Statistics assignment 1

1

Business Statistics (BUS 505) Assignment I

Question 1:

Given Data 13 15 8 16 8 4 21 11 11 3 15

Arranging them is ascending order

3 4 8 8 11 13 15 15 16 21

1. Population mean:

μX =

∑i=1

N

Xi

N

=

3+4+8+8+11+13+15+15+16+2110

=

11410

= 11.4

Taking sample 3 8 15 21

Sample average:

X =

∑i=1

n

Xi

n

=

3+4+8+8+11+13+15+15+16+2110

=

11410

=11.4

2


2. Median:

N = even so,

M =

( n2 )th+( n+2

2 )th

2

=

( 10

2 )th+ ( 10+2

2 )th

2

=

11+132

= 12

3. Mode:

Mode= 8 and 15

4. Midrange:

=X L+X S

2

=

3+212

=12

5. Geometric mean:

G =n√a1×a2×. .. . .. .. . .. .. . .. .. . ..an

3


=10√3×4×8×8×11×13×15×15×16×21

=(3×4×8×8×11×13×15×15×16×21 )

110

= 9.816

6. Harmonic mean (population):

H =

n1

a1

+1

a2

. .. . .. .. . .. .. . .. .. . .+1

an

=

1013+

14+. .. . .. .. . .. .. ..

121

=8.036

= 7

7. Range:

=X L−X S

=21-3

= 18

4


Question 2:

21 22 27 36 22 29 22 23 22 28 36 33

Averaging them in ascending order

21 22 22 22 22 23 27 28 29 33 36 36

1. Population mean:

μX =

∑i=1

N

Xi

N

=

21+22+22+22+22+23+27+28+29+33+36+3612

=

32112

=26.75

Sample

21 22 23 36

Sample mean:

X =

∑i=1

n

Xi

n

5


=

21+22+22+22+22+23+27+28+29+33+36+3612

=

32112

=26.75

2. Median:

=

( n2 )th+( n+2

2 )th

2

=

( 12

2 )th+ ( 12+2

2 )th

2

=

6th

+7th

2

=

23+272

=25

3. Mode:

Mode is 22

4. Mid-range:

=X L+X S

2

6


=

36+212 =33.5

5. Harmonic mean:

H =

n1

a1

+1

a2

. .. . .. .. . .. .. . .. .. . .+1

an

=

121

21+

122

+. .. .. . .. .. . .. ..1

36

=25.75

6. Geometric mean:

=n√a1×a2×. .. . .. .. . .. .. . .. .. . ..an

=12√21×22×22×22×22×23×27×28×29×33×36×36

=

(21×22×22×22×22×23×27×28×29×33×36×36)1

12

=26.23

7. Range:

=X L−X S

7


= 36-21

=18

Question 3:

3.6, 3.1, 3.9, 3.7, 3.5, 3.7, 3.4, 3.0, 3.6

Arranging them in ascending order

3.0, 3.1, 3.4, 3.5, 3.6, 3.6, 3.7, 3.7, 3.9

1. Population mean:

μX =

∑i=1

N

Xi

N

=

3+3 .1+3 . 4+3. 4+3 . 5+3. 6+3. 6+3 . 7+3 . 910

=3.13

Sample data,

3 3.4 3.6 3.9

Sample mean:

