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Class 12 (HSC Board)Commerce Mathematics & Statistics2026-27 Syllabus

Chapter 4 Time Series Miscellaneous Exercise 4 Solutions

Complete Maharashtra State Board Balbharati & Yuvakbharati textbook solutions for Chapter 4 Time Series Miscellaneous Exercise 4. Step-by-step solved exercises, numerical problems, and digest answers.

50 Solved Questions• 27 Diagrams• 2306 words

Balbharati Maharashtra State Board12th Commerce Maths Digest PdfChapter 4 Time Series Miscellaneous Exercise 4 Questions and Answers.

Maharashtra State Board 12th Commerce Maths Solutions Chapter 4 Time Series Miscellaneous Exercise 4

(I) Choose the correct alternative.

Question 1 Maharashtra Board Solution

Which of the following can’t be a component of a time series? (a) Seasonality (b) Cyclical (c) Trend (d) Mean

Solution & Step-by-Step Answer:

(d) Mean

Question 2 Maharashtra Board Solution

The first step in time series analysis is to (a) Perform regression calculations (b) Calculate a moving average (c) Plot the data on a graph (d) Identify seasonal variation

Solution & Step-by-Step Answer:

(c) Plot the data on a graph

Question 3 Maharashtra Board Solution

Time-series analysis is based on the assumption that (a) Random error terms are normally distributed. (b) The variable to be forecast and other independent variable are correlated. (c) Past patterns in the variable to be forecast will continue unchanged into the future. (d) The data do not exhibit a trend.

Solution & Step-by-Step Answer:

(c) Past patterns in the variable to be forecast will continue unchanged into the future.

Question 4 Maharashtra Board Solution

Moving averages are useful in identifying (a) Seasonal component (b) Irregular component (c) Trend component (d) Cyclical component

Solution & Step-by-Step Answer:

(c) Trend component

Question 5 Maharashtra Board Solution

We can use regression line for past data to forecast future data. We then use the line which (a) Minimizes the sum of squared deviations of past data from the line. (b) Minimizes the sum of deviations of past data from the line. (c) Maximizes the sum of squared deviations of past data from the line. (d) Maximizes the sum of deviation of past data from the line.

Solution & Step-by-Step Answer:

(a) Minimizes the sum of squared deviations of past data from the line

Question 6 Maharashtra Board Solution

Which of the following is a major problem for forecasting, especially when using the method of least squares? (a) The past cannot be known (b) The future is not entirely certain (c) The future exactly follows the patterns of the past (d) The future may not follow the patterns of the past

Solution & Step-by-Step Answer:

(d) The future may not follow the patterns of the past

Question 7 Maharashtra Board Solution

An overall upward or downward pattern in an annual time series would be contained in which component of the time series (a) Trend (b) Cyclical (c) Irregular (d) Seasonal

Solution & Step-by-Step Answer:

(a) Trend

Question 8 Maharashtra Board Solution

The following trend line equation was developed for annual sales from 1984 to 1990 with 1984 as base or zero year. Y1 = 500 + 60X (in 1000 Rs.) The estimated sales for 1984 (in 1000 Rs) is: (a) ₹ 500 (b) ₹ 560 (c) ₹ 1,040 (d) ₹ 1100

Solution & Step-by-Step Answer:

(a) ₹ 500

Question 9 Maharashtra Board Solution

What is a disadvantage of the graphical method of determining a trend line? (a) Provides quick approximations (b) Is subject to human error (c) Provides accurate forecasts (d) Is too difficult to calculate

Solution & Step-by-Step Answer:

(b) Is subject to human error

Question 10 Maharashtra Board Solution

Which component of time series refers to erratic time series movements that follow no recognizable or regular pattern. (a) Trend (b) Seasonal (c) Cyclical (d) Irregular

Solution & Step-by-Step Answer:

(a) Trend

(II) Fill in the blanks.

Question 1 Maharashtra Board Solution

_________ components of time series is indicated by a smooth line.

Solution & Step-by-Step Answer:

Trend

Question 2 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Seasonal

Question 3 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Cyclical

Question 4 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Irregular

Question 5 Maharashtra Board Solution

Addictive models of time series _________ independence of its components.

