WISKUNDE
NOG OEFENINGE

Statistiek : antwoorde.

MATHEMATICS
NOG OEFENINGE

1.1   Stingel / Stem    Blaar / Leave

2                  5;  6;  7

3                  4

4                  5;  7;  7;

5                  3

6                  2;  8

7                  5

8                  3;  4

Vr. / Qu. 1.1

Σ x            676
1.2.1  Gemiddelde / Mean  =  ────    =  ───
n              13

=  52                            Vr. / Qu. 1.2.1

1.2.2  Modus / Mode  =  47   (2 keer / twice)           Vr. / Qu. 1.2.2

1.2.3  Omvang / Range  =  maks. / max  − min.

=  84  −  25

=  59                                Vr. / Qu. 1.2.3

1.2.4  Lys die getalle van klein na groot.  /  List the numbers is ascending order :

25       26       27       34       45       47       47       53       62       68       75       83       84

Q1                                  Q2                                 Q3

Q2 is die middelste getal.  /  Q2 is the number exactly in the middle.

Q2  =  47

OF / OR  met die formule: / using the formulae :

2(n + 1)           2(13 + 1)           2 × 14
Posisie van / Position of Q2  =  ──────    =   ──────    =   ─────   = 7
4                       4                    4

Q2  is die 7de getal / Q2  is the 7th number  d.i. / i.e.  Q2   =  47                            Vr. / Qu. 1.2.4

1.2.5   Q1 verdeel die linkerkantse helfte van die getalle in die helfte,  /  Q1 divides the lefthand half into two equal parts.

Q1 is die getal halfpad tussen 27 en 34, die gemiddelde van 27 en 34.  /
Q1 is the number halfway between 27 and 34, i.e. the average of 27 and 34.

27 + 34
Q1  =  ──────   =  30,5
2

Q3 verdeel die regterkantse helfte van die getalle in die helfte,     /
Q3 divides the righthand half into two equal parts.

Q3 is die getal halfpad tussen 68 en 75, die gemiddelde van 68 en 75.  /
Q3 is the number halfway between 68 and 75, i.e. the average of 68 and 75.

68 + 75
Q3  =  ──────   =  71,5
2

OF / OR  gebruik die formules: / use the formulae :

(n + 1)                                                                      3(n + 1)
Posisie van / Position of Q1  =  ──────                 Posisie van / Position of Q3  =  ──────
4                                                                                 4

(13 + 1)                                                                     3(13 + 1)
=  ──────                                                               =  ──────
4                                                                                 4

14                                                                                3 × 14
=  ───    =  3,5                                                          =   ─────   = 10,5
4                                                                                    4

Q1  is die 3de getal, 27, plus die helfte van die verskil tussen die derde getal (27) en die vierde getal (34)   /
Q1  is the 3rd number, 27, plus half of the difference between the third number (27) and the fourth number (34)

34 − 27
Q1  =  27  +  ──────  =  30,5
2
Q3  is die 10de getal, 68, plus die helfte van die verskil tussen die tiende getal (68) en die elfde getal (75)  /
Q3  is the 10th number, 28, plus half of the difference between the tenth number (68) and the eleventh number (75)

75 − 68
Q3  =  68  +  ──────  =  71,5
2                                                             Vr. / Qu. 1.2.5

1.2.6  IKVW / IQR  =  Q3  −  Q1  =  71,5  −  30,5  =  41                 Vr. / Qu. 1.2.6

1.3   Vyfpuntopsomming / Five number summary :

Minimum = 25;     Q1 = 30,5;     Q2 = 47;     Q3 = 71,5;     maksimum / maximum = 86                 Vr. / Qu. 1.3

1.4

Vr. / Qu. 1.4

2.1   Stingel / Stem    Blaar / Leave

14                  8;

15                  1;  4;  5;  7;  9

16                  3;  3;  7;

17                  2;  2;  2;  3;

18                  2;  4

Vr. / Qu. 2.1

Σ x            2 427
2.2.1  Gemiddelde / Mean  =  ────    =  ───
n              15

=  164,8                           Vr. / Qu. 2.2.1

2.2.2  Modus / Mode  =  172   (3 keer / thrice)            Vr. / Qu. 2.2.2

2.2.3  Omvang / Range  =  maks. / max  − min.

=  184  −  148

=  36                                    Vr. / Qu. 2.2.3

2.2.4  Lys die getalle van klein na groot.  /  List the numbers is ascending order :

148     151     154     155     157     159     163     163     167     172     172     172     173     182     184

Q1                                             Q2                                 Q3

Q2 is die middelste getal.  /  Q2 is the number exactly in the middle.

