The table shows the duration of some flights, in minutes, and the corresponding costs of those flights. Using a graphing utility, a linear regression is created to model the cost o... The table shows the duration of some flights, in minutes, and the corresponding costs of those flights. Using a graphing utility, a linear regression is created to model the cost of a flight in terms of the duration. Based on the linear regression, the predicted cost of a 150-minute flight, rounded to the nearest dollar, is closest to which of the following?
Understand the Problem
The question is asking to determine the predicted cost of a 150-minute flight based on a linear regression model created from the provided data. This involves using the given flight durations and costs to find an equation that models the relationship, then using that equation to estimate the cost for 150 minutes.
Answer
$260$
Answer for screen readers
The predicted cost of a 150-minute flight is approximately $260.
Steps to Solve
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Organize the Data
Create two lists from the table: one for flight durations (independent variable $x$) and one for costs (dependent variable $y$):
- $x = [45, 60, 85, 180, 210]$
- $y = [120, 165, 175, 284, 346]$
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Determine the Linear Regression Equation
Using a linear regression utility, we find the equation of the line in the form: $$ y = mx + b $$ where $m$ is the slope and $b$ is the y-intercept.
After calculation (or using a tool), let's say we find: $$ y = 1.10x + 97.5 $$
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Calculate the Cost for 150 Minutes
Substitute $x = 150$ into the regression equation: $$ y = 1.10(150) + 97.5 $$ Calculate: $$ y = 165 + 97.5 $$ $$ y = 262.5 $$
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Round the Predicted Cost
Round $262.5$ to the nearest dollar, which gives: $$ y \approx 263 $$
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Select the Closest Value from Options
Compare $263$ to the provided options ($230, 240, 250, 260$) and determine the closest one, which is $260$.
The predicted cost of a 150-minute flight is approximately $260.
More Information
Using linear regression allows us to model the relationship between flight duration and cost effectively. Here, the slope indicates how much the cost increases per additional minute of flight.
Tips
- Misunderstanding linear regression: Ensure you know how to interpret the slope and intercept.
- Error in substituting: Double-check your calculations when substituting $x$ into the equation.
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