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 10.5.4

Chapter 10, Problem 10.5.4

Interpreting a Graph The accompanying graph plots the numbers of points scored in each Super Bowl from the first Super Bowl in 1967 (coded as year 1) to the last Super Bowl at the time of this writing. The graph of the quadratic equation that best fits the data is also shown in red. What feature of the graph justifies the value of R^2 = 0.205 for the quadratic model?

Verified step by step guidance

Step 1: Understand the value of R^2. The R^2 value, also known as the coefficient of determination, measures how well the quadratic model (red curve) explains the variability in the data points (blue dots). A value of R^2 = 0.205 indicates that only 20.5% of the variability in the data is explained by the quadratic model.

Step 2: Observe the graph. The blue dots represent the actual points scored in each Super Bowl, while the red curve represents the quadratic model that best fits the data. Notice that the data points are widely scattered around the red curve, indicating a weak fit.

Step 3: Analyze the scatter of data points. The large spread of the blue dots around the red curve suggests that the quadratic model does not closely follow the actual data. This lack of closeness justifies the relatively low R^2 value.

Step 4: Consider alternative models. The low R^2 value implies that the quadratic model may not be the best choice for explaining the data. A different model, such as a linear or higher-order polynomial, might provide a better fit.

Step 5: Relate the graph to the R^2 value. The feature of the graph that justifies the R^2 value is the significant deviation of the blue dots from the red curve, which shows that the quadratic model captures only a small portion of the variability in the data.

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Key Concepts

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

R-squared (R²)

R-squared, or R², is a statistical measure that represents the proportion of variance for a dependent variable that's explained by an independent variable or variables in a regression model. An R² value of 0.205 indicates that only 20.5% of the variability in Super Bowl points can be explained by the quadratic model, suggesting a weak fit. This low value implies that other factors may significantly influence the points scored, which are not captured by the model.

Quadratic Regression

Quadratic regression is a type of polynomial regression that models the relationship between a dependent variable and an independent variable using a quadratic equation (a second-degree polynomial). In the context of the Super Bowl points graph, the quadratic model attempts to capture the trend of points scored over the years, represented by the red curve. This model is useful for identifying non-linear relationships, but its effectiveness is indicated by the R² value.

Data Dispersion

Data dispersion refers to the spread of data points around a central value, often visualized in a scatter plot. In the provided graph, the blue points show considerable scatter around the red quadratic trend line, indicating variability in Super Bowl scores that the model does not account for. High dispersion can lead to a lower R² value, as it suggests that the model does not fit the data well, highlighting the complexity of predicting scores based solely on year.

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Textbook Question

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Textbook Question

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Textbook Question

Finding the Best Model

Sunspot Numbers Listed below in order by row are annual sunspot numbers beginning with 1980. Is the best model a good model? Carefully examine the scatterplot and identify the pattern of the points. Which of the models fits that pattern?

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Textbook Question

Sum of Squares Criterion In addition to the value of another measurement used to assess the quality of a model is the sum of squares of the residuals. Recall from Section 10-2 that a residual is (the difference between an observed y value and the value predicted from the model). Better models have smaller sums of squares. Refer to the U.S. population data in Table 10-7.

a. Find the sum of squares of the residuals resulting from the linear model.

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Textbook Question

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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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