Course : US04FBCA01
(Computer Based Numerical and Statistical Methods)
Question Bank
Unit-1
1
MCQ
f(a) < 0, f(b) > 0 and if x0 Є (a,b)is first approximation with f(x0) < 0 then in
bisection method,
(a) x0 is to be replaced by a (b) ais to be replaced by c
(c) bis to be replaced by x0 (d) x0 is to be replaced by b
2
For real root of an equation x3-2x-5=0, the root lies between
(a) 0 and 1 (b) 2 and 3 (c) 1 and 2 (d)none of them
3
From the following _______ method is not iterative method.
(a) False position (b) Bisection (c) Lagranges (d)none of them
4
For the function f(x):x3-2x-5 = 0 if the root of equation lies between (2,3) and if at ith
iteration c=2.5 then next approximation by bisection method is given c=
3+ 2.75
2  2.5
3+ 2.5
(a)
(b)
(c)
(d) none of them
2
2
2
5
If in a method of successive approximation, the root of equation lies between 1 and 2,
1
g  x  2
, and initial guess is 1.25 then next approximation is
x 1
(a) 0.5625 (b) 1.2177 (c) 1.7777 (d)none of them
6
From the following _______ method is the best method to obtain root of equation
f(x)=0.
(a) False position (b) Bisection (c) Newton’s Raphson (d)none of them
7
True error is defined as
(a) Present Approximation – Previous Approximation
(b) True Value – Approximate Value
(c) abs (True Value – Approximate Value)
(d) abs (Present Approximation – Previous Approximation)
8
The number 0.01850 x 103 has _000______ significant digits
(a) 3 (b) 4
(c) 5
(d) 6
9
For an equation like x2=0, a root exists at X=0. The bisection method cannot be
adopted to solve this equation in spite of the root existing at x=0 because the function
f(x) =x2
(a) is a polynomial
(c) is always non-negative
10
(b) has repeated roots at x= 0
(d) has a slope equal to zero at x= 0
If for a real continuous function f(x), f(a)f(b)<0, then in the range of [a,b] for f(x)=0,
there is (are)
(a) one root
(c) no root
(b) an undeterminable number of roots
(d) at least one root
Answers of MCQ
1.(b), 2.(b), 3.(c), 4.(b), 5.(c), 6.(c), 7.(b), 8.(d), 9.(c), 10.(d)
Short questions
1
2
3
4
5
6
1
2
3
4
5
6
7
8
9
Define Relative error and absolute error.
Describe the stopping rules to obtain approximate solution for given non-linear
equations.
Use the False Position method to obtain approximate solution of the equation X39x+1=0.
Use the secant method to obtain approximate solution of the equation X3-5x-3=0.
[initial approx. 2 & 3 ans: 0.656]
Use the Successive approximation method to obtain approximate solution of the
equation 2x-x-3=0 with interval [-3,-2] and correct to four decimal places. [initial
approx. -3
Long Questions
Write algorithm for bisection method.
Write algorithm for RegulaFalsi method.
Write algorithm for iterative method.
Define Relative error and absolute error.
Bisection method
2. x3  x 2  1  0 3. x3  4 x  9  0 4. x3  x  1  0 5. x3  2 x  5  0
RegulaFalsi OR False position method
2. x3  x  1  0 3. x3  4 x  9  0 4. x3  x  4  0
5. x3  2 x  5  0
Secant method
3. x3  2 x  5  0 4. x  12 5. x3  x  4  0
Iterative method
3. 2 x  x  3  0 Take a=-3
5. x3  x 2  1  0 Take a=1.5
Unit 2
1
MCQ
Theile defines _______as “the art of reading between the lines of a table.”
(a) Extrapolation (b) Divided difference (c) Lagrange’s (d) Interpolation
2
All the formulae of interpolation are based on the fundamental assumption that the given data
can be expressed as a _______.
(a) Polynomial (b) Equation (c) Algorithm (d) None of the above
3
__________ method is used if the estimated value lies towards the end of the difference table.
(a) Divided difference (b) Forward difference (c) Backward difference (d) None of the above
4
y depends on x can be written as _________
(a) f(x) or yx (b) f(xy) (c) f(yx) (d) none of the above
5
_________ method of interpolation is used for unequally spaced functions.
(a) Forward difference (b) backward difference
(c) Newton’s divided difference(d) None of the above
6
To estimate the value of dependent variable x for a given value of independent variable y, the
process is known as ___________.
