Does a larger movie budget lead to a higher IMDb rating?

The data contain movie budgets, ratings, and numbers of votes.

Most ratings are near the middle of the scale.

Higher-budget movies tend to have slightly higher ratings.

Correlation measures the strength of a linear relationship.

\[ r=\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2\sum_{i=1}^{n}(y_i-\bar{y})^2}} \]

For log budget and rating, \(r=\) -0.08.

Votes are included because popular movies receive more ratings.

\[ Rating_i=\beta_0+\beta_1\log_{10}(Budget_i)+ \beta_2\log_{10}(Votes_i)+\epsilon_i \]

The coefficients are estimated by least squares.

\[ \hat{\boldsymbol{\beta}}=(X^TX)^{-1}X^Ty \]

The graph compares budget, popularity, and rating together.

A movie has a budget of $50 million and 50,000 votes.

new <- data.frame(b = log10(50000000), v = log10(50000))
round(predict(fit, new), 2)

##    1 
## 6.56

The model predicts the rating shown above.

A large budget does not guarantee a high rating.