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

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#### Solution

**least square method**

#### RELATED QUESTIONS

Obtain the trend line for the above data using 5 yearly moving averages.

**Choose the correct alternative :**

What is a disadvantage of the graphical method of determining a trend line?

**State whether the following is True or False :**

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

**State whether the following is True or False :**

Least squares method of finding trend is very simple and does not involve any 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.

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

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 :**

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 :**

Following table shows the number of traffic fatalities (in a state) resulting from drunken driving for years 1975 to 1983.

Year |
1975 | 1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 |

No. of deaths |
0 | 6 | 3 | 8 | 2 | 9 | 4 | 5 | 10 |

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

**Solve the following problem :**

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010.

Year |
1980 | 1985 | 1990 | 1995 | 2000 | 2005 | 2010 |

IMR |
10 | 7 | 5 | 4 | 3 | 1 | 0 |

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

**Solve the following problem :**

Obtain trend values for data in Problem 19 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.

**Choose the correct alternative:**

Moving averages are useful in identifying ______.

The simplest method of measuring trend of time series is ______

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

**State whether the following statement is True or False:**

The secular trend component of time series represents irregular variations

**State whether the following statement is True or False:**

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

**State whether the following statement is True or False:**

Least squares method of finding trend is very simple and does not involve any calculations

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

Year |
1971 | 1972 | 1973 | 1974 | 1975 | 1976 |

Production |
1 | 0 | 1 | 2 | 3 | 2 |

Year |
1977 | 1978 | 1979 | 1980 | 1981 | 1982 |

Production |
4 | 6 | 5 | 1 | 4 | 10 |

Fit a trend line by the method of least squares

The following table gives the production of steel (in millions of tons) for years 1976 to 1986.

Year |
1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 | 1986 |

Production |
0 | 4 | 4 | 2 | 6 | 8 | 5 | 9 | 4 | 10 | 10 |

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 |

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 |

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

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010

Year |
1980 | 1985 | 1990 | 1995 |

IMR |
10 | 7 | 5 | 4 |

Year |
2000 | 2005 | 2010 | |

IMR |
3 | 1 | 0 |

Fit a trend line by the method of least squares

**Solution: **Let us fit equation of trend line for above data.

Let the equation of trend line be y = a + bx .....(i)

Here n = 7(odd), middle year is `square` and h = 5

Year |
IMR (y) |
x |
x^{2} |
x.y |

1980 | 10 | – 3 | 9 | – 30 |

1985 | 7 | – 2 | 4 | – 14 |

1990 | 5 | – 1 | 1 | – 5 |

1995 | 4 | 0 | 0 | 0 |

2000 | 3 | 1 | 1 | 3 |

2005 | 1 | 2 | 4 | 2 |

2010 | 0 | 3 | 9 | 0 |

Total |
30 |
0 |
28 |
– 44 |

The normal equations are

Σy = na + bΣx

As, Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx^{2}

As, Σx = 0, b =`square`

∴ The equation of trend line is y = `square`

Fit equation of trend line for the data given below.

Year |
Production (y) |
x |
x^{2} |
xy |

2006 | 19 | – 9 | 81 | – 171 |

2007 | 20 | – 7 | 49 | – 140 |

2008 | 14 | – 5 | 25 | – 70 |

2009 | 16 | – 3 | 9 | – 48 |

2010 | 17 | – 1 | 1 | – 17 |

2011 | 16 | 1 | 1 | 16 |

2012 | 18 | 3 | 9 | 54 |

2013 | 17 | 5 | 25 | 85 |

2014 | 21 | 7 | 49 | 147 |

2015 | 19 | 9 | 81 | 171 |

Total |
177 |
0 |
330 |
27 |

Let the equation of trend line be y = a + bx .....(i)

Here n = `square` (even), two middle years are `square` and 2011, and h = `square`

The normal equations are Σy = na + bΣx

As Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx^{2}

As Σx = 0, b = `square`

Substitute values of a and b in equation (i) the equation of trend line is `square`

To find trend value for the year 2016, put x = `square` in the above equation.

y = `square`

Complete the table using 4 yearly moving average method.

Year |
Production |
4 yearly moving total |
4 yearly centered total |
4 yearly centeredmoving average(trend values) |

2006 | 19 | – | – | |

`square` | ||||

2007 | 20 | – | `square` | |

72 | ||||

2008 | 17 | 142 | 17.75 | |

70 | ||||

2009 | 16 | `square` | 17 | |

`square` | ||||

2010 | 17 | 133 | `square` | |

67 | ||||

2011 | 16 | `square` | `square` | |

`square` | ||||

2012 | 18 | 140 | 17.5 | |

72 | ||||

2013 | 17 | 147 | 18.375 | |

75 | ||||

2014 | 21 | – | – | |

– | ||||

2015 | 19 | – | – |

**Obtain the trend values for the following data using 5 yearly moving averages:**

Year |
2000 |
2001 |
2002 |
2003 |
2004 |

Production x _{i} |
10 | 15 | 20 | 25 | 30 |

Year |
2005 |
2006 |
2007 |
2008 |
2009 |

Production x _{i} |
35 | 40 | 45 | 50 | 55 |

**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 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.