What is a model, and why is it useful? How do we evaluate the trustworthiness of a model?

This week we are going to check out linear models. When is a linear model most applicable, and how do we construct one?

Let's also take a moment to revisit Anscombe's quartet (image here)...
Here are some assumptions made when doing a linear regression. What do they mean?

The mean of the response, 𝐸(𝑌𝑖), at each set of values of the predictors, (𝑥1𝑖,𝑥2𝑖,…), is a Linear function of the predictors.

The errors, ε𝑖, are Independent.

The errors, ε𝑖, at each set of values of the predictors, (𝑥1𝑖,𝑥2𝑖,…), are Normally distributed.

The errors, ε𝑖, at each set of values of the predictors, (𝑥1𝑖,𝑥2𝑖,…), have Equal variances (see heteroscedasticity)