(a) Extrapolation (b) Inverse Interpolation (c) Interpolation (d) polynomial
7
_______ is called the forward difference operator.
(a)
(b) (c) (d)
8
_______ is called the backward difference operator.
(a)
(b) (c) (d)
9
The main disadvantage of Lagrangian interpolation is that it is difficult to find the order of the
________ to be fitted.
(a) Polynomial (b) Equation (c) Algorithm (d) None of the above
10 _______________ is not a type of interpolation method.
(a) Forward difference (b) backward difference
(c) Newton’s divided difference(d) moving average method
11 The formula for inverse interpolation is obtained from _______________interpolation
formula by changing the variable x and y=f(x).
(a) Forward difference (b) backward difference
(c) Newton’s divided difference(d) Lagrangian
12 The second order differences is denoted by ____________.
(a)
(b)
(c)
(d)
13 If population census for the years 1931, 1941, 1951, 1961 and 1971 is given and if we want to
estimate the population for the year 1935 then __________________ method is used.
(a) Forward difference (b) backward difference
(c) Newton’s divided difference(d) Lagrangian
14 If the argument for the interpolation is not an equidistant then which method of interpolation
is most appropriate.
(a) Forward difference (b) backward difference
(c) Newton’s divided difference(d) all of the above
15 The estimate obtained by interpolation method is always accurate or reliable.
(a) true (b) false (c) Not applicable
Answers of MCQ:
1.(d),
2.(a), 3.(c),
4.(a), 5.(c), 6.(b),
11.(d), 12.(b), 13.(a), 14.(c), 15.(b)
1
2
3
4
5
6
7
8
Short Questions
Define Interpolation.
Define Extrapolation.
Write algorithm for forward difference table.
Write algorithm for Backward difference table.
Write different methods of Interpolation.
Draw forward difference table.
Draw backward difference table.
Explain forward difference method.
7.(a),
8.(b), 9.(a),
10.(d),
9
10
11
12
1
2
3
4
Explain backward difference method
Explain central difference method.
Explain divided difference method.
Explain Lagrangian method.
Long Questions
Example on Interpolation
Forward and Backward Difference
If y= 2x3 – x2 + 3x +1, calculate the value of y corresponding to x = 0, 1, 2, 3, 4, 5 and form
the table of differences.
Estimate the expectation of life at the age of 16 years by using the following data:
Ans: 31.7 years
Age (In Years)
10
15
20
25
30
35
Expectation of life (In Year)
35.4 32.3 29.2 26.0 23.2 20.4
The following results are given. Using them find
Ans: 2.962
Using an appropriate formula for interpolation estimate the no. of students who obtained < 45
marks from the following. Ans: 48
Marks
0-40
40-50
50-60
60-70
70-80
No. of students 31
42
51
35
31
5
From the following table find the no. of workers falling in the earning group of Rs. 25 to Rs.
35. Ans: y35 – y25 = 396 – 218 = 178
Earning
Up to 10 Up to 20 Up to 30 Up to 40 Up to 50 Up to 60
(Rs.)
No. of
50
150
300
500
700
800
Workers
6
If Lx represents the numbers living at age x in a life table interpolate, by using newton’s
method of Lx for the values of x = 24 and x= 29. ( Ans: L24 = 486 , L29 = 447)
L20 = 512, L30= 439, L40 = 346, L50 = 243
7
The following table gives the census population of a town for the years 1931 to 1971.
Estimate the population for the year 1965 by using an appropriate interpolation formula.
