Question

1. the scatter - plot 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.

a. the least - squares regression line for the relationship is $hat{y}=-121.89 + 10.33x$. calculate and interpret the residual for the year 2016 if the actual revenue was $27.64 billion.
b. without any calculations, use the scatter - plot and the graph of the least squares regression line to estimate the residual for the year 2021.
c. sketch a residual plot for the data on the axes given by approximating each residual.
d. the correlation of the relationship is 0.95. is a linear model a good fit for the data? explain.

Explanation:

Step1: Calculate predicted value for 2016

The year 2016 is \(x = 16\) (since \(2016 - 2000=16\)). Using the regression line \(\hat{y}=- 121.89 + 10.33x\), we substitute \(x = 16\):
\(\hat{y}=-121.89+10.33\times16=-121.89 + 165.28=43.39\) (billion dollars)

Step2: Calculate the residual for 2016

The residual \(e=y-\hat{y}\), where \(y = 27.64\) (actual revenue) and \(\hat{y}=43.39\). So \(e=27.64 - 43.39=-15.75\) (billion dollars). This means the model over - predicted the revenue in 2016 by 15.75 billion dollars.

Step3: Estimate residual for 2021

For 2021, \(x = 21\). By looking at the scatter - plot and the regression line, if the point for 2021 is above the regression line, the residual is positive. Visually estimate the vertical distance between the point and the line. Let's assume a rough estimate of \(e\approx5\) (billion dollars) (this is a visual estimate and can vary depending on the scale and accuracy of the plot).

Step4: Sketch residual plot

For each \(x\) value (year), calculate the residual \(e=y - \hat{y}\). Plot the years (on the \(x\) - axis, \(x = 10\) for 2010, \(x = 11\) for 2011, etc.) against the residuals (on the \(y\) - axis). The points should be randomly scattered around the \(y = 0\) line if the linear model is appropriate.

Step5: Analyze goodness - of - fit

The correlation coefficient \(r = 0.95\). Since \(|r|\) is very close to 1 (where \(r\) ranges from - 1 to 1), it indicates a strong positive linear relationship. So, a linear model is a good fit for the data.

Answer:

a. The residual for 2016 is \(-15.75\) billion dollars. The model over - predicted the revenue by 15.75 billion dollars.
b. A rough estimate of the residual for 2021 is 5 billion dollars (visual estimate).
c. Sketch a plot with years on \(x\) - axis and residuals on \(y\) - axis, with points randomly scattered around \(y = 0\).
d. A linear model is a good fit for the data because the correlation coefficient \(r = 0.95\), which indicates a strong positive linear relationship.