MODEL ANSWER
Code: AS-2612
B.Com. (Hons.), II – Semester
Subject: Business Statistics
1. Short Answer:
i.
1. Estimation of Values: If, cause and effect relationship exists between two
variables, one (variable) can be estimated with the help of other (variable).
2. Measurement of Degree and Direction of Coefficient of Correlation:
Degree and direction of coefficient of correlation (r) can be measured with the
help of regression analysis.
ii.
Tow Properties of Regression Coefficient:
1.
The geometric mean of the two regression coefficients; i.e. regression
coefficient of x on y (bxy) and regression coefficient of y on x (byx) is equal to the
correlation coefficient. √
=√
X√
=r
2.
If one of regression coefficient is greater than unity, the other must be less
than unity, and vice versa. -1 <
X
< 1, (i.e. range of r = -1 < r < 1)
iii.
Time Series: A time series is a set of observations taken at specified times,
usually at ‘equal intervals’. Mathematically, a time series is a set of data/values
(y1, y2, y3, …..) collected/arranged according to the time (t1, t2, t3, ….). Thus, Y
(values) is a function of t (time). The unit of time may be a year, a month, a day,
an hour, etc. Symbolically, Y = f(t)
iv.
Moving Average: Moving Averages are series of consecutive arithmetic means
obtained by averaging the group of successive observations in a time series. It
may be of any period, (i.e. three months, three years, four months, four years, etc).
Another way, a moving average is a calculation to analyze data points by
creating a series of averages of different subsets of the full data set. It is also
called a Moving Mean (MM) or Rolling Mean (RM) and is a type of finite
impulse response filter.
v.
Formula for calculating the value of ‘a’ and ‘b’ applying Least Square Method:
1. Short-cut Method:
a=
,
b=
Or
2. Direct Method: By solving following two equations, values of ‘a’ and ‘b’ can
be found out∑y = Na + b∑x ---- (I), ∑xy = a∑x + b∑
---- (II)
vi.
Given,
̅̅̅ = 43,
̅̅̅ = 47,
Value of Annual Increase = ?
Value of Annual Increase =
̅
̅
1
=
t1 = 1997,
= = 1.̅
t2 = 2000,
vii.
1. Additional Theorem of Probability: If A1, A2, A3, ….. Ak be mutually
exclusive events on a sample space associated with a random experiment, then the
probability of happening one of them is the sum of their individual probabilities.
Symbolically, P(A1 + A2 + A3 + ….. + Kk) = P(A1) + P(A2) + P(A3) + ….. + P(Ak)
or
P(A1 U A2 U A3 U …..U Ak) = P(A1) + P(A2) + P(A3) + ….. + P(Ak)
or
P(A1 or A2 or A3 or …..U Ak) = P(A1) + P(A2) + P(A3) + ….. + P(Ak)
2. If two events (A and B) are independent, the probability of happening A and B
together is equal to the product of their individual probabilities. P(A∩B) =
P(A).P(B)
If tow events (A and B) dependent, the probability of happening A and B
together is equal to the product of the probability of A , P(A) and the conditional
probability P(B/A).
P(A∩B) = P(A).P(B/A) or = P(B).P(A/B)
viii.
A. Probability of drawing a queen of clubs = 1/52 and
B. Probability of drawing a king of diamond = 1/52
Required Prob. = Sum (Probability of A + Probability of B) = 1/52 + 1/52 = 2/52
ix.
1. Short-Term Forecasting: Short-term forecasting involves a few weeks or
months or one or two years. A number of simple techniques, like ‘quick method’
etc. are used in short-term forecasting.
2. Long-Term Forecasting: Long-term forecasting involves the movement of a
certain phenomenon for several years, say more than 2 years. Generally, complex
techniques are used to include more variables as compared to the techniques used
for short-term.
x.
