The birth rates of a state in different years are given in the following table. Fit a linear trend for these d

Question 33 of 41

The birth rates of a state in different years are given in the following table. Fit a linear trend for these data. Also find the estimates for birth rates in the year 2025.

Year	2018	2019	2020	2021	2022	2023	2024
Birth rate	22.2	21.8	21.3	20.9	20.6	20.2	19.9

Gujarat GsebGujarat Board (GSEB) HSC Commerce Board 2025Subjective· 5mImportance★★★★★

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Trend $\hat{y} = 20.9857 - 0.3857t$ ( $t$ from 2021); estimate for 2025 ( $t = 4$ ) $= 19.44$ .

GSEB Class-12 Statistics, Time Series (method of least squares):

There are $7$ years (2018–2024). Code time $t$ about the middle year 2021 so that $\Sigma t = 0$ .

Year	Birth rate $y$	$t$	$ty$	$t^2$
2018	22.2	-3	-66.6	9
2019	21.8	-2	-43.6	4
2020	21.3	-1	-21.3	1
2021	20.9	0	0	0
2022	20.6	1	20.6	1
2023	20.2	2	40.4	4
2024	19.9	3	59.7	9
Total	$\Sigma y = 146.9$	$\Sigma t = 0$	$\Sigma ty = -10.8$	$\Sigma t^2 = 28$

Since $\Sigma t = 0$ , the least-squares constants are:

$a = \bar{y} = \frac{\Sigma y}{n} = \frac{146.9}{7} = 20.9857$

$b = \frac{\Sigma ty}{\Sigma t^2} = \frac{-10.8}{28} = -0.3857$

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