Q2  =  163

OF / OR  met die formule: / using the formulae :

2(n + 1)           2(15 + 1)           2 × 16
Posisie van / Position of Q2  =  ──────    =   ──────    =   ─────   = 8
4                       4                    4

Q2  is die 8ste getal / Q2  is the 8th number  d.i. / i.e.  Q2   =  163                          Vr. / Qu. 2.2.4

2.2.5   Q1 verdeel die linkerkantse helfte van die getalle in die helfte.      /  Q1 divides the lefthand half into two equal parts.

Q1 is die vierde getal, d.i. Q1 = 155.  /  Q1 is the fourth number, i.e. Q1 = 155.

Q3 verdeel die regterkantse helfte van die getalle in die helfte,  /  Q3 divides the righthand half into two equal parts.

Q3 is die 12de getal, d.i. Q3 = 172  /  Q3 is the 12th number, i.e. Q3 = 172.

OF / OR  gebruik die formules: / use the formulae :

(n + 1)                                                                             3(n + 1)
Posisie van / Position of Q1  =  ──────                       Posisie van / Position of Q3  =  ──────
4                                                                                       4

(15 + 1)                                                                            3(15 + 1)
=  ──────                                                                      =  ──────
4                                                                                        4

16                                                                                      3 × 16
=  ───    =  4                                                                    =   ─────   = 12
4                                                                                           4

Q1  is die 4de getal, 155  /   Q1  is the 4th number, 155

Q3  is die 12de getal, 172  /   Q3  is the 12th number, 172                                                      Vr. / Qu. 2.2.5

2.2.6  IKVW / IQR  =  Q3  −  Q1  =  71,5  −  30,5  =  41                                                                         Vr. / Qu. 2.2.6

2.3   Vyfpuntopsomming / Five number summary :

Minimum = 25;     Q1 = 30,5;     Q2 = 47;     Q3 = 71,5;     maksimum / maximum = 86              Vr. / Qu. 2.2.6

2.4

Vr. / Qu. 2.4

3.1   Stingel / Stem    Blaar / Leave

1                  6;  8;  9

2                  0;  1;  2;  3;  4;  8;  9

Vr. / Qu. 3.1

Σ x            220
3.2.1  Gemiddelde / Mean  =  ────    =  ───
n              10

=  22                                Vr. / Qu. 3.2.1

3.2.2  Modus / Mode  =  geen / no
(al die getalle kom net een keer voor /

(all the numbers have a frequency of 1)          Vr. / Qu. 3.2.2

3.2.3  Omvang / Range  =  maks. / max  − min.

=  29  −  16

=  13                                    Vr. / Qu. 3.2.3

3.2.4  Lys die getalle van klein na groot.  /  List the numbers is ascending order :

16       18       19       20       21       22       23       24       28       29

(n + 1)              (10 + 1)              11
Posisie van / Position of Q1  =  ──────    =   ──────    =   ─────   = 2,75
4                        4                    4

Q1  =  getal 2 + 0,75(getal 3 − getal 2)  /  number 2  +  0,75(number 3 − number 2)

=  18 + 0,75(19 − 18)    =  18,75

2(n + 1)           2(10 + 1)           2 × 11
Posisie van / Position of Q2  =  ──────    =   ──────    =   ─────   = 5,5
4                       4                    4

Q2  =  getal 5 + 0,5(getal6 − getal5)  /  number5  +  0,5(number6 − number5)

=  21 + 0,5(22 − 21)    =  21,5

3(n + 1)           3(10 + 1)           3 × 11
Posisie van / Position of Q3  =  ──────    =   ──────    =   ─────   = 8,25
4                       4                    4

Q3  =  getal 8 + 0,25(getal 9 − getal 8)  /  number 8  +  0,25(number 9 − number 8)

=  24 + 0,25(28 − 24)    =  25                                                                                           Vr. / Qu. 3.2.4

3.2.5  IKVW / IQR  =  Q3  −  Q1  =  25  −  18,75  =  6,25                                                                      Vr. / Qu. 3.2.5

3.3   Vyfpuntopsomming / Five number summary :

Minimum = 16;     Q1 = 18,75;     Q2 = 21,5;     Q3 = 25;     maksimum / maximum = 29            Vr. / Qu. 3.3

3.4

Vr. / Qu. 3.4

4.1   Stingel / Stem    Blaar / Leave

3                  8

4                  3;  5;  8

5                  1;  2;  3  4;  6;  8

6                  1;  2;  3;  5

7                  1;  2

Vr. / Qu. 4.1

4.2.1  Omvang / Range  =  maks. / max  − min.

=  72  −  38

=  34                                  Vr. / Qu. 4.2.1

4.2.2  Lys die getalle van klein na groot.  /  List the numbers is ascending order :

38       43       45       48       51       52       53       54       56       58       61       62       63       65       71       72