1. Full-form of CSO is Central Statistical Organization, and
2. Full-form of NSSO is National Sample Survey Organization
Long Answers:
2. Given:
3x + 2y – 26 = 0 ----- (i),
6x + y – 31 = 0 ----- (ii)
Required to calculate: (a) Regression equation of x on y and regression equation on y on x
(b) Mean of x (x series) and y (y series)
(c) Correlation coefficient between them (r b/w x series and y series)
(d) Ratio of Standard Deviation of the two variables (series of data)
Assumption-1:
Let,
Equation (i) is for Regression equation of x on y, and
Equation (ii) is for Regression Equation of y on x
On this basis,
3x + 2y – 26 = 0 ----- (i),
6x + y – 31 = 0 ----- (ii)
X=
+
y = -6x + 31
Regression Coefficient of x on y (bxy) = -2/3
Regression Coefficient of y on x (byx) = -6
2
Geometric mean of the two Reg. coefficients (bxy & byx) is equal to the correlation coefficient (r).
r=√
=√
.√
=√
=-√ =-2
But, this assumption is not true, because ‘r’=√
.√
=-2
(as calculated above)
and the range of ‘r’ = -1< √
<1
Therefore, we are required to reverse our previous assumption.
Assumption-2:
Let,
Equation (ii) is for Regression equation of x on y, and
Equation (i) is for Regression Equation of y on x
On this basis,
3x + 2y – 26 = 0 ----- (i),
6x + y – 31 = 0 ----- (ii)
y=
+
x = -y/6 + 31/6
Regression Coefficient of y on x (byx) = -3/2
Regression Coefficient of x on y (bxy) = -1/6
Geometric mean of the two Reg. coefficients (byx & bxy) is equal to the correlation coefficient (r).
r=√
=√
.√
=-√
=-√
= - ½ = - 0.50
So, Assumption-2 is right
Equation (ii), i.e (6x + y – 31 = 0) is for Regression equation of x on y, and
Equation (i), i.e. (3x + 2y – 26 = 0) is for Regression Equation of y on x.
(b) Mean of x (x series) and y (y series)
To find-out means of x and y, we have to solve two equation3x + 2y – 26 = 0 ----- (i)
6x + y – 31 = 0 ----- (ii)
6x + 4y – 52 = 0 ----- (iii) {Multiplied equation (i) by 2}
-3y + 21 = 0 {Deducted Equation (iii) from Equation (ii)}
-3y = -21
Y = 21/3 = 7
Putting the value of Y in Equation (i)
3x + 2y – 26 = 0
3x + 2 x 7 – 26 = 0
3x = 26 – 14
X = 12/3 = 4
Mean of x or x series (̅) = 4
Mean of y or y series (̅) = 7
(c) Correlation coefficient (r) between them (r between x series and y series)
Regression Coefficient of y on x (byx) = -3/2, and Regression Coefficient of x on y (bxy) = -1/6
Geometric mean of the two Reg. coefficients (byx & bxy) is equal to the correlation coefficient (r).
r=√
=√
.√
=√
=-√
= - = -0.50
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(d) Ratio of Standard Deviation of the two variables (series of data)
bxy = r
- =- .
=
;
3. Computation of trend values without graphSales Total of Average of each
Year
(Rs.‘000) each half half/Semi Average
Operation: Yc = ̅̅̅ +
̅
:
=3:1
:
or
=1:3
̅
(t-
)
Trend
Values
43 + 1.3 (1996 – 1997) = 43+1.3(-1)
41.7
43 + 1.3 (1997 – 1997) = 43+1.3(0)
43.0
44
43 + 1.3 (1998 – 1997) = 43+1.3(1)
44.3
1999
43
43 + 1.3 (1999 – 1997) = 43+1.3(2)
45.6
2000
47
43 + 1.3 (2000 – 1997) = 43+1.3(3)
46.9
2001
51
43 + 1.3 (2001 – 1997) = 43+1.3(4)
48.2
1996
40
1997
45
1998
129
129/3 = 43
141
141/3 = 47
There are 6 years in this time series. Therefore, these 6 years has been divided into two parts of 3
years each, (i.e. 1996 to 1998 and 1999 to 2001) .
