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 10.4.10

Chapter 10, Problem 10.4.10

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

Verified step by step guidance

Step 1: Understand the problem. The goal is to determine which two predictor variables (METAL, PAPER, GLASS) should be used to predict the response variable (PLAS) in a multiple regression model. The decision will be based on statistical measures such as correlation coefficients, p-values, or adjusted R-squared values provided in the accompanying table.

Step 2: Review the accompanying table. Look for statistical metrics that indicate the strength of the relationship between each predictor variable and the response variable. For example, check the correlation coefficients to identify which predictors are most strongly correlated with PLAS.

Step 3: Evaluate the significance of each predictor variable. If p-values are provided in the table, identify the two predictor variables with the smallest p-values, as these indicate stronger statistical significance in predicting PLAS.

Step 4: Consider the adjusted R-squared value. If the table includes adjusted R-squared values for different combinations of predictor variables, identify the combination of two predictors that results in the highest adjusted R-squared value. This indicates the best fit for the model while accounting for the number of predictors.

Step 5: Make a decision. Based on the analysis of the correlation coefficients, p-values, and adjusted R-squared values, select the two predictor variables that provide the strongest and most statistically significant relationship with PLAS. Justify your choice using the data from the table.

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Key Concepts

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

Multiple Regression Analysis

Multiple regression analysis is a statistical technique used to model the relationship between a dependent variable and two or more independent variables. It helps in understanding how the independent variables collectively influence the dependent variable, allowing for predictions based on their values. In this context, the dependent variable is the weight of discarded plastic, while the independent variables are the weights of discarded metals, paper, and glass.

Variable Selection

Variable selection is the process of identifying which independent variables should be included in a regression model to optimize its predictive power. This involves evaluating the significance and contribution of each variable to the model's performance. In the given question, selecting the two most relevant predictor variables is crucial for accurately predicting the weight of discarded plastic.

Correlation and Multicollinearity

Correlation measures the strength and direction of a linear relationship between two variables. In multiple regression, it's important to assess the correlation between predictor variables to avoid multicollinearity, which occurs when independent variables are highly correlated. This can distort the regression results and make it difficult to determine the individual effect of each predictor on the dependent variable.

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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.

Population Growth Here are the values of the world population (billions) beginning with the year 2000:

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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?

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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?

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Textbook Question

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

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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.

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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?

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