Statistics & Applications

Trends & Data

Medium

applied

If $(t_1, y_1), (t_2, y_2), (t_3, y_3), ..., (t_n, y_n)$ denote the time series and $y_t$ are the trend values of the variable y, then $\sum(y - y_t)$ , the sum of the deviations of y from their corresponding trend value is equal to:

-1

✅ Correct Option: 3

Given a time series with data points $(t_1, y_1), (t_2, y_2), (t_3, y_3), ..., (t_n, y_n)$ where $y_t$ represents the trend values of variable $y$ .

When a trend line is fitted using the least squares method, it has a fundamental property: the sum of deviations of actual values from their corresponding trend values equals zero.

The expression $\sum(y - y_t)$ represents the sum of all deviations, where:

$y$ = actual values

$y_t$ = trend values (values on the fitted trend line)

$(y - y_t)$ = deviation at each point

The trend line is constructed such that positive deviations (where actual values exceed trend values) are exactly balanced by negative deviations (where actual values fall below trend values).

This occurs because:

$\sum(y - y_t) = \sum y - \sum y_t$

$= \sum y - \sum y_t$

$= 0$

Therefore, $\sum(y - y_t) = 0$

2025: 14 May Shift 2

Due to which of the following, the irregular variations in a time series are caused:

(A) Floods

(B) Rise in prices before festivals

(D) Epidemics

Choose the correct answer from the options given below:

2025: 3 June Shift 2

The break-even point is the level of production where

2025: 29 May Shift 2

The rise in prices before Diwali is an example of

2025: 2 June Shift 1

Data available for profit (₹ Thousands) of a company as

Year	2004	2005	2006	2007	2008	2009	2010
Profit(₹ 000)	114	130	126	144	138	156	164

Based on the above data using least square method the trend value for the year '2007' is

2025: 15 May Shift 2

The number of letters posted in a certain city on each day for a week is given as follows: 40, 55, 28, 25, 31, 52, 43. Which of the following is not an entry in the three-day moving averages?

2026: 25 May Shift 1

Which of the following graph plotted for the months against the three month moving average(3Yr MA) represents the given data correctly ?

Month	January	February (F)	March (M)	April (A)	May
Production of items	15	27	21	27	24

[Four line graphs labelled 1-4, each plotting 3-Yr MA against month, showing different curve shapes and values]

Solution

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