1. Intro to Stats and Collecting Data
Intro to Stats
2:38 minutesProblem 10.4.9
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).
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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 guidance1
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.
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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.
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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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