 is 5 marks)

21

A fair six-sided die is rolled three times.

(a)Find the probability that all three rolls give a 6.(2)

(b)Find the probability that at least one of the three rolls gives a 6.(3)

(Total for Question 21 is 5 marks)

22

A box contains 5 gold tokens and 3 silver tokens, and nothing else (8 tokens in total). Two tokens are drawn at random, one after another, without replacement.

(a)Find the probability that the first token drawn is gold and the second token drawn is silver.(2)

(b)Find the probability that exactly one of the two tokens drawn is silver.(3)

(c)Given that the first token drawn was gold, find the probability that the second token drawn is also gold.(1)

(Total for Question 22 is 6 marks)

23

In a survey of 50 pupils, 28 play rugby, 19 play hockey, and 12 play neither sport. You may find it helpful to draw a Venn diagram.

(a)Find the number of pupils who play both rugby and hockey.(2)

(b)A pupil is chosen at random from those who play rugby. Find the probability that this pupil also plays hockey.(2)

(Total for Question 23 is 4 marks)

24

The times, in minutes, taken by 9 employees to complete a training exercise are recorded below:
22, 24, 23, 25, 24, 23, 26, 24, 95

(a)Calculate the mean time.(2)

(b)Find the median time.(1)

(c)State, with a reason, which average (the mean or the median) better represents a typical employee's time.(1)

(Total for Question 24 is 4 marks)

Mark scheme · CE.M30 Statistics, Probability and Data Depth

Question 1

(a) B1 mode = 2 cao
(a) Answer: 2
(b) B1 range = 5 - 1 = 4 cao
(b) Answer: 4

Question 2

(a) B1 P(red) = 9/20 cao
(a) Answer: 9/20
(b) B1 P(not green) = 15/20 = 3/4 oe
(b) Answer: 3/4

Question 3

(a) B1 P(red) = 2/5 cao
(a) Answer: 2/5
(b) B1 P(not red) = 3/5 oe, e.g. 1 - 2/5
(b) Answer: 3/5

Question 4

(a) B1 mode = 0 cao
(a) Answer: 0
(b) B1 15 cao (from 8 + 5 + 2)
(b) Answer: 15

Question 5

(a) M1 sum of all 8 values = 476 seen
(a) A1 59.5 bpm cao
(a) Answer: 59.5 bpm
(b) M1 values ordered and the middle two identified: 58 and 60
(b) A1 59 bpm cao
(b) Answer: 59 bpm

Question 6

(a) B1 mode = 1 cao
(a) Answer: 1
(b) B1 14 cao (from 7 + 4 + 3)
(b) Answer: 14
(c) M1 correct products: (0x6) + (1x10) + (2x7) + (3x4) + (4x3)
(c) M1 sum of products = 48 used with total frequency = 30
(c) A1 mean = 1.6 cao
(c) Answer: 1.6 siblings

Question 7

(a) B1 16 cao
(a) Answer: 16
(b) M1 24 + 16 + 8 = 48 seen (sum of the three known bars)
(b) A1 12 cao (from 60 - 48)
(b) Answer: 12
(c) M1 24/60 seen
(c) A1 2/5 cao (simplified)
(c) Answer: 2/5

Question 8

(a) B1 28 cao (from 50 - 22)
(a) Answer: 28
(b) B1 32 cao (from 70 - 38)
(b) Answer: 32
(c) M1 38/120 seen
(c) A1 19/60 cao (simplified)
(c) Answer: 19/60

Question 9

(a) M1 number travelling by Bus = 180 - 63 - 45 - 18 = 54, and 54/180 x 360 set up
(a) A1 108 degrees cao
(a) Answer: 108 degrees
(b) M1 126/360 x 180 seen
(b) A1 63 cao
(b) Answer: 63

Question 10

(a) B1 15 degrees Celsius cao
(a) Answer: 15 degrees Celsius
(b) M1 highest = 17 (at 14:00) and lowest = 4 (at 06:00) identified
(b) A1 13 degrees Celsius cao
(b) Answer: 13 degrees Celsius
(c) B1 temperature decreases/falls, from 17 to 9 degrees Celsius oe
(c) Answer: It falls steadily from 17 degrees Celsius to 9 degrees Celsius.

Question 11

(a) B1 positive correlation oe
(a) Answer: Positive correlation
(b) B1 as temperature increases, the number of cold drinks sold tends to increase oe, in context
(b) Answer: On warmer days, more cold drinks tend to be sold.
(c) M1 gradient = (100-20)/(30-10) = 4 found, or equivalent method
(c) A1 76 cao (accept 70 to 82 if read from a drawn graph)
(c) Answer: 76 drinks
(d) B1 45 degrees Celsius is outside (an extrapolation beyond) the range of temperatures in the collected data (roughly 6 to 33 degrees Celsius), so the trend may not continue oe
(d) Answer: 45 degrees Celsius is well beyond the temperatures recorded, so using the line of best fit there would be extrapolation and may not be reliable.

