# Multiple regression coefficients

A company is interested in examining the impact that negative reviews have on the sales of a product. So the Marketing department collected data for the 3 years of the monthly operation of the company and came up with the Regression model where Y is the monthly sales, X1 is the percentage of negative reviews the company received each month, X2 is the average monthly price of the product and X3 is the company's monthly expenses for advertising.

Assume that we are only interested in the impact that negative reviews have on monthly sales and that the negative reviews relate to a specific product feature. So based to the following Excel output, how much money would you be willing to spend to improve that product feature so that to reset the negative reviews you receive monthly?

I'm not sure im understading your question, I'm assuming you got that regression equation fitting some data or from some statistics book as an example. So you have 3 random variables $$X_1$$ , $$X_2$$ and $$X_3$$ , why do you want to force $$X_1 = 0$$? What's the porpouse of the exercise you are doing? Maybe a elaborate a little more so I understand the problem.

Edit
Ok, suppose you want to reset the negative impact the reviews have on your product. The assigment asks you, how much money will you spend to improve the feature of that product to reset the effect of the negative reviews? . We can't control the negative reviews, but we can control the price of the product and the marketing budget.
Let's suppose that the inital price of the product is $$X_2^0$$ and the initial marketing budget is $$X_3^0$$. In the next month we decide lower the price of the product by an amount $$\Delta X_2$$, how are the sales affected? Using the coefficients that are given to you in the table:
$$Y = -0.030X_1 - 1.06(X_2^0 + \Delta X_2) - 0.0009 X_3^0 + 14.87$$
$$Y = -0.030X_1 -1.06\Delta X_2 - 1.06(X_2^0) - 0.0009 X_3^0 + 14.87$$ So setting $$X_1$$ to the amount of negative reviews of past month, we need to solve $$-0.030X_1 -1.06\Delta X_2 = 0$$ so: $$\Delta X_2 \approx -0.028X_1$$ That's no the complete answer because, this is how much the product price needs to be lowered such that it cancels the contribution of the negative reviews. So you would need to have additional information on how money spended on product improvement correlates to product price. Also just saying $$\Delta X_2$$ is not enough, because the coefficient in a linear fit have standard deviations. This means that we need to take into account error propagation, thats why I used the $$\approx$$ and not = in the result. I suppose you have covered error propagation in your class if not check the wikipedia page to follow the reasoning: https://en.wikipedia.org/wiki/Propagation_of_uncertainty
The correct presentation for the answer will be $$\Delta X_2 = (-0.028 \pm \sigma)X_1$$, where $$\sigma$$ is the standard error you estimate using the method in the link.
Moreover we haven't touched the marketing budget, maybe we can change that too! The problem is that the marketing budget seems negativility correlated to the sales even when you account the standard deviation, so we will leave that there.
That's all the reasoning I can offer without more information, check with your TA or Professor if that's the correct line of reasoning to follow, if you are still stucked after talking to them, make another edit and I see what can I do.

(Also note that we have asume that all the variables are indepedent, in other words that changing $$X_i$$ doesn't affect $$X_j$$)

• Thank you very much. We are asked from the excercise quudelines to do so and thus we have given the excel output. Nov 30, 2020 at 21:39
• I think maybe you have misunderstood the excercise, can you post the exact guidelines that you were given?
– Capu
Nov 30, 2020 at 22:02
• Very gladly! I have written the exact quidelines of the excercise above.Thank you very much in advance! Nov 30, 2020 at 22:25
• @Magda if have edited the answer according to the table you uploaded :)
– Capu
Nov 30, 2020 at 23:43
• Really Thank you very much, We will be in touch! Dec 1, 2020 at 5:12