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Ch. 10 - Correlation and Regression
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Triola 14th EditionCh. 10 - Correlation and Regression문제 10.2.4
10장, 문제 10.2.4

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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The linear correlation coefficient, denoted as r, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where values close to -1 or 1 indicate a strong linear relationship, and values near 0 indicate a weak or no linear relationship.
The slope of the regression line, denoted as b₁, represents the rate of change in the dependent variable (y) for a one-unit increase in the independent variable (x). It is calculated as b₁ = r * (sy / sx), where sy and sx are the standard deviations of y and x, respectively.
The relationship between r and b₁ is that the sign of r (positive or negative) determines the direction of the slope b₁. If r is positive, b₁ will also be positive, indicating an upward-sloping line. If r is negative, b₁ will be negative, indicating a downward-sloping line.
The magnitude of r affects the strength of the linear relationship but does not directly determine the value of b₁. The value of b₁ also depends on the variability (standard deviations) of the variables x and y.
In summary, while r and b₁ are related through the formula b₁ = r * (sy / sx), they describe different aspects of the relationship: r quantifies the strength and direction of the correlation, while b₁ quantifies the rate of change in y with respect to x.

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Linear Correlation Coefficient (r)

The linear correlation coefficient, denoted as r, measures the strength and direction of a linear relationship between two variables. Its value ranges from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation. Understanding r is crucial for interpreting how closely two variables are related.
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Correlation Coefficient

Slope of a Regression Line (b1)

The slope of a regression line, represented as b1, quantifies the change in the dependent variable for each unit change in the independent variable. A positive slope indicates that as the independent variable increases, the dependent variable also increases, while a negative slope indicates the opposite. The slope is a key component in understanding the nature of the relationship modeled by the regression.
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가이드 코스
05:43
Correlation Coefficient

Relationship Between r and b1

The relationship between the linear correlation coefficient r and the slope b1 is significant in regression analysis. Specifically, when both variables are standardized, the slope b1 is equal to the correlation coefficient r. This means that a strong correlation (high absolute value of r) typically corresponds to a steep slope, indicating a strong linear relationship between the variables.
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Probabilities Between Two Values
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교과서 질문

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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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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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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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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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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Coefficient of Determination Using the heights and weights described in Exercise 1, the linear correlation coefficient r is 0.394. Find the value of the coefficient of determination. What practical information does the coefficient of determination provide?

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