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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 10.2.9
Chapter 10, Problem 10.2.9

Finding the Equation of the Regression Line
In Exercises 9 and 10, use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.




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Step 1: Understand the regression line equation. The equation of a simple linear regression line is given by y = mx + b, where m is the slope and b is the y-intercept. Our goal is to calculate these values using the given data.
Step 2: Calculate the slope (m). Use the formula m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2), where n is the number of data points, Σ(xy) is the sum of the product of x and y values, Σx is the sum of x values, Σy is the sum of y values, and Σ(x^2) is the sum of the squares of x values.
Step 3: Calculate the y-intercept (b). Use the formula b = (Σy - mΣx) / n, where m is the slope calculated in the previous step, Σy is the sum of y values, Σx is the sum of x values, and n is the number of data points.
Step 4: Write the regression line equation. Substitute the calculated values of m (slope) and b (y-intercept) into the equation y = mx + b to form the regression line equation.
Step 5: Examine the scatterplot. Look for any patterns or characteristics in the data, such as outliers, clusters, or non-linear trends, that the regression line might not capture. Note these observations as they are important for interpreting the results.

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Key Concepts

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

Regression Line

The regression line is a statistical tool used to model the relationship between two variables by fitting a linear equation to observed data. It is represented by the equation y = mx + b, where m is the slope and b is the y-intercept. This line minimizes the distance between itself and the data points, providing a predictive framework for understanding how changes in one variable affect another.
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Scatterplot

A scatterplot is a graphical representation of two quantitative variables, displaying points that correspond to the values of each variable. It helps visualize the relationship between the variables, indicating patterns, trends, or correlations. By examining a scatterplot, one can identify whether the relationship is linear, non-linear, or if there are outliers that may affect the regression analysis.
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Residuals

Residuals are the differences between the observed values and the values predicted by the regression line. They provide insight into the accuracy of the regression model; smaller residuals indicate a better fit. Analyzing residuals can reveal patterns that the regression line does not capture, such as non-linearity or the presence of outliers, which are important for understanding the limitations of the model.
Related Practice
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.

Richter Scale The table lists different amounts (metric tons) of the explosive TNT and the corresponding value measured on the Richter scale resulting from explosions of the TNT.

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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.


Oscars Listed below are ages of recent Oscar winners matched by the years in which the awards were won (from Data Set 21 “Oscar Winner Age” in Appendix B). Find the best predicted age of an Oscar-winning actress given that the Oscar winner for best actor is 59 years of age. How does the result compare to the actual actress age of 60 years?


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Textbook Question

Large Data Sets

Exercises 29–32 use the same Appendix B data sets as Exercises 29–32 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 10-5.

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.

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Textbook Question

Response and Predictor Variables Using all of the Tour de France bicycle race results up to a recent year, we get this multiple regression equation: Speed = 29.2-0.00260Distance + 0.540Stages + 0.0570Finishers, where Speed is the mean speed of the winner (km/h), Distance is the length of the race (km), Stages is the number of stages in the race, and Finishers is the number of bicyclists who finished the race. Identify the response and predictor variables.

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Textbook Question

Correlation and Slope What is the relationship between the linear correlation coefficient r and the slope b1 of a regression line?

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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.

Subway and the CPI Use the subway/CPI data from the preceding exercise. What is the best predicted value of the CPI when the subway fare is \$3.00?

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