#### Online Mock Tests

#### Chapters

Chapter 1.2: Matrices

Chapter 1.3: Differentiation

Chapter 1.4: Applications of Derivatives

Chapter 1.5: Integration

Chapter 1.6: Definite Integration

Chapter 1.7: Application of Definite Integration

Chapter 1.8: Differential Equation and Applications

Chapter 2.1: Commission, Brokerage and Discount

Chapter 2.2: Insurance and Annuity

Chapter 2.3: Linear Regression

▶ Chapter 2.4: Time Series

Chapter 2.5: Index Numbers

Chapter 2.6: Linear Programming

Chapter 2.7: Assignment Problem and Sequencing

Chapter 2.8: Probability Distributions

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## Solutions for Chapter 2.4: Time Series

Below listed, you can find solutions for Chapter 2.4 of Maharashtra State Board SCERT Maharashtra Question Bank for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022.

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.4 Time Series Q.1

#### MCQ [1 Mark]

**Choose the correct alternative:**

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

Seasonality

Cyclical

Trend

Mean

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

Trend

Seasonal

Cyclical

Irregular

**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

500

560

1,040

1,100

**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?

Trend

Cyclical

Irregular

Seasonal

**Choose the correct alternative:**

Moving averages are useful in identifying ______.

Seasonal component

Irregular component

Trend component

cyclical component

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.4 Time Series Q.2

#### Fill in the blanks [1Markl]

______ 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 ______

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.4 Time Series Q.3

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

The secular trend component of time series represents irregular variations

True

False

**State whether the following statement 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

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

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

True

False

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

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

True

False

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.4 Time Series Q.4

#### Solve the followoing problems. [4 Marks]

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.

### SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 Chapter 2.4 Time Series Q.5

#### Activity based question [4 Mark]

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 | – | – |

## Solutions for Chapter 2.4: Time Series

## SCERT Maharashtra Question Bank solutions for 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2.4 - Time Series

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Concepts covered in 12th Standard HSC Mathematics and Statistics (Commerce) Maharashtra State Board 2022 chapter 2.4 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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