Ans: 96.834
Year
19931
1941
1951
1961
1971
Population
46
66
81
93
101
Estimate by Newton’s method of interpolation, the expectation of life at age 32 from the
following table. Ans: 22.0948
Age (In Years)
10
15
20
25
30
35
Expectation of life (In Year)
35.3 32.4 29.2 26.1 23.2 20.5
9 By Newton’s or by any other algebraic method find the no. of persons who probably will be
travelling if rate is 4.2. Ans : 49,712
Rate
5.0
4.5
4.0
3.5
3.0
Passenge 30,000 40,000 60,000 1,00,00 1,50,00
rs
0
0
10 Given the table of values as
x
2.0
2.25
2.50
2.75
3.0
y(x)
9.00
10.06
11.25
12.56
14.00
Find y(2.35). Ans: 10.522
11 Given the table of values as
X
2.5
3.0
3.5
4.0
4.5
y(x)
9.75
12.45
15.70
19.52
23.75
8
Find y(2.35). Ans: 21.601
Divided Difference Method
12 The observed values of a function are respectively 168, 120, 72, 63 at the four positions 3, 5,
9, 10 of the independent variable. What is the best estimate you can give for the value of the
function at the position 6 of the independent variable? Ans: 147
13 From the following table find the function f(x) assuming it to be a polynomial of 3 rd degree in
x. Ans: x3 + x2 – x + 1
x
0
1
2
3
f(x)
1
2
11
34
14 Apply Newton’s divided difference method to find the no. of persons getting Rs. 6 from the
following data. Ans: 135
Income per day
3
5
7
8
10
No. of persons
180
154
120
110
90
15 Given the table of values as (use divided difference method)
x
2.5
3.0
3.5
4.75
y(x)
8.85
11.45
20.66
22.85
Find y(2.35). Ans: 13.993
6.0
38.60
Lagrangian method
16 Given the table of values as (use Lagrangian method)
x
0
1
2
3
y(x)
0
2
8
27
Find y(2.35). Ans: 15.313
Find
the polynomial function f(x) given that f(0) = 2, f(1) = 2 , f(2) = 12 and f(3)= 35 find
17
f(5). Ans: 147
18 By using Lagrange’s method, estimate the no. of persons whose income is Rs. 19 and more
but does not exceed Rs. 25 from the following table.Ans: y25 – y19 = 228– 120 = 108
Income in
1 and not
10 and not
20 and not
29 and not
38 and not
Rs.
exceeding 9 exceeding
exceeding
exceeding
exceeding
19
28
37
46
No. of
50
70
203
406
304
persons
19 Given a function in the form of a table as
x
2.0
3.0
4.0
y(x)
6.6
9.2
8.6
Interpolate the value of y(x) using Lagrangian polynomial at x= 2.8 and x= 3.
20 Inverse Interpolation
The value of x and y= f(x) are given below
x
5
6
9
f(x)
12
13
14
Find value of x when f(x)= 15. Ans: 11.5
21 Explain Interpolation and extrapolation.
22 Explain Lagrangian method.
23 Explain forward difference and backward difference methods.
24 Explain divided and central difference method.
Unit 3
11
16
1
MCQ
We can find solution of system of linear, algebraic equations using………
(a) Newton-Raphson method
(b) Bisection method
(c) Gauss-Seidel method
(d) None of these
2
The system of linear equation AX = B can be solved by matrix inversion method only if…
(a) A  0
(b) A  0
(c) A = 0 (d) A is symmetric
3
The system of linear equation AX = B can be solved by Gauss-Seidel method only if…
(a) All diagonal elements of A are zero
(b) All diagonal elements of A are non zero
(c) All diagonal elements of A are dominant (d) A is skew symmetric
4
The system of linear equation AX = B is said to be homogeneous if…
(a) A  0
(b) B = 0
(c) A = 0 (d) A is symmetric
5
The system of linear equation AX = B is said to be non-homogeneous if…
(a) B  0
(b) B = 0
(c) A  0 (d) A is symmetric
6
Rate of change of distance with respect to time reprents….
(a) Acceleration
(b) Speed
(c) Pressure (d) None of these
7
For equally spaced tabular values, the Newton’s forward interpolation formula can be
used to find derivative at a point x if…..
(a) x is in the upper half of the table
(b) x is in the lower half of the table
(c) x is at the center of the table
(d) None of these
8
Consider the following system of linear equation
3x + y – z = 10, x + 5y + 2z = 18, x + 4 y + 9z = 16
If current approximation is x = 3.33, y = 2.93, z= 0.1, then the Gauss-Seidel method will
give next approximation as x = 2.39, y = 3.08 and z =
(a) 1.2
(b) 0.21
(c) 0.14
(d) 0.19
9
Consider the following system of linear equation
3x + y + z = 0, x + 4y – z = 1, 2x – y + 5z = 2
If current approximation is x = 0, y = 0.25, z= 0.45, then the Gauss-Seidel method will
give next approximation as x = – 0.23, y = 0.42 and z =
(a) 0.58
(b) 0.48
(c) 0.7
(d) 0.24
Answers of MCQ:
1(c), 2(b), 3(c), 4(b), 5(a), 6(b), 7(a), 8(a), 9(c)
1
Short Questions
Solve the following system of equations.
