students are often given a standardized math test at the beginning of the school year and again at the end…

students are often given a standardized math test at the beginning of the school year and again at the end of the school year to assess the growth they made during the school year. the scatterplot shows the average beginning - of - the - year score and the average end - of - year score for students in each grade level from kindergarten to 11th grade on a nationally administered test in a recent year.\na. based on the scatterplot, could a linear model be an appropriate fit for the data? explain.\nb. a least squares regression line is calculated, and the residual plot is shown. based on this new information, do you think the linear model is an appropriate fit for the data? explain.\nc. two other possible models for the data are shown. which model is the best fit for the data? give at least two reasons for your choice.

students are often given a standardized math test at the beginning of the school year and again at the end of the school year to assess the growth they made during the school year. the scatterplot shows the average beginning - of - the - year score and the average end - of - year score for students in each grade level from kindergarten to 11th grade on a nationally administered test in a recent year.\na. based on the scatterplot, could a linear model be an appropriate fit for the data? explain.\nb. a least squares regression line is calculated, and the residual plot is shown. based on this new information, do you think the linear model is an appropriate fit for the data? explain.\nc. two other possible models for the data are shown. which model is the best fit for the data? give at least two reasons for your choice.

Answer

Explanation:

Step1: Analyze scatter - plot for linear fit

The scatter - plot shows a somewhat linear trend. Points seem to follow a general upward - sloping pattern, which indicates that a linear model could potentially be appropriate as there is a visible linear - like relationship between the beginning and end of year scores.

Step2: Analyze residual plot

For a good linear fit, the residual plot should have randomly scattered points around the horizontal axis. In the given residual plot, the points are not randomly scattered. There are some patterns and clusters, which means the linear model is not a good fit as the residuals do not meet the assumption of randomness for a linear regression model.

Step3: Evaluate other models

Without seeing the two other models, in general, a good model has high $R^{2}$ value (close to 1), low residual standard error, and the residuals should be randomly distributed. If one model has a higher $R^{2}$ value and more randomly distributed residuals compared to the other, it is a better fit. Also, the model that more closely follows the overall pattern of the data points in the scatter - plot is preferable.

Answer:

a. Yes, a linear model could be an appropriate fit because the points in the scatter - plot show a somewhat linear trend. b. No, the linear model is not an appropriate fit because the points in the residual plot are not randomly scattered. c. Without seeing the two other models, we can say that the better - fitting model will have a higher $R^{2}$ value, lower residual standard error, and randomly distributed residuals, and will closely follow the pattern of the data points in the scatter - plot.