̅̅̅ = 43,
Therefore,
̅̅̅ = 47,
The Value of Annual Increase =
̅
̅
t1 = 1997,
=
=
t2 = 2000
= 1.̅
The Equation of the Trend Line:
Yc = ̅̅̅ +
̅
= 43 +
̅
(t-
)
(t – 1997)
= 43 + (t – 1997)
= 43 + 1. ̅ (t – 1997)
Here, t stands for respective year
4
Alternative: Determination of Trend-Line with the Help of Graph49Y
Y
Sales (Rs.’000)
47
45
1. 0 - X = X-Axis to Show Periods (Years)
2. 0 – Y = Y-Axis to Show Values (Sales)
43
3. 1 Unit on Y-Axis = Rs. 2000
4. 1 Unit on X-Axis = 1 Year
41
0
Year
X
The semi-average of the first half (̅̅̅ = 43) has been plotted against the mid-point (t1) of the first
half of the corresponding period of the first half (1997) and the semi-average of the second half
(̅̅̅ = 47) has been plotted against the mid-point (t2) of the corresponding period of second half
(2000).
By joining these two points, we have found-out the required trend-line as shown in the graph.
4. (i) Calculation of Trend Values:
Yc = a + bx
(Equation for Calculating trend values by the method of Least Square)
Where, Yc = Calculated Value of Y,
a = Intercept of the line on y-axis when value of x (time) is Zero,
b = Slope of the Line
Production Deviation
Operation to Calculate:
Final
2
Year (in ‘000 ton) from 1983
xy
x
Trend-Values
Trend
(y)
(x)
Yc = a + bx
Values
1979
70
-4
-280
16 88.79 + 3.51 (-4) = 88.79 – 14.04 74.75
1981
85
-2
-170
4 88.79 + 3.51 (-2) = 88.79 – 07.02 81.77
1982
94
-1
-94
1 88.79 + 3.51 (-1) = 88.79 – 03.51 85.28
1983
83
0
0
0 88.79 + 3.51 (0) = 88.79 + 00.00 88.79
1984
90
1
90
1 88.79 + 3.51 (1) = 88.79 + 03.51 92.30
1985
100
2
200
4 88.79 + 3.51 (2) = 88.79 + 07.02 95.81
1986
96
3
288
9 88.79 + 3.51 (3) =88.79 + 10.53
99.32
2
N=7
∑y = 618
∑x = -1 ∑xy = 34 ∑x = 35
Total = 618.02 ~ 618
∑y = Na + b∑x ----- (i)
∑xy = a∑x + b∑x2 ----- (ii)
Substituting the values of N, ∑y, ∑x, ∑xy and ∑x2 in the above equations, we get
5
618 = 7 (a) + -1(b) ----- (i) or
34 = -1 (a) + 35 (b) ----- (ii) or
618 = 7a – b ----- (i)
34 = - a + 35b
----- (ii)
238 = - 7a + 245b ----- (iii) ..Multiplied Equation (ii) with 7
– b ----- (i)
618 = 7a
238 = - 7a + 245b ----- (iii)
856 = 244b
b=
….. {Equation (i) and Equation (iii) added}
= 3.508 ~ 3.51
Substituting the value of ‘b’ in equation (i)
618 = 7a – b ----- (i)
618 = 7a – 3.51
7a = 618 + 3.51
a =
= 88.787 ~ 88.79
Production (in Thousand Ton)
The trend values have been calculated in the second last column of the table and they (trendvalues) have been shown in the last column of the above table.
Y
False
Base
Li n e
X
4. (ii) The ‘Annual’ increase in the production of coal:
b = 3.51,
Monthly Increase = Annual Increase/12 = 3.51/12 = 0.2925 Ton ~ 2 Quintal, 9.250 Kgs.
6
5 (a)
Table for Chances
Person A
B
Chance
Against
8
4
Favour
5
3
Total 13
7
Table for Probability
Person A
B
Prob.
Against
.
.
Favour
.
.