Solution & Step-by-Step Answer:

assume

Question 6 Maharashtra Board Solution

Multiplicative models of time series _________ independence of its components.

Solution & Step-by-Step Answer:

does not assume

Question 7 Maharashtra Board Solution

The simplest method of measuring the trend of time series is _________

Solution & Step-by-Step Answer:

graphical method

Question 8 Maharashtra Board Solution

The method of measuring the trend of time series using only averages is _________

Solution & Step-by-Step Answer:

moving average method

Question 9 Maharashtra Board Solution

The complicated but ancient method of measuring the trend of time series is _________

Solution & Step-by-Step Answer:

least-squares method

Question 10 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

trend

(III) State whether each of the following is True or False.

Question 1 Maharashtra Board Solution

The secular trend component of the time series represents irregular variations.

Solution & Step-by-Step Answer:

False

Question 2 Maharashtra Board Solution

Seasonal variation can be observed over several years.

Solution & Step-by-Step Answer:

True

Question 3 Maharashtra Board Solution

Cyclical variation can occur several times in a year.

Solution & Step-by-Step Answer:

False

Question 4 Maharashtra Board Solution

Irregular variation is not a random component of time series.

Solution & Step-by-Step Answer:

False

Question 5 Maharashtra Board Solution

The additive model of time series does not require the assumptions of independence of its components.

Solution & Step-by-Step Answer:

False

Question 6 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

True

Question 7 Maharashtra Board Solution

The graphical method of finding trends is very complicated and involves several calculations.

Solution & Step-by-Step Answer:

False

Question 8 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

False

Question 9 Maharashtra Board Solution

The least-squares method of finding trends is very simple and does not involve any calculations.

Solution & Step-by-Step Answer:

False

Question 10 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

False

(IV) Solve the following problems.

Question 1 Maharashtra Board Solution

The following table shows the productivity of pig-iron and ferro-alloys (‘000 metric tonnes) Fit a trend line to the above data by graphical method.

Solution & Step-by-Step Answer:

Question 2 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

u = , Σy = 57, Σu = 0, Σu2 = 60, Σuy = 38, n = 9 Let the equation of the trend line be Y = a + bu where u = t – 1978 ……(i) Σy = na + bΣu ………(ii) Σuy = aΣu + bΣu2 ………(iii) Substituting the values of Σu, n, Σuy, Σu2 in (ii) & (iii) we get 57 = 9a + 0 ∴ a = 6.3333 38 = 0 + b. 60 ∴ b = 0.6333. ∴ The equation of the trend line is y = 6.3333 + 0.63333u where u = t – 1978

Question 3 Maharashtra Board Solution

Obtain the trends values for the data on problem IV (1) using 5 yearly moving averages.

Solution & Step-by-Step Answer:

Question 4 Maharashtra Board Solution

The following table shows the amount of sugar production (in lac tonnes) for the years 1971 to 1982. Fit a trend line to the above data by graphical method.

Solution & Step-by-Step Answer:

Question 5 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

u = , Σy = 38, Σu = 0, Σu2 = 572, Σuy = 160, n = 12 Let the equation of the trend line be y = a + bu ……..(i) where u = u = 2t – 3953 Σy = na + bΣu ………(ii) Σuy = aΣu + bΣu2 ……..(iii) by (ii) 38 = 12o + 0 ∴ a = 3.1867 by (iii) 160 = 0 + b. 572 ∴ b = 0.2797 ∴ by (i), Equation of the trend line is Y = 3.1667 + 0.2797u where u = 2t – 3953.

Question 6 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Question 7 Maharashtra Board Solution

The percentage of girls’ enrollment in total enrollment for years 1960-2005 is shown in the following table. Fit a trend line to the above data by graphical method.

Solution & Step-by-Step Answer:

Question 8 Maharashtra Board Solution

Fit a trend line to the data in Problem 7 by the method of least squares.

