3 Marks Question

Question 13 Marks

Fit a straight line trend by the method of least square to the following data on sales (in lakhs) for the period 2011-2018.

Year	2011	2012	2013	2014	2015	2016	2017	2018
Sales (₹ lakhs)	76	80	130	144	138	120	174	190

Answer

Years $\left(t_i\right)$	Sales (₹ lakhs) $(y)$	$x_i=t_i-2014-15$	$x_i \times 2$	$x_i y_i$	$x_i^2$
2011	76	-3.5	-7	-532	49
2012	80	-2.5	-5	-400	25
2013	130	-1.5	-3	-390	9
2014	144	-0.5	-1	-144	1
2015	138	0.5	1	138	1
2016	120	1.5	3	360	9
2017	174	2.5	5	870	25
2018	190	3.5	7	1330	49
	$\Sigma y_i=1052$	$\Sigma x_i=0$		$\sum x_i y_i=1232$	$\Sigma x_i^2=168$

Now,
$\begin{array}{l}a=\frac{\Sigma y_i}{n}=\frac{1052}{8}=1315 \\ b=\frac{\Sigma x_i y_i}{\Sigma x_i^2}=\frac{1232}{168}=7.33\end{array}$
So, trend equation is
$y=131.5+7.33 x$

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Question 23 Marks

Fit a straight line trend by the method of least squares to the data given below:

Year	2012	2013	2014	2015	2016	2017	2018
Sales ( in tones)	9	11	13	12	14	15	17

Answer

Years $\left(t_i\right)$	Sales $\left(y_i\right)$	$x_i=t_i-2015$	$x_i^2$	$x_i y_i$
2012	9	-3	9	-27
2013	11	-2	4	-22
2014	13	-1	1	-13
2015	12	0	0	0
2016	14	1	1	14
2017	15	2	4	30
2018	17	3	9	51
2019	$\Sigma y_i=91$	$\Sigma x_i=0$	$\Sigma x_i^2=28$	$\Sigma x_i y_i=33$

Now, $a=\frac{\Sigma y_i}{n}=\frac{91}{7}=13$
$b=\frac{\Sigma x_i y_i}{\sum x_i^2}=\frac{33}{28}=1.179$
The equation of the straight line trend is
$y=a x+b$
$\therefore$ $y=13 x+1.179$

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Question 33 Marks

Explain the method of fitting a straight line.

Answer

The method of fitting a straight line is as follows:
(i) The straight-line trend is represented by the equation Y = a + bX .....(1)
where, Y is the actual value, X is time, a and b are constants

(ii) The constants 'a' and 'B' are estimated by solving the following two normal equations
$\Sigma Y=n a+b \Sigma X$ ....(2)
$\Sigma X Y=a \Sigma X+b \Sigma X^2$ .....(3)
where, n = number of years given in the data.

(iii) By taking the mid-point of the time as the origin, we get $\Sigma X=0$

(iv) When $\Sigma X=0$, the two normal equations reduces to
$\begin{aligned} \Sigma Y & =n a+b(0) ; a=\frac{\Sigma Y}{n}=\bar{Y} \\ \Sigma X Y & =a(0)+b \Sigma X^2 ; b=\frac{\Sigma X Y}{\Sigma X^2}\end{aligned}$
The constant 'a' gives the mean of Y and 't gives the rate of change
(slope).

(v) By substituting the values of 'a' and ' in the trend equation (1), we get the Line of Best Fit.

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Question 43 Marks

The following figures relate to the profits of a commercial concern for 8 years.

Year	2016	2017	2018	2019	2020	2021	2022	2023
Profit (₹)	15420	15470	15520	21020	26500	31950	35600	34900

Find the trend of profits by the method of three-yearly moving averages

Answer

Computation of three-yearly moving averages

Years	Profits ( ₹ )	3- yearly moving total (₹)	3- yearly moving averages (₹)
2016	15420	-	-
2017	15470	46410	15470
2018	15520	52010	17336.667
2019	21020	63040	21013.333
2020	26500	79470	26490
2021	31950	94050	31350
2022	35600	102450	34150
2023	34900	-	-

The last column gives the trend of profits.c

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Question 53 Marks

From the following time series obtain trend value by 3 yearly moving averages.

Year	Sales (in ₹ 000)	Year	Sales (in ₹ 000)
2008	8	2014	16
2009	12	2015	17
2010	10	2016	14
2011	13	2017	17
2012	15
2013	12

Answer

Calculation of trend values by three yearly moving average method.

Years	Sales (Thousand ₹)	Three-yearly Moving Totals	Three-yearly Moving Average (Trend value)
2008	8
2009	12	(8 + 12 + 10) = 30	10.00
2010	10	(12 + 10 + 13) =35	11.67
2011	13	(10 + 13 + 15) =38	12.67
2012	15	(13 + 15 + 12) =40	13.33
2013	12	(15 + 12 + 16) =43	14.33
2014	16	(12 + 16 + 17) =45	15.00
2015	17	(16 + 17 + 14) =47	15.67
2016	14	(17 + 14 + 17) =48	16.00
2017	17

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