Statistics & Applications

Trends & Data

Medium

applied

Consider the following data:

Year (x) 2010 2011 2012 2013 2014
Sale (in crore Rs.) (y) 9 18 21 29 38

A straight line trend by the method of least square is:

Year (x)	2010	2011	2012	2013	2014
Sale (in crore Rs.) (y)	9	18	21	29	38

$y = 46 + 9.6 (x-2012)$

$y = 23 + 6.9 (x-2012)$

$y = 23 + 7.5 (x-2012)$

$y = 46 + 9.6x$

✅ Correct Option: 2

The least squares method finds the "best fit" straight line through the data points.

To simplify calculations, shift the origin to $x = 2012$ by letting $X = (x - 2012)$ .

Year (x)	X = (x - 2012)	Sale (y)
2010	-2	9
2011	-1	18
2012	0	21
2013	1	29
2014	2	38

Number of data points: $n = 5$

Sum of $X$ values:

$\Sigma X = (-2) + (-1) + 0 + 1 + 2 = 0$

Sum of $y$ values:

$\Sigma y = 9 + 18 + 21 + 29 + 38 = 115$

Mean of $X$ :

$\bar{X} = \dfrac{\Sigma X}{n}$

$\bar{X} = \dfrac{0}{5} = 0$

Mean of $y$ :

$\bar{y} = \dfrac{\Sigma y}{n}$

$\bar{y} = \dfrac{115}{5} = 23$

Sum of $X$ squared:

$\Sigma X^2 = (-2)^2 + (-1)^2 + (0)^2 + (1)^2 + (2)^2$

$\Sigma X^2 = 4 + 1 + 0 + 1 + 4 = 10$

Sum of $X$ times $y$ :

X	y	X × y
-2	9	-18
-1	18	-18
0	21	0
1	29	29
2	38	76

$\Sigma Xy = -18 + (-18) + 0 + 29 + 76 = 69$

The slope is:

$b = \dfrac{\Sigma Xy}{\Sigma X^2}$

$b = \dfrac{69}{10} = 6.9$

The intercept is:

$a = \bar{y} = 23$

The least squares trend line equation is:

$y = a + b(x - 2012)$

$y = 23 + 6.9(x - 2012)$

Therefore, the answer is $y = 23 + 6.9(x - 2012)$ .

2026: 22 May Shift 1

For the given 9 values, 4,5,5,6,7,8,9,8,10, the three years moving averages are

2025: 2 June Shift 1

Calculate Three Yearly moving averages for the following data

Year	2016	2017	2018	2019	2020	2021
Production (thousand tonnes)	200	220	231	254	202	243

2025: 15 May Shift 1

Year (t)	2010	2011	2012	2013	2014	2015	2016
Sales (in Rs. crores) (y)	6	8	9	11	13	17	20

In reference to the above data, which of the following statements are correct? (where $x = t - 2013$ )

(A) If the equation of the straight line trend is $y = 12 + 2.29 x$ , then the trend value for the year 2017 is 21.16.

(B) If the equation of the straight line trend is $y = 12 + 2.29 x$ , then the trend value for the year 2013 is 11.

(D) If $(t_1, y_2),(t_2, y_2),(t_3, y_3),......,(t_n, y_n)$ denote the time series and $y_t$ are the trend values of the variable y, then $\sum(y - y_t) = 0$ .

Choose the correct answer from the options given below:

2026: 26 May Shift 2

Consider the following data:

Year( $x_i$ )	2004	2005	2006	2007	2008	2009	2010
Profit (₹ 000)( $y_i$ )	26	26	44	42	108	120	166

Using method of least squares, the trend value for year 2004 will be

2025: 22 May Shift 2

If three year moving averages for five data items given by 16, 18, 20, 10, 21 are $x$ , $y$ and $z$ respectively, then:

2025: 15 May Shift 2

The pattern and behavior of the data in any time series are based on which of the following components:

(A) Secular trend component

(B) Seasonal component

(D) Irregular component

Choose the correct answer from the options given below:

Consider the following data: Year (x)20102011201220132014Sale (in crore Rs.) (y)918212938 A straight line trend by the method of least square is:

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

Consider the following data:

Year (x) 2010 2011 2012 2013 2014
Sale (in crore Rs.) (y) 9 18 21 29 38

A straight line trend by the method of least square is: