Question

online entertainment streaming services have gained in popularity in recent years as an alternative to traditional television. one such company has seen steady growth in each period of 3 months, called a quarter, over the past 4 years. the scatterplot shows the relationship between the number of quarters since january 2014 and the log of the number of members to the streaming service. a least - squares equation that summarizes this relationship is ( overline{log subscribers}=0.026 ) (quarters) - 1.299. based on the scatterplot and residual plot, what type of model is appropriate for comparing time and subscribers? a linear model is appropriate because the residual plot shows a random scatter of points. a logarithmic model is appropriate because the log of the number of subscribers was taken. a power model is appropriate because the relationship between period and the log of subscribers is roughly linear. an exponential model is appropriate because the relationship between period and the log of subscribers is roughly linear and the residual plot shows no distinct pattern.

Explanation:

• For an exponential model \(y = ab^{x}\), taking the logarithm of both sides gives \(\log y=\log a + x\log b\). If the relationship between \(x\) (quarters) and \(\log y\) (log of subscribers) is linear (as indicated by the least - squares equation \(\widehat{\log\text{ subscribers}}=0.026(\text{quarters})- 1.299\)) and the residual plot shows no distinct pattern (which is a requirement for a good fit in regression analysis), an exponential model is appropriate.
• A linear model (\(y = mx + c\)) is not the case here because we are dealing with the log - transformed data of the response variable.
• A logarithmic model (\(y=a + b\log x\)) has the predictor variable \(x\) in a logarithmic form, which is not our situation.
• A power model \(y = ax^{b}\) when taking logarithms gives \(\log y=\log a + b\log x\), but our predictor is \(x\) (quarters), not \(\log x\).

Answer:

An exponential model is appropriate because the relationship between period and the log of subscribers is roughly linear and the residual plot shows no distinct pattern.