(a)
2x + 3y = –10
(b)
10x + 3y = 5
–x + 4y = –4
8x – 2y = 2
(c)
2
3
2x + 3y = 1
–x + 2y = 8
(d)
2x – y = 4
x + 9y = –8
List only various direct and iterative methods.
If x lies in the upper half of the table and if xk  x  xk 1 , then what is
dy ( x )
and
dx
d 2 y ( x)
?
dx 2
4
5
6
1
dy ( x )
d 2 y ( x)
and
?
dx
dx 2
dy ( x )
If x lies in the lower half of the table and if xk 1  x  xk , then what is
and
dx
d 2 y ( x)
?
dx 2
dy ( x )
d 2 y ( x)
If x lies in the lower half of the table and if x  xk , then what is
and
?
dx
dx 2
If x lies in the upper half of the table and if x  xk , then what is
Long Questions
Solve the following system of equations using matrix inversion method. (can be asked to
write in matrix form as short question)
2 x1  8 x2  2 x3  14
2 x1  2 x2  5 x3  13
2 x1  x2  x3  10
(i) x1  6 x2  x3  13
2 x1  x2  2 x3  5
2 x1  3x2  4 x3  20
(iv) 4 x1  2 x2  3x3  17
2
2 x1  3x2  20 x3  25
6
7
3x1  x2  3x3  10
x1  4 x2  9 x3  16
3x1  x2  x3  8
(v) 2 x1  3x2  2 x3  5
x1  4 x2  2 x3  17
7 x1  2 x2  5 x3  0
Solve the following system of equations using Gauss-Seidel method.
20 x1  x2  2 x3  17
10 x1  x2  2 x3  44
5 x1  2 x2  x3  12
(i) 3x1  20 x2  x3  18
3
4
5
(ii) 2 x1  3x2  4 x3  20 (iii) 3 x1  2 x2  3 x3  18
(ii) 2 x1  10 x2  x3  51 (iii) x1  4 x2  2 x3  15
x1  2 x2  10 x3  61
8 x1  2 x2  2 x3  8
x1  2 x2  x3  0
(iv) x1  8 x2  3x3  4
(v) 3x1  x2  x3  0
x1  2 x2  5 x3  20
vi)
2 x1  x2  9 x3  12
x1  x2  4 x3  3
Explain the matrix inversion method for solution of system of linear equations.
Explain the Gauss-Seidel method for solution of system of linear equations.
Write the comparison between direct and iterative methods for solution of system of
linear equations.
Given the following table
x
0.50
0.75
1.00
1.25
1.50
y  f ( x)
0.13
0.42
1.00
1.95
2.35
Find f (0.75) and f (0.85) .
The distance (s) covered as a function of time (t) by an athlete during his/her run for the
50 meter race is given in the following table
Time(Secs.)
0
1
2
3
4
5
6
Distance(Mts.)
0
2.5
8.5
15.5
24.5
36.5
50
Determine the speed of the athlete at t = 5 and t = 4.5 seconds.
8
The distance (s) covered by a car in given time (t) is given in the following table
Time(Minutes)
10
12
14
16
18
Distance(km)
12
15
20
27
37
Determine the speed of the car at t = 13 minutes.
9
The distance (s) covered by a car in given time (t) is given in the following table
Time(Miutes)
10
12
14
16
Distance(km)
12
15
20
27
Determine the acceleration of the car at t = 13 minutes.
10
Given the following table
0
1
x
y
0
2
2
2.75
Use forward difference formula to compute
11
Given the following table
0
1
x
y
0
2
2
2.75
Use backward difference formula to compute
3
3.4
18
37
4
3.2
5
2.5
6
2.2
4
3.2
5
2.5
6
2.2
dy
at 2.25.
dx
3
3.4
dy
at 6.
dx
12
The pressure of a certain gas was recorded at intervals of two seconds, and the results are
listed in the following table:
Time(t)
0
2
4
6
8
Pressure(p)
50
53
57
68
82
Estimate the rate of change of pressure at t = 2.5.
13
Given the following table
3
4
x
y
4
6.6
5
7.7
Use backward difference formula to compute
14
Given the following table
10
11
x
y
15
12.8
Estimate the value of
12
10.6
6
9.0
7
10.5
8
12.2
dy
at x = 7 and x = 7.5.
dx
13
8.5
14
6.4
dy
d2y
and
at 10, 10.5, 11, 11.2 using the forward difference
dx
dx 2
formula.
