12. Consider how to estimate the quality of restaurant from the ratings of 1 to 5 stars as in Example 7.17.

(a) What would this predict for a restaurant that has two 5-star ratings? How would you test from the data whether this is a reasonable prediction?

(b) Suppose you wanted not to optimize just for the 5-star restaurants, but for all restaurants. How can this be done?

(c) Can

c, as computed in Example 7.17, be negative? Give a scenario where this might occur.

(d) Why might we choose not to use the average rating for a0? What else might you use?

[Hint: A new restaurant may be quite different from a well-established restaurant. Hint:

Picking a restaurant at random, and then a rating at random will have a different average than picking a rating at random.]

It might be useful to try this on some real data. For example, Movielens makes a data set of movie ratings available, which can be used to test such hypotheses (albeit on movies, not restaurants).