SCERT Maharashtra solutions for Mathematics and Statistics (Commerce) [English] 12 Standard HSC chapter 2.4 - Time Series [Latest edition]

Choose the correct alternative:

Which of the following can’t be a component of a time series?

 Which component of time series refers to erratic time series movements that follow no recognizable or regular pattern?

Choose the correct alternative:

The following trend line equation was developed for annual sales from 1984 to 1990 with 1984 as base or zero year.

Y = 500 + 60X (in 1000 ₹). The estimated sales for 1984 (in 1000 ₹) is

Choose the correct alternative:

An overall upward or downward pattern in an annual time series would be contained in which component of the times series?

Choose the correct alternative:

Moving averages are useful in identifying ______.

______ components of time series is indicated by a smooth line

______ component of time series is indicated by periodic variation year after year.

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

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: 

Seasonal variation can be observed over several years

Cyclical variation can occur several times in a year.

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

Fit a trend line by the method of least squares

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

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

Obtain the trend value for the year 1990

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

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.

The following table shows the production of gasoline in U.S.A. for the years 1962 to 1976.

1. Obtain trend values for the above data using 5-yearly moving averages.
2. 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

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

The normal equations are

Σy = na + bΣx

As, Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx²

As, Σx = 0, b =`square`

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

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

Fit equation of trend line for the data given below.

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 + bΣx²

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.