Question

In a precalculus class, the students measured each student's height and arm span in centimeters. They calculated a linear regression with heights as the input values and arm spans as the output values. The given residual plot has a point labeled at coordinates ____. What does this point indicate in the context?

1) The point represents an outlier in the data.
2) The point represents a predicted arm span value for a given height.
3) The point represents the average arm span value for a given height.
4) The point represents the maximum arm span value for a given height.

Answer 1

A point on a residual plot represents how much the actual arm span of a student deviates from the arm span predicted by the regression line for a given height. Without more context, we cannot determine if the point is an outlier, as it requires the point to be more than two standard deviations away from the predicted value.

Step-by-step explanation:

When interpreting a point on a residual plot in the context of measuring student heights and arm spans, a point at given coordinates represents the difference between the actual arm span and the value predicted by the regression line for a given height. Specifically, this point on the residual plot does not represent the arm span itself (predicted, average, maximum, or outlier). Therefore, the correct interpretation of this point is:

The point represents how much the actual arm span of a student deviates from the arm span predicted by the regression line given their height.

Identifying Outliers

In regression analysis, an outlier can significantly affect the slope and correlation coefficient of the regression line. To identify outliers, we can use a rule of thumb that flags any point more than two standard deviations of the residuals from its predicted value as an outlier. If a coordinate is mentioned on the residual plot that fits this criterion, then it would represent an outlier in the data. However, without specific values or a context indicating that this point is beyond two standard deviations, we cannot immediately conclude it is an outlier from the statement alone.

Answer 2

A point on the residual plot indicates the residual for a particular height and arm span measurement, with a significant deviation from zero suggesting that the point may be an outlier in the data set. It does not reflect predicted, average, or maximum values.

Step-by-step explanation:

The coordinates given for a point on the residual plot in the context of a linear regression analysis represent the residual for a specific data point. Residuals measure the difference between the actual value of the dependent variable (in this case, arm span) and the value predicted by the regression line for the given independent variable value (height). Therefore:

• 1) If a point on the residual plot is labeled with coordinates, and these coordinates show a significant deviation from zero, this may indicate that the point represents an outlier in the data.
• 2) A point on the residual plot does not represent a predicted arm span value for a given height.
• 3) Neither does it represent the average arm span value for a given height.
• 4) Nor does it represent the maximum arm span value for a given height.

A residual plot is used to assess the fitting quality of the regression model and to identify outliers. If a point lies a significant distance from the horizontal line at zero (usually more than two standard deviations), it indicates that the actual value is significantly different from the predicted value, suggesting that the point may be an outlier.