8


X =

∑i=1

n

Xi

n

=

3+3 . 4+3 . 6+3 . 94

=3.475

2. Median:

=

( n2 )th+( n+2

2 )th

2

=

( 10

2 )th+ ( 10+2

2 )th

2

=

5th

+6th

2

=

3. 5+3 . 62

=3.55

3. Mode:

Mode= 3.4, 3.6, 3.7

4. Mid range:

9


=X L+X S

2

=

3+3 .92

=3.45

5. Hermonic mean:

H =

n1

a1

+1

a2

. .. . .. .. . .. .. . .. .. . .+1

an

=

1013+

13. 1

+ .. .. .. . .. .. . .. .1

3 . 9

=3.515

6. Geomatric mean:

=n√a1×a2×. .. . .. .. . .. .. . .. .. . ..an

=10√3∗3 .1∗3 . 4∗3 .4∗3 .5∗3 . 6∗3 .6∗3 .7∗3 .7∗3 . 9

= 3.47

10


7. Range:

=X L−X S

=3.9

= 0.9

11


Question 4

Population: 2, 4, 2, 3, 5, 4, 3, 2

1. Population Mean:

N=10 observations, so the Mean is

µx =

= 3.125

2. Median: 2, 2, 2, 3, 4, 4, 5

Median =

=

= 3

3. Mode: 2, 2, 2, 3, 4, 4, 5

Mode = 2

4. Midrange:

Midrange = XS+XL

2

= 2+5

2=7

2 = 3.5

5. Harmonic Mean:

2 + 4 + 2 + 3 + 5 + 4 + 3 + 28

∑i=1

N

x i

N

258

[ N2 ]th

+[ N+22 ]

th

2

[ 82 ]th

+[ 8+22 ]

th

2

4 th+5th

2=3+3

2

12


HM =

N1a1

+1a2

+−−−−−−−−−−−−−−−−−−−+1

aN

¿ 812+

14+

12+

13+

15+

14+

13+

12

=8

.5+.25+.5+.33+.2+.25+.33+ .5 = 2.7

6. Geometric Mean:

G. M. = N√a1×a2×a3×−−−−−−−×aN

= 8√2×4×2×3×5×4×3×2

= 2.95

7. Range:

Range = XL – XS = 5 – 2 = 3

Question No. 5

13


Population: 42, 29, 21, 37, 40, 33, 38, 26, 39, 47

1. Population Mean :

The population contains N=10 observations, so the Mean is

µx =

=

=

= 35.2

2. Median:

Ascending order: 21, 26, 29, 33, 37, 38, 39, 40, 42, 47

Median =

=

=

= 37.5

3. Mode:

The Mode of a set of observations is the value that occurs most frequently. So, there is no Mode.

4. Midrange:

Midrange = XS+XL

2

= 21+47

2=68

2 = 37.5

42 + 29 + 21 + 37 + 40 + 33 + 38 + 26 + 39 + 4710

∑i=1

N

x i

N

35210

[ N2 ]th

+[ N+22 ]

th

2

[102 ]

th

+[10+22 ]

th

25th+6 th

2=37+38

2

14


5. Harmonic Mean:

H. M. =N

1a1

+1a2

+−−−−−−−−−−−−−−−+1

aN

¿ 10142

+1

29+

121

+1

37+

140

+133

+1

38+

126

+1

39+

147

=10

.02+ .03+.04+.02+.02+.03+.02+.03+ .02+.02 = 40

6. Geometric Mean:

G. M. = N√a1×a2×a3×−−−−−−−×aN

= 10√42×29×21×37×40×33×38×26×39×47

= 34.31

7. Range:

Range = XL – XS = 47 – 21 = 26

15


Question No. 6

Population: 10.2, 3.1, 5.9, 7.0, 3.7, 2.9, 6.8, 7.3, 8.2, 4.3

1. Population Mean (Average) : The population contains N=10 observations, so the Mean is

µx =

=

=

= 5.94

2. Median: Arranging N=10 observations in ascending order, we have

Ascending Order: 2.9, 3.1, 3.7, 4.3, 5.9, 6.8, 7.0, 7.3, 8.2, 10.2

Median =

=

=

= 6.35

3. Mode: The Mode of a set of observations is the value that occurs most frequently. So, there is no Mode.

4. Midrange:

Midrange = XS+XL

2

10.2 + 3.1 + 5.9 + 7.0 + 3.7 + 2.9 + 6.8 + 7.3 + 8.2 + 4.310

∑i=1

N

x i

N

59 . 410

[ N2 ]th

+[ N+22 ]

