Higher Engineering Mathematics, Sixth Edition

540 Higher Engineering Mathematics

28 22 23 20 12 24 37 28 21 25

21 14 30 23 27 13 23 7 26 19

24 22 26 3 21 24 28 40 27 24

20 25 23 26 47 21 29 26 22 33

27 9 13 35 20 16 20 25 18 22

⎡

⎢

⎢

⎣

There is no unique solution,

but one solution is: 1–10 3;

11–19 7; 20–22 12;23–25 11;

26–28 10;29–38 5;39–48 2

⎤

⎥

⎥

⎦

5. Form a cumulative frequency distributionand
   hence draw the ogive for the frequency dis-
   tribution given in the solution to Problem 3.
   [
   20 .95 3; 21 .45 13; 21 .95 24;
   22 .45 37; 22 .95 46; 23 .45 48

]

6. Draw a histogram for the frequency distribu-
   tion given in the solution to Problem 4.
   ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
   Rectangles, touching one another,
   having mid-points of 5.5, 15,
   21, 24, 27, 33.5 and 43.5. The
   heights of the rectangles (frequency
   per unit class range) are 0.3,
   0.78, 4. 4.67, 2.33, 0.5 and 0.2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

7. The frequency distribution for a batch of
   50 capacitors of similar value, measured in
   microfarads, is:
   ⎡
   ⎣

10.5–10.9 2, 11.0–11.4 7,

11.5–11.9 10, 12.0–12.4 12,

12.5–12.9 11, 13.0–13.4 8

⎤

⎦

Form a cumulative frequency distribution for

these data.

[

( 10 .95 2), ( 11 .45 9), ( 11 .95 11),

( 12 .45 31), ( 12. 9542 ), ( 13 .45 50)

]

8. Drawanogiveforthedatagiveninthesolution
   of Problem 7.


9. The diameter in millimetres of a reel of wire
   is measured in 48 places and the results are as
   shown.

2.10 2.29 2.32 2.21 2.14 2.22

2.28 2.18 2.17 2.20 2.23 2.13

2.26 2.10 2.21 2.17 2.28 2.15

2.16 2.25 2.23 2.11 2.27 2.34

2.24 2.05 2.29 2.18 2.24 2.16

2.15 2.22 2.14 2.27 2.09 2.21

2.11 2.17 2.22 2.19 2.12 2.20

2.23 2.07 2.13 2.26 2.16 2.12

(a) Form a frequency distribution of diame-

ters having about 6 classes.

(b) Draw a histogram depicting the data.

(c) Form a cumulative frequency distribu-

tion.

(d) Draw an ogive for the data.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(a) There is no unique solution,

but one solution is:

2.05–2.09 3;2.10–21.4 10;

2.15–2.19 11;2.20–2.24 13;

2.25–2.29 9;2.30–2.34 2

(b) Rectangles, touching one

another, having mid-points of

2. 07 , 2. 12 ...and heights of

3 , 10 ,...

(c) Using the frequency

distribution given in the

solution to part (a) gives:

2 .095 3; 2 .145 13; 2 .195 24;

2 .245 37; 2 .295 46; 2 .345 48

(d) A graph of cumulative

frequency against upper

class boundary having

the coordinates given

in part (c).

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