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Question 1

Skill question

Daily step counts (in thousands) for 40 days are grouped below. Construct a less-than cumulative frequency graph and estimate the lower quartile $Q_1$ .

Data:

Class interval (thousand steps)	Frequency
0–2	3
2–4	7
4–6	15
6–8	11
8–12	4

Question 2

Skill question

Construct a less-than cumulative frequency graph for the heights (cm) of 35 students and hence estimate the median height. Use the class upper boundaries for plotting.

Data (heights in cm):

Class interval	Frequency
140–150	4
150–160	7
160–170	12
170–180	9
180–190	3

Question 3

Skill question

Lengths of parts (mm) are grouped as follows. Construct a less-than cumulative frequency graph and estimate the median length.

Data:

Class interval (mm)	Frequency
50–55	4
55–60	10
60–65	12
65–70	8
70–80	6

Question 4

Skill question

A cumulative frequency graph for the weights (kg) of 60 packages is represented by the points below. Use it to construct the estimated frequency distribution table for each class interval.

Data (less-than ogive points):

Class interval (kg)	Upper boundary	Cumulative frequency
40–50	50	3
50–60	60	11
60–70	70	26
70–80	80	44
80–90	90	57
90–100	100	60

Question 5

Skill question

A cumulative frequency curve for monthly expenses is summarized by the table below. From the graph, determine which class interval has the highest frequency.

Data:

Class interval	Upper boundary	Cumulative frequency
200–300	300	5
300–400	400	17
400–500	500	35
500–700	700	60
700–1000	1000	80

Question 6

Skill question

The cumulative frequency curve for exam scores is summarized below. Use it to reconstruct the estimated frequency distribution table for the 10-mark wide classes.

Data (ogive points):

Class interval	Upper boundary	Cumulative frequency
0–10	10	6
10–20	20	18
20–30	30	33
30–40	40	55
40–50	50	78
50–60	60	96
60–70	70	108
70–80	80	116
80–90	90	120
90–100	100	120

Question 7

Skill question

Travel times to work (minutes) for 40 commuters are grouped as follows. Construct a less-than cumulative frequency graph and estimate how many commuters take more than 45 minutes to travel.

Data:

Class interval (min)	Frequency
10–20	5
20–30	9
30–40	14
40–50	8
50–60	4

Question 8

Skill question

The lifetimes of 100 light bulbs (hours) are summarised by the cumulative frequency points below. Using the ogive, estimate the median lifetime.

Data (ogive points):

Class interval (hours)	Upper boundary	Cumulative frequency
500–600	600	5
600–700	700	18
700–800	800	40
800–900	900	68
900–1100	1100	100

Question 9

Skill question

The cumulative frequency graph of house prices (in \1000) is summarised by the table below. Use it to construct the estimated frequency distribution table for the given classes.

Data (ogive points):

Class interval (\1000)	Upper boundary	Cumulative frequency
100–150	150	4
150–200	200	12
200–250	250	27
250–300	300	41
300–400	400	59
400–600	600	80

Question 10

Skill question

The cumulative frequency graph of test scores (out of 100) is summarized by the points below (upper class boundaries with cumulative frequencies). Use the graph to estimate the interquartile range (IQR).

Data (to plot the ogive):

Class interval	Upper boundary	Cumulative frequency
0–20	20	8
20–40	40	23
40–60	60	45
60–80	80	72
80–100	100	80

Question 11

Skill question

Reaction times (s) in an experiment are grouped as below. Construct a less-than cumulative frequency graph and estimate the 90th percentile.

Data:

Class interval (s)	Frequency
0.2–0.3	3
0.3–0.4	8
0.4–0.5	14
0.5–0.7	10
0.7–0.9	5

Question 12

Skill question

Weekly study hours for 35 students are grouped below. Construct a less-than cumulative frequency graph and estimate the 65th percentile $P_{65}$ .

Data:

Class interval (hours)	Frequency
0–5	2
5–10	5
10–15	12
15–20	9
20–30	7

Question 13

Skill question

Masses of parcels (kg) are grouped below. Construct a less-than cumulative frequency graph and estimate the interquartile range (IQR).

Data:

Class interval (kg)	Frequency
0–2	6
2–4	10
4–6	18
6–8	9
8–12	7

Question Type 2: Constructing cumilative frequency graphs given the frequency distribution table Bootcamps