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A dataset consists of sales of a product in different regions alongside advertising budget in three different media: Radio, Instagram, Facebook. A model is developed to predict sales on the basis of three media budgets. The model coefficients are as follows

Coefficient Std Error p-value1
Intercept 3.365 0.3119 0.000105
Radio 0.294 0.0075 0.000086
Instagram 0.06 0.0041 0.78
Facebook 0.059 0.0011 0.000101

If the company decided to shift \\$1000 marketing spend from Facebook to Radio. How much sales increase/decrease can you expect approximately?

a) $294

b) $299

c) $352

d) $235

The question appeared on a test as is. I am trying to understand how we can actually calculate the effect and or/ how to guess the right option?

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  • $\begingroup$ Welcome to Cross Validated! We have a particular way we handle self-study questions like this. What progress have you made with this problem? $\endgroup$
    – Dave
    Commented Oct 7, 2021 at 21:28
  • $\begingroup$ I have checked the stats books I have and online but have not found a similar question. From what I understand I know what the p-value, the std error means the margin of error, and the coefficient value is 'm' in the 'y=mx+c' equation for linear regression. Intercept represents the mean value of y when x =0. @Dave $\endgroup$
    – xpwaste
    Commented Oct 7, 2021 at 21:33

1 Answer 1

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This question has a lot of details but it's actually quite simple. It only requires you to separate the wheat from the chaff. Recall that $E[y|x]=\beta^Tx$, that's all you need here.

Let the vector of initial budgets be $x=(x_r,x_i,x_f)$ and the expected sales $$E[y|x]=3.365+0.294x_r+0.06x_i+0.059x_f$$ Now we've shifted 1000 from facebook to radio so our budget vector is now $x'=(x_r+1000,x_i,x_f-1000)$. The expected sales are

$$E[y|x']=3.365+0.294(x_r+1000)+0.06x_i+0.059(x_f-1000)\\=3.365+0.294x_r+294+0.06x_i+0.059x_f-59\\=3.365+0.294x_r+0.06x_i+0.059x_f+294-59\\=E[y|x]+235$$

So the correct answer is (d).

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