An experiment was conducted to determine the effect of practice time (in seconds) on the percent of unfamiliar words recalled. Here is a Fathom scatterplot of the results with a least-squares regression line superimposed.

(a) Sketch a residual plot. Be sure to label your axes.

(b) Explain why a linear model is not appropriate for describing the relationship between practice time and the percent of words recalled.

(c) We used Fathom to transform the data in hopes of achieving linearity. The screenshots below show the results of two different transformations. Would an exponential model or a power model describe the relationship better? Justify your answer.

(d) Use the model you chose in part (c) to predict word recall for $25$seconds of practice. Show your method.

The residual is negative if the point falls below the blue regression line. If a point is well above the blue regression line, it has a positive value. The actual figure of the residual is the vertical distance between it and the blue line. The time values are on the horizontal axis, and the remainder are on the vertical axis, in the residual plot. A possible residual plot shown above.

Because the scatter plot has a curvature and the residual plot has an obvious pattern, it is clear that a linear model is not appropriate.

The top scatterplot plot represents a power model (both variables are present), while the bottom scatterplot figure represents an exponential model. The scatterplot shows that the model that best describes the relationship has by far the most linear pattern.

For the top scatterplot, the regression equation is:

$\ln y=3.48+0.293\ln x$

Where $x$represents time and$y$ represents a recollection.

Replacing the value of $x$with $x=25$

$$\begin{gathered} \ln y=3.48+0.293\ln x \\ =3.48+0.293\ln \left(25\right) \\ =4.42313 \end{gathered}$$

Taking the exponential

$$\begin{gathered} y=e^{4.42313} \\ =83.3568 \end{gathered}$$