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In "Introduction to statistical learning wit R", there's a simple linear regression fit on a 'TV' feature where the target class is the 'Sales'.

Beta 0 is 7.03 and Beta 1 is 0.0475

Now we want to predict how much sales will increase when 1000 dollars is spent on TV advertising. I naturally thought that this would

1000 * 0.0475 + 7.03 

which equals to 54.6

The book however says that it's 47.5 which i presume is because

1000 * 0.0475

I then ran my own linear regression model on the data, and asked it to predict on the 1000 dollars and it agreed with me that it's 54.6

which is it?

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predict how much sales will increase

They are asking how much sales will increase. $\beta_1$ shows how much sales will change for every dollar spent on TV advertising. So the increase is $1000 \times \beta_1 = 1000 \times 0.0475 = 47.5$.

What you have calculated is the predicted value with 1000 dollars spent total on TV advertising: $\hat{y} = \beta_0 + \beta_1X$, or $sales = 7.03 + 0.0465 \times X$.

When total dollars spent on TV advertising is 1000, then the predicted value is what you have: $\hat{y} = 7.03 + .0465 \times 1000 = 54.60$

Your answer was the predicted value when X = 1000. What you were asked for was the increase in the predicted value when X increases by 1000.

You could do it between any two predicted values:

  • When X = 5000, $\hat{y}$ = 244.53.

  • When X = 6000 (i.e., increases by 1000), $\hat{y}$ = 292.03

  • $\hat{y}_{X=6000} - \hat{y}_{X=5000}$ = 292.03 - 244.53 = 47.5.

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