Question

A business person is trying to estimate the relationship between the price of good X and the sales of good Z of certain groups of staples. Tests in similar cities throughout the country have yielded the data below:

PRICE (X)                  SALES (Z)     

$15                              3300

$20                              3900

$25                              4750

$30                              5500

$40                              6550

$50                              7250

A simple linear regression of a model SALES (Z) = b + b PRICE(X)

Was run and the computer output is shown below:

PRICE OF X / SALES OF Z

REGRESSION FUNCTION & ANOVA FOR SALES(Y)

    SALES(Z) = 1740.686 + 115.5882 PRICE(X)

    R-Squared                              = 0.977573

    Adjusted R-Squared               = 0.971966

    Standard error of estimate      = 255.2152

    Number of cases used            = 6

Analysis of Variance

                                                                                             p-value

    Source             SS               df         MS                    F Value        Sig Prob

    Regression 1.13565E+07   1        1.13565E+07    174.35450     0.000190

    Residual      260539.20000 4        65134.80000

    Total            1.16171E+07   5

PRICE OF X / SALES OF Z

REGRESSION COEFFICIENTS FOR SALES(Z)

                                                                                                p-value

     Variable    Coefficient       Std Error         t Value            Sig Prob           

     Constant     1740.68600    282.52800      6.16111           0.003522

     PRICE(X)    115.58820      8.75381          13.20434         0.000190 *

     Standard error of estimate = 255.2152

     Durbin-Watson statistic    = 1.240299

QUESTIONS

What is the estimated equation of the model:

SALES(Z) = b + b PRICE(X)?

What sort of relationship exists between SALES OF Z and the PRICE OF X? Does this relationship make sense? Why or why not?

What can you say about GOOD Z and GOOD X (a good can be an item, a commodity, etc.). Give an example of good X and good Z that can display this kind of relationship

At a level of significance, ? = 0.05, test for the significance of the relationship.

SALES(Z) = b + b PRICE(X)

H : b = 0

H : b ? 0

What is your conclusion?

Answer 1

we look at the estimates of regression coefficients given in the following table

REGRESSION COEFFICIENTS FOR SALES(Z)
					p-value
	Variable	Coefficient	Std Error	t Value	Prob Sig
	Constant	1740.68600	282.52800	6.16111	0.003522
	PRICE(X)	115.58820	8.75381	13.20434	0.000190

From the above table we know that the value of an estimate of intercept is

The value of an estimate slope coefficient is

The estimated equation of the model is

The slope coefficient of PRICE(X) is positive. This indicates that as the price of X increases the sales of product Z increases. In fact for $1 increase in the price of X, the sales of Z increases by 115.58 units

This relationship makes sense only if products Z and X are substitues. That is if consumers can use Z in place of X. That is when the price of X increases the consumers are perfectly willing to use Z in place of X.

There are many examples of substitute goods. Some are listed below

• Beef Vs Chicken
• Cable TV Vs on demand streaming service such as Netflix/Prime (to an extent)
• Tea Vs Coffee
• Tide Vs Surf
• Air travel Vs Driving car

Let the regression model that needs to be estimated is

We want to test the significance of the sope coefficient

We want to test the hypotheses

The test statistics and the p-values are given in the table above.

The test statistics t-value = 13.20434. The corresponding p-value = 0.000190

this p-value is less than the level of significance alpha = 0.05. That means we reject the null hypothesis.

We conclude that there is sufficicient evidewnce to support the claim that the relationship between Sales(Z) and price(X) is significant