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Exam practice

Every question from the end of every unit, with hints and answers to open. After you check an answer, mark how you did, and come back to the ones you didn’t get.

26 questions

Unit 17 · Classification and Presentation of Data

Exercise 17.1

Question 1

a.

Present the marks as a discrete series.

Answer
Marks Tally Frequency
2 |||| 4
4 |||| | 6
6 |||| |||| 10
8 |||| || 7
10 ||| 3
N = 30
b.

Make its cumulative frequency table.

Answer
Marks Frequency Cumulative frequency
2 4 4
4 6 10
6 10 20
8 7 27
10 3 30

So, for example, 20 students scored 6 or less.

Question 2

2.

Present this data as a continuous series, with the first class 5–10:
19, 5, 14, 17, 20, 21, 35, 39, 30, 31, 6, 8, 14, 28, 27, 39, 30, 31, 32, 25, 26, 10, 11, 12, 15, 28, 30, 31, 24, 22

Answer

Classes of 5 from 5 to 40 cover the lowest value, 5, and the highest, 39:

Class Tally Frequency
5–10 ||| 3
10–15 |||| 5
15–20 ||| 3
20–25 |||| 4
25–30 |||| 5
30–35 |||| || 7
35–40 ||| 3
N = 30

(10 is counted in 10–15, 15 in 15–20, 20 in 20–25, and so on.)

Question 3

a.

Make the less-than cumulative frequency table.

Answer
Class Cumulative frequency
Less than 10 3
Less than 15 8
Less than 20 11
Less than 25 15
Less than 30 20
Less than 35 27
Less than 40 30
b.

Make the more-than cumulative frequency table.

Answer
Class Cumulative frequency
5 or more (≥ 5) 30
10 or more (≥ 10) 30 − 3 = 27
15 or more (≥ 15) 27 − 5 = 22
20 or more (≥ 20) 22 − 3 = 19
25 or more (≥ 25) 19 − 4 = 15
30 or more (≥ 30) 15 − 5 = 10
35 or more (≥ 35) 10 − 7 = 3

Question 4

4.

Make a cumulative frequency table from this discrete data:

Wage (Rs) Workers
1,000 3
2,000 7
3,000 10
4,000 8
5,000 7
Answer
Wage (Rs) Workers Cumulative frequency
1,000 3 3
2,000 7 10
3,000 10 20
4,000 8 28
5,000 7 35
N = 35

Question 5

a.

Make the less-than cumulative frequency table.

Answer
Hours Cumulative frequency
Less than 6 100
Less than 10 350
Less than 14 675
Less than 18 875
Less than 22 1,025
b.

Make the more-than cumulative frequency table.

Answer
Hours Cumulative frequency
2 or more (≥ 2) 1,025
6 or more (≥ 6) 1,025 − 100 = 925
10 or more (≥ 10) 925 − 250 = 675
14 or more (≥ 14) 675 − 325 = 350
18 or more (≥ 18) 350 − 200 = 150

Exercise 17.2

Question 1

a.
Marks Students
30–40 4
40–50 6
50–60 8
60–70 10
70–80 8
80–90 4
Answer

x-axis 1 cm for 10 marks from 30 (a zigzag after 0); y-axis 1 cm for 2 students.

0246810students4681084030405060708090marks
Figure 17.11
b.
Age (years) 0–10 10–20 20–30 30–40 40–50
Patients 15 12 8 20 27
Answer

The classes start at 0, so no zigzag is needed.

0481216202428patients15128202701020304050age (years)
Figure 17.12
c.
Wage (Rs 100) 50–60 60–70 70–80 80–90 90–100
Workers 20 40 30 60 7
Answer
0102030405060workers20403060705060708090100wage (Rs 100)
Figure 17.13
d.
Class interval 0–5 5–10 10–15 15–20 20–25
Frequency 3 8 12 7 2
Answer
024681012frequency3812720510152025class interval
Figure 17.14

Question 2

a.
Marks 0–10 10–20 20–30 30–40 40–50
Students 7 3 8 10 2
Answer

Mid-values 5, 15, 25, 35, 45, with the empty class 50–60 (mid-value 55) on the right. On the left the empty class would be −10–0, so the line runs from (5, 7) towards (−5, 0) and stops at the y-axis, at height 3.5.

0246810students051525354555marks (mid-values)
Figure 17.15
b.
Cost of a pen (Rs) 10–20 20–30 30–40 40–50 50–60
Pens 15 20 30 25 5
Answer

Points (5, 0), (15, 15), (25, 20), (35, 30), (45, 25), (55, 5), (65, 0).

051015202530pens05152535455565cost (Rs) (mid-values)
Figure 17.16
c.
Income (Rs 100) Frequency
20–35 10
35–50 7
50–65 6
65–80 5
80–95 2
Answer

The classes are 15 wide, so the mid-values are 20+352=27.5 \frac{20 + 35}{2} = 27.5 , 42.5, 57.5, 72.5, 87.5, with the empty classes 5–20 (12.5) and 95–110 (102.5). Points (12.5, 0), (27.5, 10), (42.5, 7), (57.5, 6), (72.5, 5), (87.5, 2), (102.5, 0).