Solution & Step-by-Step Answer:

u = , Σy = 51, Σu = 0, Σu2 = 330, Σuy = 157, n = 10 Let the equation of the trend line be Y = a + bu where u = …….(i) Σy = na + bΣu ……..(ii) Σuy = aΣu + bΣu2 ……..(iii) Substituting the values of Σy, Σu, n, Σuy, Σu2 We get 51 = 10a + 0 ∴ a = 5.1 and 157 = 0 + 6.330 ∴ b = 0.4758 by (i) equation of the trend line is Y = 5.1 + 0.4758u where u =

Question 9 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Question 10 Maharashtra Board Solution

The following data shows the number of boxes of cereal sold in the years 1977 to 1984. Fit a trend line to the above data by graphical method.

Solution & Step-by-Step Answer:

Question 11 Maharashtra Board Solution

Fit a trend line to data in Problem 10 by the method of least squares.

Solution & Step-by-Step Answer:

u = , Σy = 39, Σu = 0, Σu2 = 168, Σuy = 79, n = 8 Let the equation of the trend line by Y = a + bu Where u = 2t – 3961 …….(i) Σy = na + bΣu …….(ii) Σuy = aΣu + bΣu2 ……..(iii) Substituting the values of Σy, n, Σu, Σuy, Σu2, in (ii) & (iii) 39 = 8a + 0 ∴ a = 4.875 79 = 0 + b (168) ∴ b = 0.4702 by (i) the equation of the trend line is Y = 4.875 + 0.4702u Where u = 2t – 3961.

Question 12 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Question 13 Maharashtra Board Solution

The following table shows the number of trade fatalities (in a state) resulting from drunken driving for the years 1975 to 1983. Fit a trend line to the above data by graphical method.

Solution & Step-by-Step Answer:

Question 14 Maharashtra Board Solution

Fit a trend line to data in Problem 13 by the method of least squares.

Solution & Step-by-Step Answer:

u = , Σy = 47, Σu = 0, Σu2 = 60, Σuy = 40, n = 9 Let the equation of the trends line be Y = a + bu where u = t – 1979 …….(i) Σy = na + bΣu ………(ii) Σuy = aΣu + bΣu2 ……..(iii) Substituting values of Σy, n, Σu, Σuy, Σu2 in (ii) & (iii) We get 47 = 9a + 0 ∴ a = 5.2222 and 40 = 0 + b(60) ∴ b = 0.6667 ∴ by (i) the equation of the trend line is Y = 5.2222 + 0.6667u Where u = t – 1979.

Question 15 Maharashtra Board Solution

Obtain trend values for data in Problem 13 using 4-yearly moving averages.

Solution & Step-by-Step Answer:

Question 16 Maharashtra Board Solution

The following table shows the all India infant mortality rates (per ‘000) for the years 1980 to 2000. Fit a trend line to the above data by graphical method.

Solution & Step-by-Step Answer:

Question 17 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

u = , Σy = 30, Σu = 0, Σu2 = 70, Σuy = -70, n = 7 Let the equation of the trend line be Y = a + bu Where u = …..(i) Σy = na + bΣu ……(ii) Σuy = aΣu + bΣu2 …….(iii) Substituting values of Σy, n, Σu, Σuy & Σu2 in (ii) & (iii) we get 30 = 7a + 0 ∴ a = 4.2857 -70 = 0 + 6(70) ∴ b = -1 ∴ by (i) the equation of the trend line is y = 4.2857 – 1(u) Where u =

Question 18 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

Question 19 Maharashtra Board Solution

the following table shows the wheat yield (‘000 tonnes) in India for the years 1959 to 1968. Fit a trend line to the above data by the method of least squares.

Solution & Step-by-Step Answer:

u = , Σy = 24, Σu = 0, Σu2 = 330, Σuy = 94, n = 10 Let the equation of the trend line be y = a + bu where u = ……(i) i.e. u = 2t – 3927 Σy = na + bΣu …….(ii) Σuy = aΣu + Σu2 …..(iii) Substituting values of Σy, n, Σu, Σuy & Σu2 in (ii) & (iii) we get 24 = 10a + 0 ∴ a = 2.4 94 = 0 + 6.330 ∴ b = 0.2848 ∴ Equation of the trend line is y = 2.4 + (0.2848)u where u = 2t – 3927

Question 20 Maharashtra Board Solution

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

Solution & Step-by-Step Answer:

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Chapter 4 Time Series Ex 4.1

Chapter 5 Index Numbers Ex 5.1