https://file.aiquickdraw.com/mathbot/file/172672575168528wktk7r.jpg

The questions in the image revolve around analyzing a linear regression model fitted to Facebook's annual revenue data from 2010 to 2022. Here are the solutions:

### a. Calculating the Residual for 2016

The regression equation provided is:
\[ p = -121.89 + 10.33x \]
where $ x $ represents the year since 2000, so for 2016, $ x = 2016 - 2000 = 16 $.

First, calculate the predicted revenue for 2016:
\[ p = -121.89 + 10.33 \times 16 \]

Let's compute this:

\[ p = -121.89 + 10.33 \times 16 \]The predicted revenue for 2016 using the regression model is $43.39 billion.

Next, we'll calculate the residual, which is the difference between the actual revenue and the predicted revenue. The actual revenue for 2016 was $27.64 billion.

\[ \text{Residual} = \text{Actual Revenue} - \text{Predicted Revenue} \]
\[ \text{Residual} = 27.64 - 43.39 \]

Let's compute this residual value.The residual for the year 2016 is $-15.75$ billion dollars. This indicates that the actual revenue was $15.75 billion less than what the regression model predicted.

### b. Estimating the Residual for 2021

To estimate the residual for 2021 visually from the scatterplot and the regression line:
- Identify the actual revenue point for 2021 on the scatterplot.
- Compare this point with the corresponding value on the regression line for 2021.
- The residual is the vertical distance between these two points, with actual revenue appearing to be slightly below the regression line.

### c. Sketching a Residual Plot

To sketch a residual plot:
- Plot the years on the x-axis.
- Plot the residuals (the differences between actual and predicted revenues) on the y-axis.
- Each year's residual is estimated by assessing the vertical distance from the actual data point to the regression line on the scatterplot.

### d. Linear Model Fit Analysis

The correlation of the relationship is 0.95, which is very high, indicating a strong linear relationship. Typically, a correlation above 0.8 suggests a good fit for a linear model. However, the suitability of a linear model should also be judged by the pattern in the residuals. If the residuals show no systematic pattern and are relatively small compared to the revenue values, the linear model is considered a good fit. In this case, a visual check of a residual plot would be crucial.

Would you like to explore further details or have any questions about these calculations?

Related Questions:
1. How do residuals help in validating a regression model?
2. What would be the implications of a systematic pattern in the residuals?
3. How could a non-linear model possibly fit this data better?
4. How does the choice of model affect predictions for future revenues?
5. Can outliers significantly impact the correlation coefficient?

Tip: When interpreting regression analysis, always consider the possibility of outliers or influential points that might skew the results significantly.

The scatterplot shows the total annual revenue (in billions of dollars) for the social media website Facebook, for each year from 2010 to 2022 recorded as years since 2000. Calculate and interpret the residual for the year 2016 if the actual revenue was $27.64 billion.

Learn how to calculate and interpret the residual for the year 2016 in Facebook's revenue regression model using the least-squares regression line. This problem involves key concepts in linear regression, residual analysis, and correlation, suitable for high school students in Grades 10-12.

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