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Ch. 10 - Correlation and Regression
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 10 - Correlation and RegressionProblem 10.2.3
Chapter 10, Problem 10.2.3

Best-Fit Line


What is a residual?
In what sense is the regression line the straight line that “best” fits the points in a scatterplot?

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A residual is the difference between the observed value of the dependent variable (y) and the predicted value (ŷ) from the regression line. Mathematically, it is expressed as: e=y-ŷ.
The regression line is considered the 'best fit' because it minimizes the sum of the squared residuals. This is known as the 'least squares criterion,' which ensures that the total squared differences between observed and predicted values are as small as possible.
To calculate the regression line, the slope (m) and intercept (b) are determined using formulas derived from the least squares method. The line is represented as: ŷ=mx+b.
The slope (m) indicates the rate of change of the dependent variable with respect to the independent variable, while the intercept (b) represents the predicted value of the dependent variable when the independent variable is zero.
The regression line is optimal in the sense that it provides the best linear approximation of the relationship between the variables, reducing prediction errors and providing a clear summary of the trend in the data.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Residuals

A residual is the difference between the observed value of a dependent variable and the value predicted by a regression model. It quantifies the error in the prediction for each data point, indicating how far off the model's predictions are from the actual data. Residuals are crucial for assessing the accuracy of a regression model and can be analyzed to identify patterns or potential issues in the model.

Best-Fit Line

The best-fit line, or regression line, is the straight line that minimizes the sum of the squared residuals in a scatterplot. This line represents the relationship between the independent and dependent variables, providing the most accurate predictions based on the available data. The method of least squares is commonly used to determine the slope and intercept of this line, ensuring it best captures the trend of the data points.
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Scatterplot

A scatterplot is a graphical representation of two variables, where each point represents an observation in the dataset. It allows for visual assessment of the relationship between the variables, helping to identify trends, correlations, or outliers. The arrangement of points in a scatterplot can indicate whether a linear model is appropriate for the data, guiding the selection of the best-fit line.
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Related Practice
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.

Finding a Prediction Interval For a car weighing 4000 pounds (x = 4000) identify the 95% prediction interval estimate of the highway fuel consumption. Write a statement interpreting that interval.

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Textbook Question

Regression and Predictions

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1.


Find the regression equation, letting the first variable be the predictor (x) variable.

Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5.


Taxis Use the distance/fare data from Exercise 15 and find the best predicted fare amount for a distance of 3.10 miles. How does the result compare to the actual fare of \$15.30?

Textbook Question

Appendix B Data Sets

In Exercises 29–32, use the data from Appendix B to construct a scatterplot, find the value of the linear correlation coefficient r, and find either the P-value or the critical values of r from Table A-6 using a significance level of α = 0.05. Determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.

Taxis Repeat Exercise 15 using all of the time/tip data from the 703 taxi rides listed in Data Set 32 “Taxis” from Appendix B. Compare the results to those found in Exercise 15.

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.



Testing for Correlation Use the information provided in the display to determine the value of the linear correlation coefficient. Is there sufficient evidence to support a claim of a linear correlation between weights of large cars and the highway fuel consumption amounts?

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?

Textbook Question

Finding the Best Model

In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.

Sound Intensity The table lists intensities of sounds as multiples of a basic reference sound. A scale similar to the decibel scale is used to measure the sound intensity.