Case-I: Person ‘A’ live but Persons ‘B’ died
P(i) = Probability of Person ‘A’ live and Probability of Person ‘B’ died
= P(A) ∩ P ( ̅ ) = P(A) . P( ̅ ) = . =
Case-II: Person ‘B’ live but Persons ‘A’ died
P(ii) = Probability of Person ‘B’ live and Probability of Person ‘A’ died
= P(B) ∩ P ( ̅) = P(B) . P( ̅) = . . =
Case-III: Both the Persons (A & B) live
P(iii) = Probability of Person ‘A’ live and Probability of Person ‘B’ died
= P(A) ∩ P (B) = P(A) . P(B) = . =
Prob. of at least one person would be alive 30 years hence
= Prob. of Case-I + Prob. of Case-II + Prob. of Case-III
Required Probability = P(i) + P(ii) + P(iii) =
+
+
=
5 (b) There are 6 letters in the word of ‘Jaipur’ and we have to formulate different words with
the letters of ‘JAIPUR’.
In this way,
n
n = 6, r = 6;
Pr = 6P6 = (
)
=
=
= 6! = 6.5.4.3.2.1 = 720
i. Each word begins with ‘J’
Therefore, the first place of 6 letters would be occupied by ‘J’
J
After filling the first place with ‘J’ there are remaining 5 letters (i.e. AIPUR).
These 5 letters (AIPUR) can be arranged in following ways. In this way,
(n-1)
P(r-1) = (6-1)P(6-1) = 5P5 = (
)
=
=
= 5! = 5.4.3.2.1 = 120 Ways
ii. Arrangement of all letters of word ‘Jaipur’ without caring any rule
– Arrangement of all letters of word ‘Jaipur’ following Case-I (letters begins with ‘J’)
6
P6 - 5P5 = (
)
-(
)
=(
)
-
=
-
= 6! – 5! = (6.5.4.3.2.1) – (5.4.3.2.1) = 720 – 120 = 600 Ways
7
6.
Balls in the Urns
Urns
I
II
Total
1
2
3
2
1
3
Color of Balls
Black
White
A ball is drawn from the first urn and to be
put in urn second may be black or white.
Case-i: Let the drawn ball is black. Then,
Probability of drawing a black ball from Urn-I =
=
After drawing a black ball from Urn-I and putting it into Urn-II, the number of balls in both
the Urns would be as shown bellow –
Now, there are 3 Black and 1white balls
Urns
I
II
Color of Balls
in Urn-II.
Black
1 – 1 = 0 2 + 1 = 3 The probability of drawing a white ball
White
2
1
from Urn-II=
=
Total
2
4
Therefore, Probability of this compound event = . =
Case-i: Let the drawn ball is White. Then,
Probability of drawing a white ball from Urn-I =
=
After drawing a white ball from Urn-I and putting it into Urn-II, the number of balls in both
the Urns would be as shown bellow –
Now, there are 2 Black and 2white balls
Urns
I
II
Color of Balls
in Urn-II.
Black
1
2
The probability of drawing a white ball
White
2 – 1 = 1 1 + 1 = 2 from Urn-II=
=
Total
2
4
Therefore, Probability of this compound event = . =
Required Probability = Prob. of Comp. Event of Case-I + Prob. of Comp. Event of Case-II
=
7.
+
=
….. (It is proved)
What is the concept of business forecasting?
Write Meaning, two definitions of authors then a definition in your own words.
Explain the methods of business forecasting.
Explain briefly any three methods of forecasting from the following,
a. Simple and Naïve Method
8
b. Trend Projection (Time Series Method)
c. Methods used in Predicting Business Cycles
d. Causal / Econometric Forecasting Method
e. Qualitative vs. Quantitative Method
f. Judgment Method
g. Artificial Intelligence Method
h. Other Methods
8. Write a detail note on-
(i) CSO,
(ii) NSSO
CSO:
Write about 50 words for each of the following mentioned points, i.e.
Meaning,
History,
Organization,
Functions and
Conclusion
NSSO:
Write about 50 words for each of the following mentioned points, i.e.
Meaning,
History,
Divisions,
Objectives,
Functions, and
Conclusion
Jamaluddeen
Assistant Professor (Ad-hoc)
Department of Commerce
Gugu Ghasidas Vishwavidyalaya
Central University, Bilaspur - 495009
Chhattisgarh, India
9