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## Solutions for Chapter 4: Time Series

Below listed, you can find solutions for Chapter 4 of Maharashtra State Board Balbharati for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board.

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Time Series Exercise 4.1 [Pages 66 - 67]

The following data gives the production of bleaching powder (in '000 tons) for the years 1962 to 1972.

Year |
1962 | 1963 | 1964 | 1965 | 1966 | 1967 | 1968 | 1969 | 1970 | 1971 | 1972 |

Production |
0 | 0 | 1 | 1 | 4 | 2 | 4 | 9 | 7 | 10 | 8 |

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

Use the method of least squares to fit a trend line to the data in Problem 1 above. Also, obtain the trend value for the year 1975.

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

The following table shows the index of industrial production for the period from 1976 to 1985, using the year 1976 as the base year.

Year |
1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 |

Index |
0 | 2 | 3 | 3 | 2 | 4 | 5 | 6 | 7 | 10 |

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

Fit a trend line to the data in Problem 4 above by the method of least squares. Also, obtain the trend value for the index of industrial production for the year 1987.

Obtain the trend values for the data in using 4-yearly centered moving averages.

Year |
1976 | 1977 | 1978 | 1979 | 1980 | 1981 | 1982 | 1983 | 1984 | 1985 |

Index |
0 | 2 | 3 | 3 | 2 | 4 | 5 | 6 | 7 | 10 |

The following table gives the production of steel (in millions of tonnes) 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 |

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

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.

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

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) |
10 | 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.

### Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board Chapter 4 Time Series Miscellaneous Exercise 4 [Pages 67 - 69]

**Choose the correct alternative :**

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

Seasonality

Cyclical

Trend

Mean

**Choose the correct alternative :**

The first step in time series analysis is to _______.

perform regression calculations

calculate a moving average

plot the data on a graph

identify seasonal variation

**Choose the correct alternative :**

Time-series analysis is based on the assumption that _______.

random error terms are normally distributed.

the variable to be forecast and other independent variables are correlated.

past patterns in the variable to be forecast will continue unchanged into the future.

the data do not exhibit a trend.

Moving averages are useful in identifying _______.

Seasonal component

Irregular component

Trend component

Cyclical component

**Choose the correct alternative :**

We can use regression line for past data to forecast future data. We then use the line which_______.

minimizes the sum of squared deviations of past data from the line

minimizes the sum of deviations of past data from the line

maximizes the sum of squared deviations of past data from the line

maximizes the sum of deviations of past data from the line

**Choose the correct alternative :**

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

The past cannot be known.

The future is not entirely certain.

The future exactly follows the patterns of the past.

The future may not follow the patterns of the past.

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

Trend

Cyclical

Irregular

Seasonal

**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 Rs). The estimated sales for 1984 (in 1000 Rs) is:

Rs 500

Rs 560

Rs 1,040

Rs 1,100

**Choose the correct alternative :**

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

Provides quick approximations

Is subject to human error

Provides accurate forecasts

Is too difficult to calculate

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

Trend

Seasonal

Cyclical

Irregular

**Fill in the blank :**

_______ component of time series is indicated by a smooth line.

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

_______ component of time series is indicated by a long wave spanning two or more years.

**Fill in the blank :**

_______ component of time series is indicated by up and down movements without any pattern.

**Fill in the blank :**

Addictive models of time series _______ independence of its components.

**Fill in the blank :**

Multiplicative models of time series _______ independence of its components.

**Fill in the blank :**

The simplest method of measuring trend of time series is _______

**Fill in the blank :**

The method of measuring trend of time series using only averages is _______

**Fill in the blank :**

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

**Fill in the blank :**

The graph of time series clearly shows _______of it is monotone.

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

The secular trend component of time series represents irregular variations.

True

False

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

Seasonal variation can be observed over several years.

True

False

Cyclical variation can occur several times in a year.

True

False

Irregular variation is not a random component of time series.

True

False

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

Additive model of time series does not require the assumption of independence of its components.

True

False

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

Multiplicative model of time series does not require the assumption of independence of its components.

True

False

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

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

True

False

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

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

True

False

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

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

True

False

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

All the three methods of measuring trend will always give the same results.

True

False

**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 data in Problem 19 using 3-yearly moving averages.

**Solve the following problem :**

Obtain trend values for the following data using 5-yearly moving averages.

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 :**

Fit a trend line to data in Problem 4 by the method of least squares.

**Solve the following problem :**

Obtain trend values for data in Problem 4 using 4-yearly centered moving averages.

**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 the data in Problem 7 by the method of least squares.

**Solve the following problem :**

Obtain trend values for the data in Problem 7 using 4-yearly moving averages.

**Solve the following problem :**

Fit a trend line to the data in Problem IV (1) by the method of least squares.

**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 :**

Obtain trend values for data in Problem 10 using 3-yearly moving averages.

**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 :**

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 :**

Following table shows the all India infant mortality rates (per ‘000) for years 1980 to 2010.

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

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

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

**Solve the following problem :**

Fit a trend line to data in Problem 16 by the method of least squares.

**Solve the following problem :**

Obtain trend values for data in Problem 16 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.

**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.

## Solutions for Chapter 4: Time Series

## Balbharati solutions for Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 - Time Series

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Concepts covered in Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board chapter 4 Time Series are Introduction to Time Series, Uses of Time Series Analysis, Components of a Time Series, Mathematical Models, Measurement of Secular Trend.

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