th

2

[102 ]

th

+[10+22 ]

th

2

5th+6 th

2=5 .9+6 .8

2

16


= 2.9+10.2

2=13.1

2 = 6.55

5. Harmonic Mean:

H. M. =

N1a1

+1a2

+−−−−−−−−−−−−−−−+1

aN

¿ 101

10.2+

13.1

+1

5.9+

17.0

+1

3.7+

12.9

+1

6.8+

17.3

+1

8.2+

14.3

=10

.09+.32+.16+.14+ .27+.34+.14+.13+.12+.23 = 4.85

6. Geometric Mean:

G. M. = N√a1×a2×a3×−−−−−−−×aN

= 10√10.2×3.1×5.9×7.0×3.7×2.9×6.8×7.3×8.2×4.3 = 5.48

7. Population Range:

Range = XL – XS = 10.2 – 2.9 = 7.3

17


Question No. 7

Population: 7.3, 10.2, 13.1, 15.0, 15.8, 16.9, 18.2, 24.7, 25.3, 28.4, 29.3, 34.7

1. Population Mean: The population contains N=12 observations, so the Mean is

µx =

=

=

= 19.90


Ascending Order: 7.3, 10.2, 13.1, 15.0, 15.8, 16.9, 18.2, 24.7, 25.3, 28.4, 29.3, 34.7

Median =

=

=

= 17.55


4. Population Midrange:

Midrange = XS+XL

2

= 7.3+34.7

2=42

2 = 21

Here,XS = Smallest observations

XL = Largest observations

15.8 + 7.3 + 28.4 + 18.2 + 15.0 + 24.7 + 13.1 + 10.2 + 29.3 + 34.7 + 16.9 + 25.312

∑i=1

N

x i

N

238 . 912

[ N2 ]th

+[ N+22 ]

th

2

[122 ]

th

+[12+22 ]

th

2

6th+7th

2=16 . 9+18 . 2

2

18


5. Harmonic Mean:

H. M. =

N1a1

+1a2

+−−−−−−−−−−−−−−−+1

aN

¿ 121

15.8+

17.3

+1

28.4+

118.2

+1

15+

124.7

+1

13.1+

110.2

+1

29.3+

134.7

+1

16.9+

125.3

=12

.06+.13+.03+.05+.06+.04+.07+.09+.03+.02+.05+.03 = 18.18

6. Geometric Mean: G. M. = N√a1×a2×a3×−−−−−−−×aN

= 12√15.8×7.3×28.4×18.2×15.0×24.7×13.1×10.2×29.3×34.7×16.9×25.3 = 18.15

7. Range:

Range = XL – XS = 34.7 – 7.3 = 27.4

19


Question No. 16

Population: 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 21

1. Population Mean:

The population contains N=12 observations, so the Mean is

µx =

=

=

= 9.83


Ascending Order: 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 21

Median =

=

=

= 9.5


4. Midrange:

Midrange = XS+XL

2

= 3+21

2=24

2 = 12

12 + 7 + 4 + 16 + 21 + 5 + 9 + 3 + 11 + 14 + 10 + 612

∑i=1

N

x i

N

11812

[ N2 ]th

+[ N+22 ]

th

2[122 ]

th

+[12+22 ]

th

2

6th+7th

2=9+10

2

20


5. Harmonic Mean:

H. M. =

N1a1

+1a2

+−−−−−−−−−−−−−−−+1

aN

¿ 12112

+17+

14+

116

+121

+15+

19+

13+

111

+1

14+

110

+16

=12

.08+.14+.25+ .06+.04+.2+.11+.13+.09+.07+.1+.16 = 7.40

6. Geometric Mean:

G. M. = N√a1×a2×a3×−−−−−−−×aN

= 12√12×7×4×16×21×5×9×3×11×14×10×6 = 6.95

7. Range:

Range = XL – XS = 21 – 3 = 18

Education

Statistics assignment 1