A maker of baby foods has found a high correlation between the aggregate company

Question

A maker of baby foods has found a high correlation between the aggregate company sales (in $100,000) and the number of births nationally the preceding year. Suppose that the sales and the birth figures during the past (9) years are the following:

                  

  Year >>> 02     03 04 05     06 07 08 09 10

Sales                                6.1     6.4     8.3     8.8     5.1     9.2     7.3     12.5 9.4

US Births (millions) 2.9     3.4     3.5     3.1     3.8     2.8     4.2 3.9     4.1

Determine a linear regression equation for predicting sales based on births.

What is the forecast for sales revenue in year 2011?

If the births in year 2011 are 3.6 million, what is the forecast for sales revenue in year 2012?

Explanation / Answer

Sales(y-axis) is dependent on births(x-axis) of the preceding year.

x = 27.6,   y = 67,  xy = 237.95, x2 = 96.96,

n = 8

x bar = x / n = 27.6 / 8 = 3.45

y bar = y / n = 67 / 8 = 8.375

a = (nxy -xy) / (nx2– (x)2)

= ( (8*237.95) - (27.6*67) ) / ( 8*96.96 - (27.6)2)

= ( 1903.6 - 1849.2 ) / ( 775.68 - 761.76 )

= 54.4 / 13.92

= 3.908

b = y bar – (a * xbar)

= 8.375 - (3.908*3.45)

= 8.375 - 13.4826

= - 5.1076

A linear regression equation is :-

y = ax + b

y = 3.908 x - 5.1076

Forecast for sales revenue in year 2011 :-

y = 3.908 (4.1) - 5.1076

= $10.9152 * 100,000

The forecast for sales revenue in year 2012 :-

y = 3.908 (3.6) - 5.1076

= $8.9612 * 100,000

Births (x) Sales (y) x*y x^2 2.9 6.4 18.56 8.41 3.4 8.3 28.22 11.56 3.5 8.8 30.8 12.25 3.1 5.1 15.81 9.61 3.8 9.2 34.96 14.44 2.8 7.3 20.44 7.84 4.2 12.5 52.5 17.64 3.9 9.4 36.66 15.21 Total 27.6 67 237.95 96.96

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