Question

The data obtained on X, the length of time in weeks that a promotional project has been in progress at a small business, and Y, the percentage increase in weekly sales over the period just prior to the beginning of the campaign.

X	1	2	3	4	1	3	1	2	3	4	2	4
Y	10	10	18	20	11	15	12	15	17	19	13	16

Find the equation of the regression line to predict the percentage increase in sales if the campaign has been in progress for 1.5 weeks.

Answer 1

Here, X = Length of time in weeks,
Y = Percentage increase in weekly sales

X = xi	Y = yi	`x_i^2`	xi yi
1	10	1	10
2	10	4	20
3	18	9	54
4	20	16	80
1	11	1	11
3	15	9	45
1	12	1	12
2	15	4	30
3	17	9	51
4	19	16	76
2	13	4	26
4	16	16	64
30	176	90	479

From the table, we have

n = 12, ∑ xi = 30, ∑ yi = 176, `sum x_i^2 = 90`, ∑ x_i y_i = 479

∴ `bar x = (sum x_i)/"n" = 30/12 = 2.5`

`bar y = (sum y_i)/"n" = 176/12 = 14.67`

Now, `"b"_"YX" = (sum"x"_"i" "y"_"i" - "n" bar "x" bar "y")/(sum "x"_"i"^2 - "n" bar "x"^2)`

`= (479 - 12 xx 2.5 xx 14.67)/(90 - 12 xx (2.5)^2)`

`= (479 - 440.1)/(90 - 75) = 38.9/15 = 2.59`

Also, 

`"a" = bar y - "b"_"YX"  bar x`

= 14.67 - 2.59 × 2.5 = 14.67 - 6.475 = 8.195

∴ a ≈ 8.2

∴ The regression equation of percentage increase in weekly sales (Y) on length of weeks (X) is

Y = a + b_YX + X

i.e., Y = 8.2 + 2.59 X

For X = 1.5, we get

Y = 8.2 + 2.59(1.5) = 8.2 + 3.885 = 12.085

∴ Increase in sales is 12.085% if the campaign has been in progress for 1.5 weeks.