changed the order of the animal names so they correspond to the question
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No, you shouldn't add all of the coefficients together. You essentially have the model

$$ {\rm lifespan} = \beta_{0} + \beta_{1} \cdot {\rm pig} + \beta_{2} \cdot {\rm wolf} + \beta_{3} \cdot {\rm fox} + \beta_{4} \cdot {\rm weight} + \varepsilon $$$$ {\rm lifespan} = \beta_{0} + \beta_{1} \cdot {\rm fox} + \beta_{2} \cdot {\rm pig} + \beta_{3} \cdot {\rm wolf} + \beta_{4} \cdot {\rm weight} + \varepsilon $$

where, for example, ${\rm pig} = 1$ if the animal was a pig and 0 otherwise. So, to calculate $\beta_{0} + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4}$ as you've suggested for getting the overall average when ${\rm weight}=1$ is like saying "if you were a pig, a wolf, and a fox, and your weight was 1, what is your expected lifespan?". Clearly since each animal is only one of those things, that doesn't make much sense.

You will have to do this separately for each animal. For example, $\beta_{0} + \beta_{1} + \beta_{4}$$\beta_{0} + \beta_{2} + \beta_{4}$ is the expected lifespan for a pig when its weight is 1.

No, you shouldn't add all of the coefficients together. You essentially have the model

$$ {\rm lifespan} = \beta_{0} + \beta_{1} \cdot {\rm pig} + \beta_{2} \cdot {\rm wolf} + \beta_{3} \cdot {\rm fox} + \beta_{4} \cdot {\rm weight} + \varepsilon $$

where, for example, ${\rm pig} = 1$ if the animal was a pig and 0 otherwise. So, to calculate $\beta_{0} + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4}$ as you've suggested for getting the overall average when ${\rm weight}=1$ is like saying "if you were a pig, a wolf, and a fox, and your weight was 1, what is your expected lifespan?". Clearly since each animal is only one of those things, that doesn't make much sense.

You will have to do this separately for each animal. For example, $\beta_{0} + \beta_{1} + \beta_{4}$ is the expected lifespan for a pig when its weight is 1.

No, you shouldn't add all of the coefficients together. You essentially have the model

$$ {\rm lifespan} = \beta_{0} + \beta_{1} \cdot {\rm fox} + \beta_{2} \cdot {\rm pig} + \beta_{3} \cdot {\rm wolf} + \beta_{4} \cdot {\rm weight} + \varepsilon $$

where, for example, ${\rm pig} = 1$ if the animal was a pig and 0 otherwise. So, to calculate $\beta_{0} + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4}$ as you've suggested for getting the overall average when ${\rm weight}=1$ is like saying "if you were a pig, a wolf, and a fox, and your weight was 1, what is your expected lifespan?". Clearly since each animal is only one of those things, that doesn't make much sense.

You will have to do this separately for each animal. For example, $\beta_{0} + \beta_{2} + \beta_{4}$ is the expected lifespan for a pig when its weight is 1.

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No, you shouldn't add all of the coefficients together. You essentially have the model

$$ {\rm lifespan} = \beta_{0} + \beta_{1} \cdot {\rm pig} + \beta_{2} \cdot {\rm wolf} + \beta_{3} \cdot {\rm fox} + \beta_{4} \cdot {\rm weight} + \varepsilon $$

where, for example, ${\rm pig} = 1$ if the animal was a pig and 0 otherwise. So, to calculate $\beta_{0} + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4}$ as you've suggested for getting the overall average when ${\rm weight}=1$ is like saying "if you were a pig, a wolf, and a fox, and your weight was 1, what is your expected lifespan?". Clearly since each animal is only one of those things, that doesn't make much sense.

You will have to do this separately for each animal. For example, $\beta_{0} + \beta_{1} + \beta_{4}$ is the expected lifespan for a pig when its weight is 1.