Evaluating the Fit of a Model â Calculate residuals and create and ...
Evaluating the Fit of a Model – Calculate residuals and create and interpret a residual plot Check for Understanding The data given below show the height (in cm) at various ages (in months) for a group of children. 1. What is the value of the residual for a child of 19 months?
2. What is the value of the residual for a child of 29 months? 3. In an AP Statistics class, the teacher wanted to show the students that humans are not good at guessing or estimating quantities. She displayed a series of clear containers with different numbers of gumballs inside and asked each student to guess the number of gumballs in one of the containers. A large number of students participated in this study. The data was then used to produce the following residual plot:
Comment on the appropriateness of a linear model for these data.
Answers 1. First, you have to perform a linear regression for the data. Using the STAT->CALC->4: LINREG function on the calculate, the equation of the least-squares regression line appears to by approximately y=64.045+0.6339x. To calculate the residual, the process is “observed value – predicted value”. In this case, the observed value at 19 months is 77.1, and the predicted value (using the equation above) is 76.9887. Therefore, the residual is -.1113. The negative sign on the residual indicates that the observed value was smaller than the predicted value. 2. Using the same steps as above, we can determine that the residual at 29 months is negative .1723. 3. Sample Response: The residual plot does not show a discernable pattern, which provides strong evidence that a linear model is an appropriate model. The residual plot also shows that as the actual number of gumballs in a container increases, the least squares regression line is a less accurate predictor due to the increasing size of the residuals.
2. What is the value of the residual for a child of 29 months? 3. In an AP Statistics class, the teacher wanted to show the students that humans are not good at guessing or estimating quantities. She displayed a series of clear containers with different numbers of gumballs inside and asked each student to guess the number of gumballs in one of the containers. A large number of students participated in this study. The data was then used to produce the following residual plot:
Comment on the appropriateness of a linear model for these data.
Answers 1. First, you have to perform a linear regression for the data. Using the STAT->CALC->4: LINREG function on the calculate, the equation of the least-squares regression line appears to by approximately y=64.045+0.6339x. To calculate the residual, the process is “observed value – predicted value”. In this case, the observed value at 19 months is 77.1, and the predicted value (using the equation above) is 76.9887. Therefore, the residual is -.1113. The negative sign on the residual indicates that the observed value was smaller than the predicted value. 2. Using the same steps as above, we can determine that the residual at 29 months is negative .1723. 3. Sample Response: The residual plot does not show a discernable pattern, which provides strong evidence that a linear model is an appropriate model. The residual plot also shows that as the actual number of gumballs in a container increases, the least squares regression line is a less accurate predictor due to the increasing size of the residuals.
