The number of members of a co-operative society (in hundreds) over seven years is given below. Fit a straight-line trend by the method of least squares and forecast the number of members for 2018 and 2020.

| Year | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
|---|---|---|--

Question

The number of members of a co-operative society (in hundreds) over seven years is given below. Fit a straight-line trend by the method of least squares and forecast the number of members for 2018 and 2020.

| Year | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 |
|---|---|---|---|---|---|---|---|
| Members | 13 | 18 | 17 | 20 | 21 | 26 | 25 |

Accepted Answer

There are $n = 7$ years (odd), so take **2014 as origin** with $X = -3, -2, -1, 0, 1, 2, 3$; then $\sum X = 0$ and $a = \dfrac{\sum Y}{n},\ b = \dfrac{\sum XY}{\sum X^2}$.

| Year | $Y$ | $X$ | $XY$ | $X^2$ |
|---|---|---|---|---|
| 2011 | 13 | $-3$ | $-39$ | 9 |
| 2012 | 18 | $-2$ | $-36$ | 4 |
| 2013 | 17 | $-1$ | $-17$ | 1 |
| 2014 | 20 | 0 | 0 | 0 |
| 2015 | 21 | 1 | 21 | 1 |
| 2016 | 26 | 2 | 52 | 4 |
| 2017 | 25 | 3 | 75 | 9 |
| **Total** | **140** | **0** | **56** | **28** |

So
$$a = \frac{140}{7} = 20, \qquad b = \frac{56}{28} = 2,$$
giving the trend line $Y_t = 20 + 2X$ (origin 2014, unit = 1 year).

**Forecast.** Count $X$ forward from 2014: for **2018**, $X = 4$, so $Y_t = 20 + 2(4) = 28$; for **2020**, $X = 6$, so $Y_t = 20 + 2(6) = 32$. …

Year	$Y$	$X$	$XY$	$X^2$
2011	13	$-3$	$-39$	9
2012	18	$-2$	$-36$	4
2013	17	$-1$	$-17$	1
2014	20	0	0	0
2015	21	1	21	1
2016	26	2	52	4
2017	25	3	75	9
Total	140	0	56	28

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