Is this an interaction model or a quadratic? The price of the “Spirit of the Dev

Question

Is this an interaction model or a quadratic? The price of the “Spirit of the Devil” brand beer fluctuates from month to month due to the change in raw material prices and other factors. The company that sells this beer also advertises several times a month. The company wishes to investigate the impact of price and advertising on beer sales and collects monthly sales data (in thousands of dollars), number of advertisements per month and the average monthly price of beer (in dollars). The company suspects that higher prices lead to lower sales and more advertising leads to greater sales. In addition, the company also expects that the negative impact of prices on sales is affected by the level of advertising such that higher levels of advertising may lower the negative effect of price. Express the regression model that the company must estimate to test their intuition, state the relevant hypothesis, estimate the model and provide interpretation. Given the results of your estimation, predict the beer sales when the company spends on 12 advertisements and prices the beer at $1.25. The data is in spirit.sav .

Sales Price Ads

159.14   .80   13
33.08 .75   3
156.04   .90   13
93.86 1.00   8
27.02 1.10   3
130.69   1.00   11
104.17   1.20   9
117.84   1.25   10
32.88 .90   3
145.77   1.50   12
91.65   1.00   8
94.03   1.25   8

Explanation / Answer

We can analyse this using the open source statistical package R

# read the data into R dataframe
data.df<- read.csv("C:\Users\586645\Downloads\Chegg\sales.csv",header=TRUE)
str(data.df)

### fit the regression for interaction

fit <- lm(Sales ~ Price + Ads + Price*Ads, data=data.df)
summary(fit)

### fit the regression for quadratic

fit <- lm(Sales ~ Price + Ads + Price*Ads, data=data.df)
summary(fit)

The summary of the model is

summary(fit)

Call:
lm(formula = Sales ~ Price + Ads + Price * Ads, data = data.df)

Residuals:
Min 1Q Median 3Q Max
-1.8416 -1.1530 -0.4664 0.9358 2.2313

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 12.5355 7.3894 1.696 0.1282
Price -21.3923 7.7874 -2.747 0.0252 *
Ads 10.9362 0.7087 15.432 3.09e-07 ***
Price:Ads 1.8573 0.7429 2.500 0.0369 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.666 on 8 degrees of freedom
Multiple R-squared: 0.9991,   Adjusted R-squared: 0.9988 ## the rsquare value of the model is 0.9991
F-statistic: 2935 on 3 and 8 DF, p-value: 1.671e-12 , as the p value is less tha 0.05 , hence the model is statistically signficant and is not by chance alone

r2 : The model has a very high square of 99.91% , this means that the model can really explain the variation of the data well.. Hence the model is very good

The regression equation is
Sales =12.53 -21.39*Price + 10.93*Ads +1.85*PriceAds

Put the given values in the regression equation

Sales =12.53 -21.39*1.25 + 10.93*12 +1.85*1.25*12 = 144.7

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