Question

A retailer wanted to estimate the monthly fixed and variable selling expenses. As first step, she collected data from the past 8 months. The total selling expenses ($1000) and the total sales ($1000) were recorded as listed below:

 

20	14
40	16
60	18
50	17
50	18
55	18
60	18
70	20

 

1. Compute the covariance, the coefficient of correlation, and the coefficient of determination and describe what these statistics tell you.
2. Determine the least squares line and use it to produce the estimates the retailer wants.

 

Answer 1

SOLUTION QUESTION A

 

Step 1

 

The following table is the necessary value based on the given data,

Total Sales, x	Selling Expense, y	(x-x̄)	(y-ȳ)	(x-x̄) * (y-ȳ)
20	14	-30.625	-3.375	103.3594
40	16	-10.625	-1.375	14.60938
60	18	9.375	0.625	5.859375
50	17	-0.625	-0.375	0.234375
50	18	-0.625	0.625	-0.39063
55	18	4.375	0.625	2.734375
60	18	9.375	0.625	5.859375
70	20	19.375	2.625	50.85938
x̄ = 50.625	ȳ = 17.375			∑ = 183.125

 

 

Step 2

 

• To calculate Covariance we use the following formula,

= 183.125 / (8-1)

= 26.161

 

Step 3

• To calculate Coefficient of Correlation we use the following formula,

 

 

• To calculate Sx we use the following formula,

= 15.2216

 

• To calculate Sy we use the following formula,

= 1.7678

 

• Thus, the Coefficient of Correlation is,

= 0.9722

 

Step 4

• The Coefficient of Determination is calculated by,

= 0.9452

= 0.9452 * 100%

= 94.52%

 

Step 5

Discussion: The data resulted in a Covariance equal to 26.161. This is because when the variable x and y moved in similar patterns, the Covariance will be a large positive number. For this case, the variables x and y moved in almost similar patterns, this concluded a large positive number of 26.161. Next, we will use the Coefficient of Correlation to determine if there is a strong linear relationship between the two variables or no relationship at all. In this case, the Coefficient of Correlation from the Covariance is 0.9722, this indicates that there is a strong positive linear relationship between the total sales and the selling expenses as it is close to +1. Moving on, we will use the Coefficient of Determination to measure the amount of variation in the dependent variable that is explained by the variation in the independent variable. For this case, the Coefficient of Determination which is a square of the Coefficient of Correlation resulted in 94.52%. Therefore, 94.52% of the variation in the dependent variable can be explained by the independent variable.

 

SOLUTION QUESTION B

Step 1

To determine the least square line, we must first identify the variables. In this case,

Total sale is the independent variable (x)

Selling expense is the dependant variable (y)

 

• So, the least square line to find the Selling Expense (y) is based on the variable Total Sales (x) and can be defined as the following:

 

Step 2

• To calculate b₁ we use the following formula,

= 26.161 / 231.696

= 0.1129

 

Step 3

 

• To calculate  b₀ we use the following formula,

= 17.375 - (0.1129) (50.625)

= 11.659

 

Thus, the least square line is,

 

Selling Expense = 11.6259 + 0.1129x

Through this least square line, we can estimate the Selling Expense for any value of Total Sales by substituting that said value in place of x.

• • Covariance = 26.161
  • Coefficient of Correlation = 0.9722
  • Coefficient of Determination = 0.9452

Selling Expense = 11.6259 + 0.1129x