Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.4.4

Chapter 10, Problem 10.4.4

Interpreting R^2 For the multiple regression equation given in Exercise 1, we get R^2 = 0.897. What does that value tell us?

Verified step by step guidance

Understand the concept of R-squared (R^2): R-squared is a statistical measure that represents the proportion of the variance for a dependent variable that's explained by an independent variable or variables in a regression model.

Recognize that R^2 values range from 0 to 1: An R^2 value of 0 indicates that the model explains none of the variability of the response data around its mean, while an R^2 of 1 indicates that the model explains all the variability.

Interpret the given R^2 value: In this problem, R^2 = 0.897, which means that 89.7% of the variability in the dependent variable can be explained by the independent variables in the model.

Consider the context of the data: A high R^2 value like 0.897 suggests a strong relationship between the dependent and independent variables, indicating that the model fits the data well.

Be cautious of overfitting: While a high R^2 is generally desirable, it's important to ensure that the model is not too complex and overfitting the data, which can lead to poor predictive performance on new data.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

R-squared (R^2)

R-squared, or the coefficient of determination, measures the proportion of variance in the dependent variable that is predictable from the independent variables in a regression model. An R^2 value of 0.897 indicates that 89.7% of the variability in the dependent variable can be explained by the model, suggesting a strong fit.

Multiple Regression

Multiple regression is a statistical technique that models the relationship between a dependent variable and two or more independent variables. It extends simple linear regression by allowing for the analysis of the impact of multiple factors simultaneously, providing a more comprehensive understanding of the relationships within the data.

Interpreting R-squared in Context

Interpreting R-squared involves understanding its implications for the model's explanatory power. An R^2 of 0.897 suggests that the model is highly effective in explaining the variability of the outcome variable. However, it is crucial to consider other factors such as the potential for overfitting and the significance of individual predictors to ensure the model's reliability.

Recommended video:

Guided course

01:35

Step 1: Write Hypotheses Example 1

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.

Landing on the Moon When the Apollo spacecraft landed on the Moon, the rocket engine would typically cut off at about 1.3 meters above the surface so that hot gases and dust and other surface materials would not cause damage. The landing module was in freefall starting at about 1 meter above the surface. The table below lists the time t (seconds) after being dropped and the distance d (meters) travelled by an object dropped near the surface of the Moon.

views

Textbook Question

Testing Hypotheses About Regression Coefficients If the coefficient has a nonzero value, then it is helpful in predicting the value of the response variable. If it is not helpful in predicting the value of the response variable and can be eliminated from the regression equation. To test the claim that use the test statistic Critical values or P-values can be found using the t distribution with degrees of freedom, where k is the number of predictor variables and n is the number of observations in the sample. The standard error is often provided by software. For example, see the accompanying StatCrunch display for Example 1, which shows that (found in the column with the heading of “Std. Err.” and the row corresponding to the first predictor variable of height). Use the sample data in Data Set 1 “Body Data” and the StatCrunch display to test the claim that Also test the claim that What do the results imply about the regression equation?

" style="" width="600">

198

views

Textbook Question

Finding the Best Model

Earthquakes Listed below are earthquake depths (km) and magnitudes (Richter scale) of different earthquakes. Find the best model and then predict the magnitude for the last earthquake with a depth of 3.78 km. Is the predicted value close to the actual magnitude of 7.1?

views

Textbook Question

Garbage: Finding the Best Multiple Regression Equation

In Exercises 9–12, refer to the accompanying table, which was obtained by using the data from 62 households listed in Data Set 42 “Garbage Weight” in Appendix B. The response (y) variable is PLAS (weight of discarded plastic in pounds). The predictor (x) variables are METAL (weight of discarded metals in pounds), PAPER (weight of discarded paper in pounds), and GLASS (weight of discarded glass in pounds).

[IMAGE]

If exactly two predictor (x) variables are to be used to predict the weight of discarded plastic, which two variables should be chosen? Why?

172

views

Textbook Question

se Notation Using Data Set 1 “Body Data” in Appendix B, if we let the predictor variable x represent heights of males and let the response variable y represent weights of males, the sample of 153 heights and weights results in se = 16.27555 cm. In your own words, describe what that value of se represents.

190

views

Textbook Question

Interpreting r

In Exercises 5–8, use a significance level of α = 0.05 and refer to the accompanying displays.

Bear Weight and Chest Size Fifty-four wild bears were anesthetized, and then their weights and chest sizes were measured and listed in Data Set 18 “Bear Measurements” in Appendix B; results are shown in the accompanying Statdisk display. Is there sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes? When measuring an anesthetized bear, is it easier to measure chest size than weight? If so, does it appear that a measured chest size can be used to predict the weight?

169

views