Question

A store manager believes items with discounts (10% off, 20% off, etc.) sell better than items offered at full price. She also believes items with well-known brand names sell better than items not associated with a well-known brand. Furthermore, she believes that when the prices of brand-name items are discounted sales are much higher than normal.

You think this sounds like she is describing an interaction between brand-named and discounts and you want to use regression to test the effects of discounts, brands, and their interaction.

Assume that the manager provides the following IVs:

• sales                     number of sales (the DV
• item_cost            cost of the item (a continuous IV)
• brand (0,1)          brand=1 for items associated with a well-known brand and 0 otherwise
• discount (0,1)      discount = 1 for items with discounted prices and 0 otherwise

Question: What specific evidence would you need to conclude that the interaction does affect sales?

Answer 1

In the Problem we have to use Regression model with interaction term of categorical data.

Let Y : Sales items cost Number of sales cost of the items.

X1 : Brand of an item represented by an indicator variable.

brand=1 for items associated with a well-known brand and 0 otherwise.

X2: Discount (0,1) represented by indicator variable.

discount = 1 for items with discounted prices and 0 otherwise

Now to study interaction between brand-named and discounts , I have introduced third variable in the regression model given below,

Let X3 = Brand name × Discount

Then the regression model becomes,

Where a: estimated mean sales when data contains 0.

b1, b2, b12 are the regression coefficients.

Then we fit the multiple regression model including brands , discounts and their interactions.

Then we conduct a hypothesis test to determine whether there is a significant linear relationship between an independent variable X1, X2 , X3 and a dependent variableY.

The test procedure consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

Step 1) : Set null hypothesis: H0= b12=0 Vs H1 = b12 0

If the null hypothesis is rejected, then we say that iteration term X3 affect on Y (sales).

Step 2) formulate an analysis plan: It consists 2 steps:

• Significance level. choose significance level equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
• Test method. Use a linear regression t-test (described in the next section) to determine whether the slope of the regression line differs significantly from zero.

Step 3) Analyze sample data:

i) Find the standard errors of regression coefficients a , b1, b2 , b12 by using the following formula.

Similarly calculate for b2 and b12.

ii) Find DF: The degrees of freedom is equal to

DF = n-2

iii) Test statistic. The test statistic is a t statistic (t) defined by the following equation.

t1 = b1 / SE(b1)

where b1 is the slope of the sample regression line, and SE is the standard error of the slope.

iv)

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a t statistic, use the t distribution calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.

Step v) Interpret the results: This involves comparing the P-value to the significance level, and rejecting the null hypothesis when the P-value is less than the significance level.

All above steps can be summarised in table given below,

Regression equation: Y = a+b1 X1+ b2 X2 + b12X3 +€

Predictors	Coeff	SEcoeff	Test statistic	P value
Constant		SE(a)	to	P0
Brands(X1)		SE(b1)	t1	P1
Discounts(X2)		SE(b2)	t2	P2
Interaction (X3)		SE(b12)	t3	P3