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महाराष्ट्र राज्य शिक्षण मंडळएचएससी वाणिज्य (इंग्रजी माध्यम) इयत्ता १२ वी

Solve the following problem : Fit a trend line to data by the method of least squares. Year 1977 1978 1979 1980 1981 1982 1983 1984 Number of boxes (in ten thousands) 1 0 3 8 10 4 5 8 - Mathematics and Statistics

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प्रश्न

Solve the following problem :

Fit a trend line to data by the method of least squares.

Year 1977 1978 1979 1980 1981 1982 1983 1984
Number of boxes (in ten thousands) 1 0 3 8 10 4 5 8
बेरीज
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उत्तर

In the given problem, n = 8 (even), two middle t – values are 1980 and 1981, h – 1

u = `"t - mean of two middle values"/("h"/2) = ("t" - 1980.5)/(1/2)` = 2(t – 1980.5)

We obtain the following table.

Year
t
No. of boxes (in ten thousands) 
yt
u = 2(t –  1980.5) u2 uyt Trend Value
1977 1 –7 49 –7 1.5836
1978 0 –5 25 0 2.5240
1979 3 –3 9 –9 3.4644
1980 8 –1 1 –8 4.4048
1981 10 1 1 10 5.3452
1982 4 3 9 12 6.2856
1983 5 5 25 25 7.2260
1984 8 7 49 56 8.1664
Total 39 0 168 79  

From the table, n = 8, `sumy_"t" = 39, sumu = 0, sumu^2 = 168,sumuy_"t" = 79`

The two normal equations are: `sumy_"t" = "na"' + "b"' sumu  "and" sumuy_"t", = a'sumu + b'sumu^2`

∴ 39 = 8a' + b'(0)            ...(i)   and
79 = a'(0) + b'(168)         ...(ii)

From (i), a' = `(39)/(8)` = 4.875

From (ii), b' = `(79)/(168)` = 0.4702
∴  The equation of the trend line is yt = a' + b'u
i.e., yt = 4.875 + 0.4702 u, where u = 2(t – 1980.5).

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Measurement of Secular Trend
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 4: Time Series - Miscellaneous Exercise 4 [पृष्ठ ७०]

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बालभारती Mathematics and Statistics 2 (Commerce) [English] Standard 12 Maharashtra State Board
पाठ 4 Time Series
Miscellaneous Exercise 4 | Q 4.11 | पृष्ठ ७०

संबंधित प्रश्‍न

Fit a trend line to the data in Problem 7 by the method of least squares. Also, obtain the trend value for the year 1990.


The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

Year 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976
Production
(Million Barrels)
0 0 1 1 2 3 4 5 6 7 8 9 8 9 10

i. Obtain trend values for the above data using 5-yearly moving averages.
ii. Plot the original time series and trend values obtained above on the same graph.


Choose the correct alternative :

We can use regression line for past data to forecast future data. We then use the line which_______.


The simplest method of measuring trend of time series is ______.


Fill in the blank :

The method of measuring trend of time series using only averages is _______


Fill in the blank :

The complicated but efficient method of measuring trend of time series is _______.


State whether the following is True or False :

Moving average method of finding trend is very complicated and involves several calculations.


Solve the following problem :

The following table shows the production of pig-iron and ferro- alloys (‘000 metric tonnes)

Year 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 0 4 9 9 8 5 4 8 10

Fit a trend line to the above data by graphical method.


Solve the following problem :

Obtain trend values for the following data using 5-yearly moving averages.

Year 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 0 4 9 9 8 5 4 8 10

Solve the following problem :

Following table shows the amount of sugar production (in lac tonnes) for the years 1971 to 1982.

Year 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982
Production 1 0 1 2 3 2 3 6 5 1 4 10

Fit a trend line to the above data by graphical method.


Solve the following problem :

Fit a trend line to data in Problem 4 by the method of least squares.


Solve the following problem :

Obtain trend values for the data in Problem 7 using 4-yearly moving averages.


Solve the following problem :

Obtain trend values for data in Problem 10 using 3-yearly moving averages.


Solve the following problem :

Fit a trend line to data in Problem 13 by the method of least squares.


Solve the following problem :

Obtain trend values for data in Problem 16 using 3-yearly moving averages.


Solve the following problem :

Following tables shows the wheat yield (‘000 tonnes) in India for years 1959 to 1968.

Year 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968
Yield 0 1 2 3 1 0 4 1 2 10

Fit a trend line to the above data by the method of least squares.


