###### Advertisements

###### Advertisements

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

The secular trend component of time series represents irregular variations

#### Options

True

False

###### Advertisements

#### Solution

**False**

#### APPEARS IN

#### RELATED QUESTIONS

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 :**

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

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 :**

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 |

**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 :**

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

**Solve the following problem :**

Obtain trend values for data in Problem 19 using 3-yearly moving averages.

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:**

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

Obtain trend values for data, using 4-yearly centred moving averages

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 |

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.

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 | – | – |

**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 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.

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`