Question

1. the scatterplot displays the relationship between the number of spotify streams and u number of youtube views for 20 popular songs. the least - squares regression line is $hat{y}=5.024 + 0.318x$, where $x$ = number of spotify streams (in millions) for the song and $y$ number of youtube views (in millions) for the song

a. describe the relationship between the two variables.
b. interpret the $y$-intercept of the least - squares regression line in context.
c. use the least - squares regression line to predict the number of youtube views for a song with 250 million spotify streams. show your work.
d. the song with 250 million spotify streams has 32.229 million views on youtube. calculate and interpret the residual.
e. the correlation between these two variables is 0.77. can we conclude that an increase in spotify streams will cause an increase in youtube views for these popular songs? explain.

Explanation:

Step1: Predict YouTube views

Given the regression line \(\hat{y}=5.024 + 0.318x\), substitute \(x = 250\) (million Spotify streams) into the equation.

$$
LATEXBLOCK0
$$

Step2: Calculate the residual

The formula for the residual \(e=y-\hat{y}\). We know \(y = 32.229\) (actual YouTube views in millions) and \(\hat{y}=84.524\) (predicted YouTube views in millions)

$$ e=32.229 - 84.524=- 52.295 $$

The residual of \(-52.295\) means that the model over - predicted the number of YouTube views for this song by \(52.295\) million.

Step3: Analyze causation from correlation

Correlation (\(r = 0.77\)) shows a strong positive association. But correlation does not imply causation. There could be lurking variables (e.g., marketing efforts, song quality, artist popularity) that affect both Spotify streams and YouTube views simultaneously.

Answer:

a. There is a moderately strong, positive linear relationship between the number of Spotify streams and YouTube views with a few potential outliers.
b. When the number of Spotify streams (\(x = 0\)) is zero, the predicted number of YouTube views is \(5.024\) million.
c. Predicted YouTube views: \(84.524\) million.
d. Residual \(=-52.295\); the model over - predicted the number of YouTube views for this song by \(52.295\) million.
e. No. Correlation shows an association, not causation. There could be other factors (lurking variables) influencing both variables.