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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`

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#### Solution

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

Here n = 7(odd), middle year is **1995** and h = 5

x = `("t" - "middle year")/"h"`

= `("t" - 1995)/5`

**We obtain the following table:**

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 |

From the table, n = 7, Σy_{t} = 30, Σx = 0, Σx^{2} = 28, Σxy_{t} = – 44

The normal equations are

Σy = na + bΣx

∴ 30 = 7a + bΣx

As, Σx = 0, a = `30/7` = **4.2857**

Also, Σxy = aΣx + bΣx^{2 }

∴ – 44 = aΣx + b′(28)

As, Σx = 0, b =`(-44)/28` = ** – 1.5714**

∴ The equation of trend line is y = a + bx

∴ The equation of trend line is y = **4.2857 – 1.5714x**

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#### RELATED QUESTIONS

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

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

**Choose the correct alternative :**

Which of the following is a major problem for forecasting, especially when using the method of least squares?

**Fill in the blank :**

The simplest method of measuring trend of time series 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 :**

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

**Solve the following problem :**

Following data shows the number of boxes of cereal sold in years 1977 to 1984.

Year |
1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 |

No. of boxes in ten thousand |
1 | 0 | 3 | 8 | 10 | 4 | 5 | 8 |

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

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

The simplest method of measuring trend of time series is ______

**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

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 |

- 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.

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

Solution:

Year |
IMR |
3 yearlymoving total |
3-yearly movingaverage (trend value) |

1980 | 10 | – | – |

1985 | 7 | `square` | 7.33 |

1990 | 5 | 16 | `square` |

1995 | 4 | 12 | 4 |

2000 | 3 | 8 | `square` |

2005 | 1 | `square` | 1.33 |

2010 | 0 | – | – |

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:**