https://file.aiquickdraw.com/mathbot/file/17307799422677sg051f2.jpg

Let's solve each part of the question based on the information in the image.

### Part (a)
The question is asking us to interpret the relationship between rental revenue and theater revenue relative to their means.

- When rental revenue \( y \) is **less than the mean** of rental revenue, it tends to be paired with theater revenue \( x \) values that are **less than the mean** of theater revenue.
- Similarly, when rental revenue \( y \) is **greater than the mean**, it is often paired with theater revenue values that are also **greater than the mean**.

Since the trend is positively correlated, the answer for part (a) should be:
> **"less than"**

### Part (b)
In part (b), we are asked to determine the change in rental revenue \( y \) for an increase of one million dollars in theater revenue \( x \), according to the regression equation.

The regression equation given is:
\[
\hat{y} = 3.30 + 0.15x
\]
In this equation:
- \( 3.30 \) is the y-intercept, meaning it's the base rental revenue when theater revenue is zero.
- \( 0.15 \) is the slope, which indicates the rate of change in \( y \) (rental revenue) for each 1 million dollar increase in \( x \) (theater revenue).

Thus, for every increase of one million dollars in theater revenue, the rental revenue increases by:
> **0.15 million dollars**

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### Summary of Answers:
1. **(a)** "less than" the mean of theater revenue.
2. **(b)** 0.15 million dollars.

Would you like more details on any part of this solution, or do you have additional questions?

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### Related Questions for Further Exploration:
1. How would the interpretation change if the slope of the regression line was negative?
2. What would be the rental revenue if theater revenue was zero, based on the regression equation?
3. Can the regression line be used to accurately predict rental revenue for very high theater revenue values?
4. How does the correlation coefficient relate to the interpretation of "less than" and "greater than" in part (a)?
5. What assumptions underlie the use of a linear regression model in this scenario?

### Tip:
When interpreting regression equations, the slope tells us the change in the dependent variable (rental revenue, \( y \)) for each unit increase in the independent variable (theater revenue, \( x \)). This can help understand relationships even if the line doesn’t pass through every data point exactly.

Can movie rental revenue be predicted? A movie studio wishes to determine the relationship between the revenue from rental of comedies on streaming services and the revenue generated from the theatrical release of such movies. The studio has the following bivariate data from a sample of fifteen comedies released over the past five years. These data give the revenue x from theatrical release (in millions of dollars) and the revenue y from streaming service rentals (in millions of dollars) for each of the fifteen movies. Also shown are the scatter plot and the least-squares regression line for the data. The equation for this line is y-hat = 3.30 + 0.15x. Based on the studio's data and the regression line, complete the following.

Explore how to predict movie rental revenue based on theater revenue using a linear regression model. This example includes bivariate data analysis, scatter plots, and the least-squares regression line y-hat = 3.30 + 0.15x, ideal for Grades 10-12.

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