Unit 4
1
MCQ
Forecasts ______________________________________.
a. become more accurate with longer time horizons
b. are rarely perfect
c. are more accurate for individual items than for groups of items
d. all of the above
2
One purpose of short-range forecasts is to determine __________________.
a. production planning
b. inventory budgets
c. facility location
d. job assignments
3
Forecasts are usually classified by time horizon into three categories ______________.
a. short-range, medium-range, and long-range
b. finance/accounting, marketing, and operations
c. strategic, tactical, and operational
d. exponential smoothing, regression, and time series
4
A forecast with a time horizon of about 3 months to 3 years is typically called a ________.
a. long-range forecast
b. medium-range forecast
c. short-range forecast
d. weather forecast
5
Gradual, long-term movement in time-series data is called _____________________.
a. seasonal variation
b. cycles
c. trends
d. exponential variation
6
A time series is a set of data recorded_________________.
a. periodically
b. at time or space intervals
c. at successive points of time
d. all the above
7
The time series analysis helps:
a. to compare the two or more series
b. to know the behavior of business
c. to make predictions
d. all the above
8
A time series consists of:
a. two components
b. three components
c. four components
d. five components
9
The component of a time series attached to long-term variations is term as:
a. Cyclic Variations
b. Secular Trend
c. Irregular Variation
d. all of the above
10
The component of a time series which is attached to short-term fluctuations is:
a. seasonal variation
b. cyclic variation
c. irregular variation
d. all of the above
11
A lock-out in a factory for a month is associated with the component of a time series:
a. irregular movement
b. secular trend
c. cyclic variation
d. none of the above
The general decline in sales of cotton clothes is attached to the component of the time series:
a. secular trend
b. cyclical variation
c. seasonal variation
d. all of the above
12
13
The sales of departmental store on Dushera and Diwali are associated with the component of
a time series:
a. secular trend
b. seasonal variation
c. irregular variation
d. all the above
14
The consistent increase in production of cereals constitutes the component of a time series:
a. secular trend
b. seasonal variation
c. cyclical variation
d. all the above
15
Secular trend is indicative of long-term variation towards:
a. increase only
b. decrease only
c. either increase or decrease
d. none of the above
16
Linear trend of a time series indicates towards:
a. constant rate of change
b. constant rate of growth
c. change in geometric progression
d. all the above
17
Seasonal variation means the variation occurring within:
a. a number of years
b. parts of a year
c. parts of a month
d. none of the above
18
Salient factors responsible for seasonal variation are:
a. weather
b. social customs
c. festivals
d. all the above
19
Cyclic variations in a time series are caused by:
a. lockouts in a factory
b. war in a country
c. floods in the states
d. none of the above
20
Irregular variations in a time series are caused by:
a. lockouts and strikes
b. epidemics
c. floods
d. all the above
21
In ratio to moving average method for seasonal indices, the ratio of an observed value to the
moving average remove the influence of:
a. trend
b. cyclic variation
c. trend and cyclic variation both
d. none of these
22
The moving averages in a time series are free from the influences of:
a. seasonal and cyclic variations
b. seasonal and irregular variations
c. trend and cyclical variations
d. trend and random variations
Answers of MCQ
1.(b),
2.(d),
3.(a),
11.(a), 12.(a), 13.(b),
21.(c),
22.(b)
4.(b),
5.(c),
14.(a), 15.(c),
6.(d),
7.(d), 8.(c), 9.(b), 10.(d),
16.(a), 17.(b), 18.(d), 19.(d), 20.(d),
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Short Questions
What is Time Series?
What do you mean by Secular trend?
Differentiate between linear and nonlinear trend.
What time series analysis consists?
List the utilities of time series.
List the component of Time series.
What do you mean by Cyclic Variation?
What do you mean by random or irregular Variation?
Write objectives for studying seasonal pattern in time series.
Write limitation of Simple Average Method.
Write merits and demerits of ‘Ratio to Moving Average’ method.
Write a note on Forecasting by the use of Time Series Analysis.
Write a note on Casual Model of Forecasting.
Write a note on Survey Method of Forecasting.
