Question

the equation of the line of best fit is ( f = 1.5h - 4.3 ), where ( f ) is foot length in millimeters and ( h ) is height in centimeters.
explain the meaning of the slope in the context of the data.
is the ( y )-intercept reasonable for this context? explain.
use the model above to predict the height of a students with a foot measuring 245 millimeters.

Explanation:

1. For the slope:

• In the linear equation \(F = 1.5h−4.3\) (where \(F\) is foot - length and \(h\) is height), the slope is \(1.5\). In the context of the data (foot - length and height), the slope represents the rate of change. Since \(F\) is the dependent variable (foot - length) and \(h\) is the independent variable (height), for every 1 - centimeter increase in height (\(h\)), the foot - length (\(F\)) increases by 1.5 millimeters.

1. For the \(y\) - intercept:

• The \(y\) - intercept is \(- 4.3\) in the equation \(F = 1.5h−4.3\). When \(h = 0\) (height is 0 centimeters), \(F=-4.3\) millimeters. A height of 0 centimeters is not a valid physical height for a human (as we are talking about students, who are at least infants or children with some positive height). And a negative foot - length (\(F=-4.3\) mm) is also not physically meaningful. So, the \(y\) - intercept is not reasonable for this context.

1. For predicting the height:

• We are given \(F = 245\) mm and the equation \(F = 1.5h−4.3\).
• Substitute \(F = 245\) into the equation:
• \(245=1.5h−4.3\).
• First, add 4.3 to both sides of the equation: \(245 + 4.3=1.5h\), so \(249.3 = 1.5h\).
• Then, solve for \(h\) by dividing both sides by 1.5: \(h=\frac{249.3}{1.5}=166.2\) centimeters.

Answer:

1. The foot length increases 1.5 millimeters for every 1 centimeter of height growth.
2. The \(y\) - intercept (\(-4.3\)) is not reasonable. A height of 0 cm (when finding the \(y\) - intercept, \(h = 0\)) gives a negative foot - length (\(F=-4.3\) mm), which is not physically meaningful for students.
3. \(166.2\) centimeters.