###### Advertisements

###### Advertisements

The simplest method of measuring trend of time series is ______

###### Advertisements

#### Solution

**graphical**

#### APPEARS IN

#### RELATED QUESTIONS

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

**Fill in the blank :**

The simplest method of measuring trend of time series is _______

**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 data in Problem 7 using 4-yearly moving averages.

**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 13 using 4-yearly moving averages.

**Solve the following problem :**

Fit a trend line to data in Problem 16 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 ______

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

The secular trend component of time series represents irregular variations

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

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 |

**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.

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 |

**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 x_{t} |
u = t - 1979 |
u^{2} |
ux_{t} |

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 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 = 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 x_{t} = `square`