Students use the least-squares regression line to make predictions.
When the relationship between two numerical variables and is linear, a straight line can be used to describe the relationship. Such a line can then be used to predict the value of based on the value of . When a prediction is made, the prediction error is the difference between the actual -value and the predicted -value.
The prediction error is called a residual, and the residual is calculated as residual = actual y-value – predicted y-value. The least-squares line is the line that is used to model a linear relationship. The least-squares line is the “best” line in that it has a smaller sum of squared residuals than any other line.
Example 1: Using a Line to Describe a Relationship
Kendra likes to watch crime scene investigation shows on television. She watched a show where investigators used a shoe print to help identify a suspect in a case. She questioned how possible it is to predict someone’s height is from his shoe print.
To investigate, she collected data on shoe length (in inches) and height (in inches) from 10 adult men. Her data appear in the table and scatter plot below.
1. Is there a relationship between shoe length and height?
2. How would you describe the relationship? Do the men with longer shoe lengths tend be taller?
Example 2: Using Models to Make Predictions
When two variables and are linearly related, you can use a line to describe their relationship. You can also use the equation of the line to predict the value of the y-variable based on the value of the x-variable.
For example, the line y = 25.3 + 3.66x might be used to describe the relationship between shoe length and height, where x represents shoe length and y represents height. To predict the height of a man with a shoe length of 12, you would substitute 12 in for xin the equation of the line and then calculate the value of y.
1. The scatter plot below displays the elevation and mean number of clear days per year of 14 U.S. cities. Two lines are shown on the scatter plot. Which represents the least-squares line? Explain your choice.2. Below is a scatter plot of foal birth weight and mare’s weight.
a. The equation of the least squares line for the data is: y = 19.6 + 0.248x, where = mare’s weight (in kg) and = foal’s birth weight (in kg). What foal birth weight would you predict for a mare who weighs 520 kg?
b. How would you interpret the value of the slope in the least-squares line?
c. Does it make sense to interpret the value of the -intercept in this context? Explain why or why not.
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