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 10.4.9

Chapter 10, Problem 10.4.9

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 only one predictor (x) variable is used to predict the weight of discarded plastic, which single variable is best? Why?

Verified step by step guidance

Step 1: Understand the problem. The goal is to determine which single predictor variable (METAL, PAPER, or GLASS) is the best for predicting the response variable PLAS (weight of discarded plastic). This involves analyzing the relationship between each predictor and the response variable.

Step 2: Review the accompanying table or data. Look for statistical measures such as correlation coefficients, p-values, or regression coefficients for each predictor variable. These values indicate the strength and significance of the relationship between the predictor and the response variable.

Step 3: Identify the variable with the strongest correlation to PLAS. The predictor variable with the highest absolute value of the correlation coefficient is likely the best single predictor, as it shows the strongest linear relationship with the response variable.

Step 4: Consider statistical significance. Check the p-values associated with each predictor variable. A smaller p-value (typically less than 0.05) indicates that the predictor variable has a statistically significant relationship with the response variable.

Step 5: Conclude which variable is best. Based on the correlation coefficients and p-values, select the predictor variable that has both the strongest correlation and statistical significance. Explain why this variable is the best choice for predicting PLAS.

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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 one dependent variable and two or more independent variables. It helps in understanding how the independent variables collectively influence the dependent variable, allowing for predictions and insights into the data. In this context, it is essential for determining which predictor variable best explains the variation in the weight of discarded plastic.

Coefficient of Determination (R²)

The coefficient of determination, denoted as R², measures the proportion of variance in the dependent variable that can be explained by the independent variable(s) in a regression model. A higher R² value indicates a better fit of the model to the data, suggesting that the predictor variable has a strong relationship with the response variable. This concept is crucial for identifying which single predictor variable is most effective in predicting the weight of discarded plastic.

Correlation

Correlation quantifies the degree to which two variables are related, ranging from -1 to 1. A positive correlation indicates that as one variable increases, the other also tends to increase, while a negative correlation suggests an inverse relationship. Understanding correlation is vital in this scenario to assess which of the predictor variables (METAL, PAPER, or GLASS) has the strongest linear relationship with the weight of discarded plastic, guiding the selection of the best single predictor.

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Correlation Coefficient

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.

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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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Textbook Question

Testing for a Linear Correlation

In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Taxis Using the data from Exercise 15, is there sufficient evidence to support the claim that there is a linear correlation between the distance of the ride and the fare (cost of the ride)?

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Textbook Question

Finding a Prediction Interval

In Exercises 13–16, use the following paired data consisting of weights of large cars (pounds) and highway fuel consumption (mi/gal) from Data Set 35 “Car Data” in Appendix B. (These are the same data used in Exercises 9-12.) Let x represent the weight of the car and let y represent the corresponding highway fuel consumption. Use the given weight and the given confidence level to construct a prediction interval estimate of highway fuel consumption.

Cars Use x = 3800 pounds with a 99% confidence level.

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Textbook Question

Finding the Best Model

Deaths from Motor Vehicle Crashes Listed below are the numbers of deaths in the United States resulting from motor vehicle crashes. Use the best model to find the projected number of such deaths for the year 2025.

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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.

Cars Sales and the Super Bowl Listed below are the annual numbers of cars sold (thousands) and the numbers of points scored in the Super Bowl that same year. What is the best predicted number of Super Bowl points in a year with sales of 8423 thousand cars? How close is the predicted number to the actual result of 37 points?

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