(n + 1)              (16 + 1)              17
Posisie van / Position of Q1  =  ──────    =   ──────    =   ─────   = 4,25
4                       4                    4

Q1  =  getal 4 + 0,25(getal 5 − getal 4)  /  number 4  +  0,25(number 5 − number 4)

=  48 + 0,25(51 − 48)    =  48,75

2(n + 1)           2(16 + 1)           2 × 17
Posisie van / Position of Q2  =  ──────    =   ──────    =   ─────   = 8,5
4                       4                    4

Q2  =  getal 8 + 0,5(getal9 − getal8)  /  number8  +  0,5(number9 − number8)

=  54 + 0,5(56 − 54)    =  55

3(n + 1)           3(16 + 1)           3 × 17
Posisie van / Position of Q3  =  ──────    =   ──────    =   ─────   = 12,75
4                       4                    4

Q3  =  getal 12 + 0,75(getal 13 − getal 12)  /  number 12  +  0,75(number 13 − number 12)

=  62 + 0,75(63 − 62)    =  62,75                                                                                     Vr. / Qu. 4.2.2

4.2.3  IKVW / IQR  =  Q3  −  Q1  =  62,75  −  48,75  =  14                                                                  Vr. / Qu. 4.2.3

4.3   Vyfpuntopsomming / Five number summary :

Minimum = 16;     Q1 = 18,75;     Q2 = 21,5;     Q3 = 25;     maksimum / maximum = 29          Vr. / Qu. 4.3

4.4

Vr. / Qu. 4.4

omvang / range = maksimum / maximum  −  minimum
a = minimum =  38 − 24
= 14
d = Q3  =  2 × minimum
= 2 × 14   = 28
IKVW / IQR = Q3 − Q1
b = Q1  =  Q3 − IKVW / IQR
= 28 − 8 = 20
c = Q2  =  Q1 + 4
= 20 + 4 = 24

a = 14;    b = 20;    c = 24;    d = 28;    e = 38

Vraag / Question 5.

omvang / range = maksimum / maximum  −  minimum
36 = e  −  18
e = 36 + 18   =  54
c = Q2  =  e − 18
= 54 − 18   = 36
d = Q3  =  Q2 + 6
= 36 + 6 = 42
IKVW / IQR = Q3 − Q1
b = Q1  =  Q3 − IKVW / IQR
= 42 − 11 = 31

a = 18;    b = 31;    c = 36;    d = 42;    e = 54

Vraag / Question 6.

omvang / range = maksimum / maximum  −  minimum
32 = e  −  33
e = 32 + 33   =  65
c = e − 12
= 65 − 12   = 53
d = 2a − 4
= 2 × 33 − 4   = 62
b = d − 20
= 62 − 20   = 42

a = 33;    b = 42;    c = 53;    d = 62;    e = 65

Vraag / Question 7.

8.1  25%

8.2  50%

8.3  omvang / range  =  68 − 4    = 64

8.4  IKVW / IQR = Q3 − Q1
=  50  −  32    =  18

Vraag / Question 8.

9.1

 Klas middelpunt (x) Class midpoint (x) Frekwensie f Frequency f . x 0 ≤ m < 10 5 2 10 10 ≤ m < 20 15 6 90 20 ≤ m < 30 25 12 300 30 ≤ m < 40 35 15 525 40 ≤ m ≤ 50 45 10 450

Σfx               1 375
9.2  gemiddelde / mean  =  ────    =    ─────  =    30,56
n                    45

9.3  modale klas /
modal class  =  30  ≤  m  <  40     ( f = 15 )

n
9.4.1  Posisie van / Position of  Q2   =  ───
4

45
=    ────
4

=  11,25
Q2  is die 12de getal en dus in die
interval  20 ≤ m < 30  /
Q2  is the 12th number and thus in
the interval  20 ≤ m < 30

3n
9.4.3  Posisie van / Position of  D3   =  ───
10

3 × 45
=  ─────
10

=  13,5
Die derde desiel is die 14de getal en is
in die interval  20 ≤ m < 30  /

The third decile is the 14th number and thus
in the interval  20 ≤ m < 30

70n
9.4.2  Posisie van / Position of  P70   =  ───
100

70 × 45
=  ──────
100

=  31,5
Die 70ste  persentiel is die 32ste getal en is
in die interval  30 ≤ m < 40  /