1
Long Questions
Calculate three yearly moving averages for the following data
YEAR
Y
2
1950
242
1951
250
1952
252
1953
249
1954
253
1955
255
1956
251
1957
257
1958
260
1959
265
1960
262
Calculate the trend values by the method of moving average, assuming a four-yearly cycle
from the following data relating to sugar production in India.
YEAR
1971
1972
1973
1974
1975
1976
SUGAR PRODUCTION
(lakh tones)
37.4
31.1
38.7
39.5
47.9
42.6
YEAR
1977
1978
1979
1980
1981
1982
SUGAR PRODUCTION
(lakh tones)
48.4
64.4
58.4
38.6
51.4
84.4
31. Obtain the trend from the time series given below by method of moving average of [i] 3 years
[ii] 5 years
Year
1961 1962 1963 1964 1965 1966 1967 1968 1969 1970
Y
500 540 550 530 520 560 600 640 620 640
42. Obtain the trend from the time series given below by method of moving average of 4 years
Year
1958 1959 1960 1961 1962 1963 1964 1965 1966 1967
Y
50.0 36.5 43.0 44.5 38.9 38.1 32.6 41.7 41.1 33.8
53. Obtain seasonal indices using simple average method.
Year summer monsoon Autumn Winter
1
30
81
62
119
2
33
104
86
171
3
42
153
99
221
4
56
172
129
235
5
67
201
136
302
64. Obtain seasonal indices using simple average method.
Year Q-1 Q-II Q-III Q-IV
1974 30 81
62
119
1975 33 104 86
171
1976 42 153 99
221
1977 56 172 129 235
75. Obtain seasonal indices using ratio to moving average.
6.
Year Q-1 Q-II Q-III Q-IV
1970 25 30
21
32
1971 27 28
25
34
1972 22 27
21
30
1973 24 25
20
33
7.
8.
89. Obtain seasonal indices using ratio to moving average.
10.
Year Q-1 Q-II Q-III Q-IV
1990 30 40
36
34
1991 34 52
50
44
1992 40 58
54
48
1993 54 76
68
62
9 Use the method of monthly averages to determine the monthly indices for the following data
of production of a commodity for the year 1979, 1980, 1981.
1979
1980
1981
MONTH
(Production in lakhs of tones)
JANUARY
12
15
16
FEBRUARY
11
14
15
MARCH
10
13
14
APRIL
14
16
16
MAY
15
16
15
JUNE
15
15
17
JULY
16
17
16
AUGUST
13
12
13
SEPTEMBER
11
13
10
OCTOBER
10
12
10
NOVEMBER
12
13
11
DECEMBER
15
14
15
10
Calculate seasonal indices by ‘ratio to moving average method’ for the following data.
YEAR
I Quarter
II Quarter
III Quarter
IV Quarter
1971
68
61
61
63
1972
65
58
66
61
1973
11
12
13
14
15
68
63
63
67
Calculate the seasonal indices by the ‘ratio to moving average’ method from the following.
Year
Quarter
Y
4-Quarterly Moving Average
I
75
II
60
1972
III
54
63.375
IV
59
65.375
I
86
67.125
II
65
70.875
1973
III
63
74.000
IV
80
75.375
I
90
76.625
II
72
77.625
1974
III
66
79.500
IV
85
81.500
I
100
83.000
II
78
84.750
1975
III
72
IV
93
-
The number (in hundreds) of letters posted in certain city on each day in a typical period of
five weeks was as follows. Find the seasonal indices by simple average method.
Week
Sun.
Mon.
Tue.
Wed.
Thu.
Fri.
Sat.
1st
18
161
170
164
153
181
76
nd
2
18
165
179
157
168
195
85
3rd
21
162
169
153
139
185
82
th
4
24
171
182
170
162
179
95
5th
27
162
186
170
170
182
120
Compute the seasonal average and seasonal indices for the following time-series.
Month
1974
1975
1976 Month
1974
1975
1976
JANUARY
JULY
15
23
25
20
22
30
FEBRUARY
AUGUST
16
22
25
28
28
34
MARCH
SEPTEMBER
18
28
35
29
32
38
APRIL
OCTOBER
18
27
36
33
37
47
MAY
NOVEMBER
23
31
36
33
34
41
JUNE
DECEMBER
23
28
30
38
44
53
Calculate the seasonal index from the following data using the simpleaverage method.
