4. Probability
Basic Concepts of Probability
1:53 minutesProblem 10.3.11
Textbook Question
Interpreting a Computer Display
In Exercises 9–12, refer to the display obtained by using the paired data consisting of weights (pounds) and highway fuel consumption amounts (mi/gal) of the large cars included in Data Set 35 “Car Data” in Appendix B. Along with the paired weights and fuel consumption amounts, StatCrunch was also given the value of 4000 pounds to be used for predicting highway fuel consumption.
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Predicting Highway Fuel Consumption Using a car weight of x = 4000 (pounds), what is the single value that is the best predicted amount of highway fuel consumption?
Verified step by step guidance1
Identify the regression equation provided in the computer display. The regression equation typically has the form y = b0 + b1 * x, where y is the dependent variable (highway fuel consumption), x is the independent variable (car weight), b0 is the y-intercept, and b1 is the slope.
Substitute the given value of x = 4000 (pounds) into the regression equation. This means replacing x in the equation with 4000.
Simplify the equation by performing the multiplication and addition operations to calculate the predicted value of y (highway fuel consumption).
Interpret the result as the best predicted amount of highway fuel consumption for a car weight of 4000 pounds, based on the regression model.
Verify that the prediction falls within the range of observed data to ensure it is reasonable and consistent with the model's assumptions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Regression
Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In this context, highway fuel consumption is the dependent variable, while car weight is the independent variable. The goal is to find the best-fitting line that predicts fuel consumption based on weight, allowing for predictions at specific values, such as 4000 pounds.
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Prediction Equation
The prediction equation in linear regression is derived from the regression line, typically expressed in the form y = mx + b, where y is the predicted value, m is the slope, x is the independent variable, and b is the y-intercept. This equation allows us to input a specific weight (e.g., 4000 pounds) to calculate the expected highway fuel consumption, providing a single predicted value.
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Paired Data
Paired data refers to two related sets of observations, in this case, weights of cars and their corresponding fuel consumption. Each pair consists of a weight and its associated fuel consumption, allowing for analysis of the relationship between the two variables. Understanding paired data is crucial for interpreting the results of regression analysis and making accurate predictions.
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