0246810frequency012.527.542.557.572.587.5102.5income (Rs 100) (mid-values)
Figure 17.17
d.
Class interval Frequency
11.5–19.5 7
19.5–27.5 17
27.5–35.5 10
35.5–43.5 4
43.5–51.5 1
51.5–59.5 1
Answer

The classes are 8 wide, so the mid-values are 15.5, 23.5, 31.5, 39.5, 47.5, 55.5, with the empty classes’ 7.5 and 63.5. Points (7.5, 0), (15.5, 7), (23.5, 17), (31.5, 10), (39.5, 4), (47.5, 1), (55.5, 1), (63.5, 0).

048121620frequency07.515.523.531.539.547.555.563.5class interval (mid-values)
Figure 17.18

Question 3

a.
Class interval Frequency
0–5 7
5–10 10
10–15 20
15–20 13
20–25 17
25–30 10
30–35 14
Answer
Limit Less than This or more
0 0 91
5 7 84
10 17 74
15 37 54
20 50 41
25 67 24
30 77 14
35 91 0

The less-than ogive goes through (upper limit, less-than cf), the more-than ogive through (lower limit, more-than cf). Drawn here on one graph (draw them on two if the question asks):

020406080100cumulative frequencyless thanmore than05101520253035class limit
Figure 17.19
b.
Marks Frequency
40–50 4
50–60 6
60–70 16
70–80 20
80–90 30
90–100 24
Answer
Limit Less than This or more
40 0 100
50 4 96
60 10 90
70 26 74
80 46 54
90 76 24
100 100 0
020406080100studentsless thanmore than0405060708090100marks
Figure 17.20
c.
Periods a week Teachers
0–6 3
6–12 10
12–18 20
18–24 10
24–30 5
30–36 2
Answer
Limit Less than This or more
0 0 50
6 3 47
12 13 37
18 33 17
24 43 7
30 48 2
36 50 0
01020304050teachersless thanmore than061218243036periods a week
Figure 17.21
d.
Hours a week 0–4 4–8 8–12 12–16 16–20
Frequency 28 35 66 40 31
Answer
Limit Less than This or more
0 0 200
4 28 172
8 63 137
12 129 71
16 169 31
20 200 0
050100150200cumulative frequencyless thanmore than048121620hours a week
Figure 17.22

Question 4

4.

Present the data of this histogram in a frequency table.

02468101214frequency030405060708090100marks
Figure 17.23 Read each bar’s class from the x-axis and its height from the y-axis.
Answer
Marks Frequency
30–40 4
40–50 8
50–60 10
60–70 12
70–80 14
80–90 6
90–100 2

N = 56.

Question 5

5.

Present the data of this cumulative frequency curve in a frequency table.

020406080students0102030405060708090marks
Figure 17.24 The points are at 10, 20, 30, … 80 marks.
Hint

Does the curve rise or fall? So which ogive is it, and are the marks on the x-axis upper or lower limits?

Answer

The curve falls, so it is a more-than ogive, and its points are (lower limit, “this or more”): 75 students scored 10 or more, 70 scored 20 or more, then 55, 40, 30, 25, 10 and 5.
Each class’s frequency is the drop from one point to the next: 75 − 70 = 5 scored 10–20, 70 − 55 = 15 scored 20–30, and so on. The last point says 5 scored 80 or more, the class 80–90.

Marks Students
10–20 5
20–30 15
30–40 15
40–50 10
50–60 5
60–70 15
70–80 5
80–90 5

N = 75 ✓ (the first point).

Question 6

6.

The water used in a month by 200 families of a neighborhood:

Water (thousand liters) 0–4 4–8 8–12 12–16 16–20
Families 28 35 66 40 31

Draw a histogram of the data.

Answer
010203040506070families2835664031048121620water (thousand liters)
Figure 17.25

Most families, 66, use 8,000 to 12,000 liters a month.

Question 7

a.

Present the data as a continuous series with the classes 10–15, 15–20, and so on.

Answer
Protein (g) Tally Frequency
10–15 ||| 3
15–20 |||| 4
20–25 |||| |||| 9
25–30 |||| |||| 10
30–35 |||| 4
35–40 |||| 5
40–45 |||| 4
45–50 0
50–55 0
55–60 | 1
N = 40

Two classes are empty, but keep them: the 57 is far from the rest, and the table should show the gap.

b.

Draw a histogram of it.

Answer
0246810foods34910454101015202530354045505560protein (g)
Figure 17.26
c.

Draw both the less-than and the more-than ogive on the same graph.

Answer
Limit Less than This or more
10 0 40
15 3 37
20 7 33
25 16 24
30 26 14
35 30 10
40 35 5
45 39 1
50 39 1
55 39 1
60 40 0
010203040foodsless thanmore than≈ 26.901015202530354045505560protein (g)
Figure 17.27

The curves cross where each is 20, at about 27 g: half the foods have less protein than that.