Question

6 A market manager condacted a study to determine whether there is a linear relatioreship betwecn money spent and advertising and company sales. Determine if there is a linear correlation or not. Uise a-0.05 Advertising Expnscs 24 1.6 2.0 2.6 14 16 20 22 Company184 Sales 225 184 220 240 180 184 186 215 (Each one times $1000) Find: a) The correlation coefficient r b) In your opinion, is there a strong linear correlation? Weak lincar correlation? Explain c) What is the centroid? d) Find the regression line equation for the advertising expenditures and company sales data e) Use this equation to predict the expected company sales when 1,5 thousand dollars have been used in advertising.

Answer 1

Solution

Note:

Final answers are given first. Back-up Theory and Calculation Details follow at the end.

Part (a)

Correlation coefficient, r = 0.9129 Answer 1

Part (b)

Since r is very close to 1, the linear correlation is strong. Answer 2

Part (c)

Centroid = point (xbar, ybar) = (1.975, 204.25) Answer 3

Part (d)

Regression line equation:

Company Sales ($’000) = 104.061 + 50.729Advertising Expenses ($’000) Answer 4

Part (e)

Substituting x = 1.5,

Prediction for expected sales = 180.15 ($’000) Answer

DONE

Back-up Theory and Calculation Details

The linear regression model Y = β₀ + β₁X + ε, …………………………….………………………..…..(1)

where ε is the error term, which is assumed to be Normally distributed with mean 0 and variance σ².

Estimated Regression of Y on X is given by: Y_cap = β_0cap + β_1capX, …………………………………….(2)

where β_1cap = Sxy/Sxx and β_0cap = Ybar – β_1cap.Xbar..……………………………………..……….…..(3)

Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i ………………………………………………..…….……….….(4)

Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i ………………………………………………..…….……….….(5)

Sxx = Σ(i = 1 to n)(x_i – Xbar)² ………………………………………………………..…..…………....(6)

Syy = Σ(i = 1 to n)(y_i – Ybar)² ………………………………………………………....………………(7)

Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)} ………………………………………..………………….(8)

Correlation coefficient, r = Sxy/sqrt(Sxx. Syy) …………………………………………….……….. (10)

Now, to work out the solution…

Let

y = Company Sales ($’000)and x = Advertising Expenses ($’000)

Data

i	xi	yi
1	2.4	225
2	1.6	184
3	2	220
4	2.6	240
5	1.4	180
6	1.6	184
7	2	186
8	2.2	215

Summary of Excel Calculations

n	8
Xbar	1.9750
ybar	204.2500
Sxx	1.235
Syy	3813.5
Sxy	62.65
β1cap	50.72874494
β0cap	104.0607287
r	0.912905285

Complete