The 70th  percentile is the 32nd number and thus
in the interval 30 ≤ m < 40

10.1

 Klas middelpunt (x) Class midpoint (x) Frekwensie f Frequency f . x 40 ≤ m < 50 45 6 180 50 ≤ m < 60 55 11 550 60 ≤ m < 70 65 9 595 70 ≤ m < 80 75 2 150

Σfx               1 610
10.2  gemiddelde / mean  =  ────    =    ─────  =    57,5
n                    28

10.3  modale interval /
modal interval  =  50  ≤  m  <  60     ( f = 11 )

2n
10.4.1  Posisie van / Position of  Q2   =  ───
4

2 × 28
=  ─────
4

=  14
Q2  is die 14de getal en dus in die interval
50 ≤ m < 60  /
Q2  is the 14th number and thus in the interval
50 ≤ m < 60

2n
10.4.3  Posisie van / Position of  D2   =  ───
10

2 × 28
=  ─────
10

=  5,6
D2  is die 6de getal en dus in die interval
40 ≤ m < 50  /
D2  is the 6th number and thus in the interval
40 ≤ m < 50

80n
10.4.2  Posisie van / Position of  P80   =  ───
100

80 × 28
=  ─────
100

=  22,4
P80  is die 23ste getal en dus in die interval
60 ≤ m < 70  /
P80  is the 23rd number and thus in the interval
60 ≤ m < 70

11.1

 Lengte  h (cm) Height Frekwensie f Frequency Klas middelpunt (x) Class midpoint f . x 120 ≤ h < 140 4 130 520 140 ≤ h < 160 15 150 2 250 160 ≤ h < 180 10 170 1 700 180 ≤ h < 200 3 190 570

Σfx               5 040
11.2  gemiddelde / mean  =  ────    =    ─────  =    157,5
n                    28

11.3  modale interval /
modal interval  =  140  ≤  h  <  160     ( f = 15 )

11.4.1  Die gemiddelde is in die interval 140 ≤ h < 160 /

The mean is in the interval   140  ≤  h  <  160

3n
11.4.3  Posisie van / Position of  Q3   =  ───
4

3 × 32
=  ──────
4

=  24
Q3  is die 24ste getal en dus in die
interval  160 ≤ h < 180  /
Q3  is the 24th number and thus in the
interval  160 ≤ h < 180

8n
11.4.5  Posisie van / Position of  D8   =  ───
10

8 × 32
=  ─────
10

=  25,6
D8  is die 26ste getal en dus in die
interval  160 ≤ h < 180  /
D8  is the 26th number and thus in the
interval  160 ≤ h < 180

n
11.4.2  Posisie van / Position of  Q1   =  ───
4

32
=  ───
4

=  8
Q1  is die 8ste getal en is in
die interval  140 ≤ h < 160  /
Q1  is the 8th number and thus
in interval  140 ≤ h < 160

30n
11.4.4  Posisie van / P30   =  ───
100

30 × 32
=  ─────
100

=  9,6
P30  is die 10de waarde en is in die
interval  140 ≤ h < 160  /
P30  is the 10th value and thus
in interval  140 ≤ h < 160

11.5  Die aantal dogters is 19  /  There are 19 girls.

Vraag / Question 11.

12.1

 Tyd t (minute) Time Frekwensie f Frequency Klas middelpunt (x) Class midpoint f . x 5 ≤ t < 10 1 7,5 7,5 10 ≤ t < 15 5 12,5 62,5 15 ≤ t < 20 8 17,5 140 20 ≤ t < 25 4 22,5 90 25 ≤ t < 30 2 27,5 55

Σfx               355
12.2  gemiddelde / mean  =  ────    =    ─────  =    17,75
n                    20

12.3  modale interval / modal interval  =  15  ≤  t  <  20
( f = 8 )

n
12.4.1  Posisie van / Position of  Q1   =  ───
4

20
=  ───
4

=  5
Q1  is die 5de waarde en is in
die interval  10 ≤ t < 15  //
Q1  is the 5th value and thus in
interval  10 ≤ t < 15

4n
12.4.3  Posisie van / Position of  D4   =  ───
10

4 × 16
=  ─────
10

=  6,4
D4  is die 7de waarde en is in
die interval  15 ≤ t < 20  /
D4  is the 7th value and thus
in interval  15 ≤ t < 20

80n
12.4.2  Posisie van / Position of  P80   =  ────
100

80 × 20
=  ───────
100

=  16
P80  is die 16de waarde en is in
die interval  20 ≤ t < 25  /
P80  is the 16th value and thus in
interval  20 ≤ t < 25