Question 12

(a) B1 0.3 cao
(a) Answer: 0.3
(b) M1 0.3 x 40 seen, ft from part a
(b) A1 12 cao
(b) Answer: 12

Question 13

(a) B1 all 8 outcomes listed: (1,H)(1,T)(2,H)(2,T)(3,H)(3,T)(4,H)(4,T)
(a) Answer: (1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T)
(b) M1 identify 2 favourable outcomes, (2,H) and (4,H), out of 8
(b) A1 1/4 cao (simplified from 2/8)
(b) Answer: 1/4
(c) M1 identify 6 favourable outcomes out of 8, e.g. (3,H)(3,T)(4,H)(4,T)(1,T)(2,T), allowing for the overlap at (3,T) and (4,T)
(c) A1 3/4 cao (simplified from 6/8)
(c) Answer: 3/4

Question 14

(a) B1 42/60 = 7/10 or 0.7 oe
(a) Answer: 7/10 (0.7)
(b) M1 150 x 7/10 (or 150 x 0.7) seen, ft from part a
(b) A1 105 cao
(b) Answer: 105 times

Question 15

(a) B1 4 ≤ h < 6 cao
(a) Answer: 4 ≤ h < 6
(b) M1 midpoints 1, 3, 5, 7, 9 used
(b) M1 sum of (midpoint x frequency) = 180 used with total frequency = 40
(b) A1 4.5 hours cao
(b) Answer: 4.5 hours
(c) B1 the exact values within each class are unknown, so midpoints are used as an approximation oe
(c) Answer: The exact time each pupil spent is not known, only the class it falls in, so midpoints are used instead of the real values.

Question 16

(a) B1 127 cm cao
(a) Answer: 127 cm
(b) B1 33 cm cao (145 - 112)
(b) Answer: 33 cm
(c) B1 5 cao
(c) Answer: 5 pupils

Question 17

(a) M1 both totals correct: Osei = 67.0, Ffion = 67.0
(a) A1 both means correct: 13.4 s each cao
(a) Answer: Osei: 13.4 s, Ffion: 13.4 s
(b) M1 correct method for at least one range: max - min
(b) A1 Osei = 0.7 s and Ffion = 0.4 s, both cao
(b) Answer: Osei: 0.7 s, Ffion: 0.4 s
(c) B1 Ffion, because although the means are equal, Ffion has the smaller range (0.4 s < 0.7 s), ft from parts a and b
(c) Answer: Ffion, because she has the smaller range even though the means are the same.

Question 18

(a) M1 correct regions found: Art only = 17, Music only = 10, both = 8
(a) A1 neither = 10 cao
(a) Answer: 10 pupils
(b) M1 17/45 seen, ft from part a
(b) A1 17/45 cao (already in simplest form)
(b) Answer: 17/45

Question 19

(a) M1 0.4 x 0.4 seen
(a) A1 0.16 cao
(a) Answer: 0.16
(b) M1 identifies the two ways this can happen, red-then-blue and blue-then-red, each with probability 0.4 x 0.6 = 0.24
(b) M1 adds the two probabilities, dependent on previous method
(b) A1 0.48 cao
(b) Answer: 0.48
(c) M1 P(no red) = 0.6 x 0.6 = 0.36 seen
(c) A1 0.64 cao, ft from their P(no red)
(c) Answer: 0.64

Question 20

(a) M1 4/10 x 3/9 seen
(a) A1 2/15 cao (simplified from 12/90)
(a) Answer: 2/15
(b) M1 P(both black) = 6/10 x 5/9 = 30/90 (= 1/3) seen
(b) M1 adds P(both blue) from part a to P(both black), dependent on previous method
(b) A1 7/15 cao (from 42/90)
(b) Answer: 7/15

Question 21

(a) M1 1/6 x 1/6 x 1/6 seen
(a) A1 1/216 cao
(a) Answer: 1/216
(b) M1 P(no 6 on one roll) = 5/6 seen
(b) M1 (5/6)³ = 125/216 seen, dependent on previous method
(b) A1 91/216 cao
(b) Answer: 91/216

Question 22

(a) M1 5/8 x 3/7 seen
(a) A1 15/56 cao
(a) Answer: 15/56
(b) M1 P(silver then gold) = 3/8 x 5/7 = 15/56 seen
(b) M1 adds P(gold then silver) from part a to P(silver then gold), dependent on previous method
(b) A1 15/28 cao (from 30/56)
(b) Answer: 15/28
(c) B1 4/7 cao
(c) Answer: 4/7

Question 23

(a) M1 at least one sport = 50 - 12 = 38 used with 28 + 19 - 38
(a) A1 9 cao
(a) Answer: 9 pupils
(b) M1 9/28 seen, ft from part a
(b) A1 9/28 cao
(b) Answer: 9/28

Question 24

(a) M1 sum of all 9 values = 286 seen
(a) A1 31.8 minutes cao (3 s.f., accept 31.78 or exact 286/9)
(a) Answer: 31.8 minutes (3 s.f.)
(b) B1 24 minutes cao
(b) Answer: 24 minutes
(c) B1 median, because the mean is distorted/inflated by the extreme value of 95, which the median is not affected by, ft from parts a and b
(c) Answer: The median (24 minutes), because the one very large value of 95 minutes pulls the mean up, but does not affect the median.

Statistics, Probability and Data Depth - Worksheets, Questions and Revision

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