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**Choose the correct alternative:**

Moving averages are useful in identifying ______.

#### Options

Seasonal component

Irregular component

Trend component

cyclical component

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#### Solution

Moving averages are useful in identifying **trend component**.

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#### RELATED QUESTIONS

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 :**

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

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 :**

The percentage of girls’ enrollment in total enrollment for years 1960-2005 is shown in the following table.

Year |
1960 | 1965 | 1970 | 1975 | 1980 | 1985 | 1990 | 1995 | 2000 | 2005 |

Percentage |
0 | 3 | 3 | 4 | 4 | 5 | 6 | 8 | 8 | 10 |

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 13 using 4-yearly moving averages.

The complicated but efficient 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

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

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 |

- 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`

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 | – | – |

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 |

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

Fit a trend line to the following data by the method of least square :

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

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

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 x_{t} |
t |
u = t - 5 |
u^{2} |
u.x_{t} |

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 x_{t} =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

x_{t} = `square`