Question

#1. It seems logical that restaurant chains with more restaurants (units) would have greater sales. This is mitigated by several possibilities. Use the data below to answer whether there is a positive relationship between the number of restaurants and sales. Provide a precise quantitative explanation.

Chain	Sales ($ Billions)	No. of Restaurants (1000s)
McDonald's	17.1	12.4
Burger King	7.9	7.5
Taco Bell	4.8	6.8
Pizza Hut	4.7	8.7
Wendy's	4.6	4.6
KFC	4.0	5.1
Subway	2.9	11.2
Dairy Queen	2.7	5.1
Hardee's	2.7	2.9

#2. It seems logical that restaurant chains with more restaurants (units) would have greater sales. This is mitigated by several possibilities. Speculate on why this relationship may not hold. Give at least two possibilities.

Answer 1

Solution

#1

Final answers are given below. Back-up Theory and Details of calculations follow at the end.

There is a positive relationship between the number of restaurants and sales. Answer 1

Quantitatively,

Correlation coefficient = 0.6361 Answer 2

and

the regression equation is:

sales = - 0.8636 + 0.9203 x number of restaurants Answer 3

Back-up Theory and Details of calculations

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_hat = β_0hat + β_1hatX, ………….......................................................………….(2)

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

Correlation coefficient, r = Sxy/sqrt(Sxx. Syy) ……………………………………................................................................. (4)

where

[Mean X = Xbar = (1/n) Σ(i = 1 to n)x_i ; Mean Y = Ybar = (1/n) Σ(i = 1 to n)y_i Sxx = Σ(i = 1 to n)(x_i – Xbar)²

Syy = Σ(i = 1 to n)(y_i – Ybar)² ; Sxy = Σ(i = 1 to n){(x_i – Xbar)(y_i – Ybar)} ] ........................................................................ (5)

n	9
Xbar	7.1444
ybar	5.7111
Sxx	79.5822
Syy	166.5489
Sxy	73.2356
β1cap	0.9203
β0cap	-0.8636
r	0.6361

#2

Two two possibilities are:

Sales volume (quantity) variation among restaurants of the same company

Price per unit variation among restaurants of the same company.

The relationship that has been established is between sales value and number of restaurants.

Sales value is directly dependent on both sales volume and unit price. Thus, if a chain has large number of restaurants, but a few of them have low volume, low unit price, or both, the total sales value would drastically come down upsetting the relationship. Answer 4

DONE