Year
1st Quarter
2nd Quarter
3rd Quarter
4th Quarter
1974
72
68
80
70
1975
76
70
82
74
1976
74
66
84
80
1977
74
74
84
78
1978
78
74
86
82
Calculate the seasonal index number for each of the four quarters for following data.
Year
1st Quarter
2nd Quarter
3rd Quarter
4th Quarter
1980
106
124
104
90
1981
84
114
107
88
1982
90
112
101
85
1983
76
94
91
76
1984
80
104
95
83
16
17
18
19
20
21
22
23
1985
104
112
102
84
Given the following quarterly sales figures in thousands of rupees for the years 1966-1969,
find the specific seasonal by the method of moving averages.
Year
I
II
III
IV
1966
290
280
285
310
1967
320
305
310
330
1968
340
321
320
340
1969
370
360
362
380
What is time series? Explain various components of time series.
Explain/Write steps of Moving Average method with its merits and demerits.
Explain/Write steps of Simple Average method.
Explain/Write steps of ‘Ratio to moving Average’ Method.
List Various Forecasting models. Explain any three of them.
Explain Extrapolation model in forecasting.
Explain Exponential smoothing model in forecasting.
Write algorithm for forward difference table.
Begin
For j=1 to (n-1) by 1 do
For i=1 to (n-j) by 1 do
If (j=1) then
Set dij = yi+1 – y i
else
Set dij = d(i+1)(j-1) – di (j-1)
Endif
Endfor
Endfor
End
Write algorithm for Backward difference table.
Begin
For j=1 to (n-1) by 1 do
For i = (j+1) to n by 1 do
If (j=1) then
Set dij = y i - yi+1
else
Set dij = di (j-1) - d(i+1)( j-1)
Endif
Endfor
Endfor
End
Write different methods of Interpolation.
Methods for equally spaced functions
 Newton’s forward interpolation formula
 Newton’s backward interpolation formula
 Gauss’s formula
 Stirling’s formula
 Bessel’s formula
 Laplace – Everett’s formula
Method for unequally spaced functions
 Lagrangian interpolation
 Newton’s divided difference interpolation formula
Define Extrapolation.
The process of estimating the value of independent variable y for a given value of x outside the range
x1 ≤ x ≤x n is known as extrapolation.
Write limitation of Simple Average Method.
The method of simple average, though very simple to apply gives only approximate estimates of the
pattern of seasonal variations in the series.
It assumes that the data do not contain any trend and cyclical fluctuations at all or their effect on the
time series value is not quite significant. This is very serious limitation since most of the economic
and business time series exhibit definite trends and are affected to a great extent by cycles.
Accordingly, the indices obtained by this method do not truly represent the seasonal swings in the
data because they include the influence of trend and cyclic variations also.
This method tries to eliminate the random or irregular component by averaging the monthly values
over different years.
In order to arrive at any meaningful seasonal indices, first of all trend effects should be eliminated
from the given values.
Write merits and demerits of ‘Ratio to Moving Average’ method.
‘Ratio to Moving Average’ is the most satisfactory and widely used method for estimating the
seasonal fluctuations in a time series because it includes both trend and cyclical components from
the indices of seasonal variations.
However, it should be kept in mind that it will give true seasonal indices provided the cyclical
fluctuations are regular in periodicity as well as amplitude.
A drawback of this method is that there is loss of some trend values in the beginning and at the end
and accordingly seasonal indices for first six months of the first year and last six months of the last
year cannot be determined.
Write objectives for studying seasonal pattern in time series.
Some major purposes of the statistical analysis of time
series are:
 To understand the variability of the time series.
 To identify the regular and irregular fluctuation of the time series.
 To describe the characteristics of these fluctuation
 To understand the physical processes that give rise to each of these fluctuation
SECULAR TREND MEANS LONG TERM TREND
Differentiate between linear and nonlinear trend.
 If the time series values plotted on graph lies nearer to a straight line, the trend exhibited by the time
series is termed as Linear otherwise Non-Linear.
 In straight line trend, the time series values increases of decreases more or less by a constant absolute
amount, i.e., the rate of growth is constant.
 Although, in practice, linear trend is commonly used, it is rarely observed in economic and business
data.
 In an economic and business phenomenon, the rate of growth or decline is not of constant nature but
varies considerably in different sector of time.
 Usually, in the beginning, the growth is slow, then rapid which is further accelerated for quite some
time after which it becomes stationary or stable some period and finally decline slowly.
Linear Trend
Non-Linear Trend