Residual Radar Worksheet
AP / Intro Statistics Linear Regression
Residual Calculations Practice
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Directions: For each word problem below, calculate the predicted value \(\hat{y}\) from the regression line, determine the residual, and state whether the actual observation lies above, below, or directly on the line. Show all work in the workspace provided.
Problem 1: Food Truck Revenue
\(\hat{y} = 14.5x + 320\)
A taco truck manager models daily revenue using the regression line \(\hat{y} = 14.5x + 320\), where \(x\) is the noon temperature in \(^\circ\text{F}\) and \(\hat{y}\) is predicted revenue in dollars. On a festival Saturday with a noon temperature of \(78^\circ\text{F}\), actual revenue reached \(\$1,515\). Calculate the residual and determine whether this point lies above, below, or on the regression line.
Problem 2: Smartphone Battery Drain
\(\hat{y} = -12.5x + 95\)
An electronics lab assesses battery performance during continuous gaming with \(\hat{y} = -12.5x + 95\), where \(x\) is active gaming time in hours and \(\hat{y}\) is predicted remaining battery percentage. A phone tested after \(4\text{ hours}\) of high-performance gaming had an actual remaining battery of \(38\%\). Calculate the residual and determine whether this point lies above, below, or on the regression line.
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Residual Calculations Practice • Page 2
Name: Period:
Problem 3: Hydroponic Seedling Growth
\(\hat{y} = 3.2x + 8.4\)
In a botany lab, students model seedling height using \(\hat{y} = 3.2x + 8.4\), where \(x\) represents nutrient solution added in milliliters (\(\text{mL}\)) and \(\hat{y}\) is predicted height in centimeters (\(\text{cm}\)). A tomato seedling administered \(5\text{ mL}\) of solution reached an actual measured height of \(24.4\text{ cm}\). Calculate the residual and determine whether this point lies above, below, or on the regression line.
Problem 4: Electric Vehicle Range
\(\hat{y} = -2.8x + 310\)
An automotive testing team models an electric vehicle's highway driving range using \(\hat{y} = -2.8x + 310\), where \(x\) is average speed in miles per hour (\(\text{mph}\)) and \(\hat{y}\) is predicted range in miles. During a test run at an average speed of \(70\text{ mph}\), the car achieved an actual range of \(122\text{ miles}\). Calculate the residual and determine whether this point lies above, below, or on the regression line.
Residual Calculations Practice • Linear Modeling Page 2 of 2 • End of Worksheet
Residual Radar Answer Key
Teacher Answer Key AP / Intro Statistics
Residual Calculations Answer Key
Solutions • Page 1 \(\text{Residual} = y - \hat{y}\)
Problem 1: Food Truck Revenue
Above the Line (+ Residual)
Model: \(\hat{y} = 14.5x + 320\), with \(x = 78^\circ\text{F}\) and observed actual revenue \(y = \$1,515\).
1. Predicted Value (\(\hat{y}\)): \(\hat{y} = 14.5(78) + 320 = \mathbf{\$1,451}\)
2. Residual (\(y - \hat{y}\)): \(\text{Residual} = 1,515 - 1,451 = \mathbf{+\$64}\)
Position: Above the regression line
Model Assessment: Underestimated actual revenue
Concise Explanation: The food truck generated $64 more revenue than projected by the regression line. A positive residual indicates that the actual observed point lies above the line of best fit (the model underestimated sales).
Problem 2: Smartphone Battery Drain
Below the Line (- Residual)
Model: \(\hat{y} = -12.5x + 95\), with \(x = 4\text{ hours}\) and observed actual battery \(y = 38\%\).
1. Predicted Value (\(\hat{y}\)): \(\hat{y} = -12.5(4) + 95 = \mathbf{45\%}\)
2. Residual (\(y - \hat{y}\)): \(\text{Residual} = 38 - 45 = \mathbf{-7\%}\)
Position: Below the regression line
Model Assessment: Overestimated battery level
Concise Explanation: The phone retained 7 percentage points less battery than anticipated. A negative residual indicates that the actual observation sits below the regression line (the model overestimated battery life).
Residual Calculations Answer Key • Linear Modeling Page 1 of 2 • Continued on next page →
Residual Calculations Answer Key • Page 2 Teacher Reference
Problem 3: Hydroponic Seedling Growth
On the Line (Zero Residual)
Model: \(\hat{y} = 3.2x + 8.4\), with \(x = 5\text{ mL}\) and observed actual height \(y = 24.4\text{ cm}\).
1. Predicted Value (\(\hat{y}\)): \(\hat{y} = 3.2(5) + 8.4 = \mathbf{24.4\text{ cm}}\)
2. Residual (\(y - \hat{y}\)): \(\text{Residual} = 24.4 - 24.4 = \mathbf{0.0\text{ cm}}\)
Concise Explanation: The seedling grew to the exact height projected by the regression line. A residual of zero demonstrates that the data point lies directly on the line of best fit (zero prediction error).
Problem 4: Electric Vehicle Range
Above the Line (+ Residual)
Model: \(\hat{y} = -2.8x + 310\), with \(x = 70\text{ mph}\) and observed actual range \(y = 122\text{ miles}\).
1. Predicted Value (\(\hat{y}\)): \(\hat{y} = -2.8(70) + 310 = \mathbf{114\text{ miles}}\)
2. Residual (\(y - \hat{y}\)): \(\text{Residual} = 122 - 114 = \mathbf{+8\text{ miles}}\)
Concise Explanation: The car traveled 8 miles farther than projected. Because the actual range exceeds the line's estimate, the residual is positive (+8 miles) and the point lies above the regression line (underestimate).
At-a-Glance Scoring & Answer Summary
| Problem |
Actual (\(y\)) |
Predicted (\(\hat{y}\)) |
Residual (\(y - \hat{y}\)) |
Line Position |
Model Error |
| 1. Food Truck |
$1,515 |
$1,451 |
+$64 |
Above |
Underestimated |
| 2. Smartphone |
38% |
45% |
-7% |
Below |
Overestimated |
| 3. Seedling |
24.4 cm |
24.4 cm |
0.0 cm |
On Line |
Exact Match |
| 4. EV Range |
122 mi |
114 mi |
+8 mi |
Above |
Underestimated |
Residual Calculations Answer Key • Linear Modeling Page 2 of 2 • End of Key
