Question

As a data scientist of a company, you want to analyze the following data collected by your company which relates the advertising expenditure A in thousands of dollars to total sales S in thousands of dollars. The following table shows this relationship

Advertising Expenditure (A)	Total Sales (S)
18.6	312
18.8	322
18.8	333
18.8	317
19	301
19	320
19.2	305

Using Advertising expenditure (A) as the domain and Total Sales (S) as the range, the data is not a function because the value 18.8 and 19 appear in the domain more than once with a different corresponding value of the range each time.

--Interpret the slope and y-intercept of this equation.

--Express this equation as a function S of A and find its domain.

--Predict the sales if the advertising expenditure is $25000.

Answer 1

Solution:
intercept of the regression line can be calculated as
a = (summation(y) * summation(X^2) - summation(x)*summation(xy))/n*summation(x^2)-(summation(x))^2

X	Y	XY	X2	Y2
18.6	312	5803.2	345.96	97344
18.8	322	6053.6	353.44	103684
18.8	333	6260.4	353.44	110889
18.8	317	5959.6	353.44	100489
19	301	5719	361	90601
19	320	6080	361	102400
19.2	305	5856	368.64	93025
132.2	2210	41731.8	2496.92	698432

So a = (2210*2496.92)-(132.2*41731.8) / (7*2496.92)-(132.2*132.2) = 780.775
Slope of the line can be calculated as
b = n*summation(XY) - summation(X)*summation(Y)/n*summation(X^2) - (summation(x))^2
b = (7*41731.8) - (132.2*2210) / (7*2496.92)-(132.2*132.2) = -24.625
So the equation is in terms of S and A as follows:
S = 780.775 -24.625*A

From the slope we can say that As advertising expenditure increase by 1 unit than sales will decrease by 24.625unit. and intercept tells that if Expenditure is 0 than sales would be 780.775

If Advertising expenditure = $25,000
than Sales. = 780.775 -24.625*25 = 780.775 - 615.625 = 165.15 or $165000