The complicated but efficient method of measuring trend of time series is ______


State whether the following statement is True or False:

The secular trend component of time series represents irregular variations


Obtain the trend values for the data, using 3-yearly moving averages

Year 1976 1977 1978 1979 1980 1981
Production 0 4 4 2 6 8
Year 1982 1983 1984 1985 1986  
Production 5 9 4 10 10  

Use the method of least squares to fit a trend line to the data given below. Also, obtain the trend value for the year 1975.

Year 1962 1963 1964 1965 1966 1967 1968 1969
Production
(million barrels)
0 0 1 1 2 3 4 5
Year 1970 1971 1972 1973 1974 1975 1976  
Production
(million barrels)
6 8 9 9 8 7 10  

The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

Year 1962 1963 1964 1965 1966 1967 1968 1969
Production
(million barrels)
0 0 1 1 2 3 4 5
Year 1970 1971 1972 1973 1974 1975 1976  
Production
(million barrels)
6 7 8 9 8 9 10  
  1. Obtain trend values for the above data using 5-yearly moving averages.
  2. Plot the original time series and trend values obtained above on the same graph.

Following table shows the amount of sugar production (in lakh tonnes) for the years 1931 to 1941:

Year Production Year Production
1931 1 1937 8
1932 0 1938 6
1933 1 1939 5
1934 2 1940 1
1935 3 1941 4
1936 2    

Complete the following activity to fit a trend line by method of least squares:


The following table shows gross capital information (in Crore ₹) for years 1966 to 1975:

Years 1966 1967 1968 1969 1970
Gross Capital information 20 25 25 30 35
Years 1971 1972 1973 1974 1975
Gross Capital information 30 45 40 55 65

Obtain trend values using 5-yearly moving values.


The publisher of a magazine wants to determine the rate of increase in the number of subscribers. The following table shows the subscription information for eight consecutive years:

Years 1976 1977 1978 1979
No. of subscribers
(in millions)
12 11 19 17
Years 1980 1981 1982 1983
No. of subscribers
(in millions)
19 18 20 23

Fit a trend line by graphical method.


Complete the following activity to fit a trend line to the following data by the method of least squares.

Year 1975 1976 1977 1978 1979 1980 1981 1982 1983
Number of deaths 0 6 3 8 2 9 4 5 10

Solution:

Here n = 9. We transform year t to u by taking u = t - 1979. We construct the following table for calculation :

Year t Number of deaths xt u = t - 1979 u2 uxt
1975 0 - 4 16 0
1976 6 - 3 9 - 18
1977 3 - 2 4 - 6
1978 8 - 1 1 - 8
1979 2 0 0 0
1980 9 1 1 9
1981 4 2 4 8
1982 5 3 9 15
1983 10 4 16 40
  `sumx_t` =47 `sumu`=0 `sumu^2=60` `square`

The equation of trend line is xt= a' + b'u.

The normal equations are,

`sumx_t = na^' + b^' sumu`              ...(1)

`sumux_t = a^'sumu + b^'sumu^2`      ...(2)

Here, n = 9, `sumx_t = 47, sumu= 0, sumu^2 = 60`

By putting these values in normal equations, we get

47 = 9a' + b' (0)       ...(3)

40 = a'(0) + b'(60)      ...(4)

From equation (3), we get a' = `square`

From equation (4), we get b' = `square`

∴ the equation of trend line is xt = `square`


Following table gives the number of road accidents (in thousands) due to overspeeding in Maharashtra for 9 years. Complete the following activity to find the trend by the method of least squares.

Year 2008 2009 2010 2011 2012 2013 2014 2015 2016
Number of accidents 39 18 21 28 27 27 23 25 22

Solution:

We take origin to 18, we get, the number of accidents as follows:

Year Number of accidents xt t u = t - 5 u2 u.xt
2008 21 1 -4 16 -84
2009 0 2 -3 9 0
2010 3 3 -2 4 -6
2011 10 4 -1 1 -10
2012 9 5 0 0 0
2013 9 6 1 1 9
2014 5 7 2 4 10
2015 7 8 3 9 21
2016 4 9 4 16 16
  `sumx_t=68` - `sumu=0` `sumu^2=60` `square`

The equation of trend is xt =a'+ b'u.

The normal equations are,

`sumx_t=na^'+b^'sumu             ...(1)`

`sumux_t=a^'sumu+b^'sumu^2      ...(2)`

Here, n = 9, `sumx_t=68,sumu=0,sumu^2=60,sumux_t=-44`

Putting these values in normal equations, we get

68 = 9a' + b'(0)     ...(3)

∴ a' = `square`

-44 = a'(0) + b'(60)          ...(4)

∴ b' = `square`

The equation of trend line is given by

xt = `square`


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