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 10.5.15

Chapter 10, Problem 10.5.15

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.
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?

Verified step by step guidance

Step 1: Construct a scatterplot by plotting the given earthquake depths (x-axis) against their corresponding magnitudes (y-axis). This visual representation helps to observe the relationship between depth and magnitude.

Step 2: Analyze the scatterplot to identify the pattern or trend. Check if the points suggest a linear, quadratic (parabolic), logarithmic, exponential, or power relationship between depth and magnitude.

Step 3: Fit each type of model (linear, quadratic, logarithmic, exponential, power) to the data using appropriate regression techniques. For example, use linear regression for a linear model, polynomial regression for quadratic, and transformations for logarithmic, exponential, and power models.

Step 4: Compare the goodness of fit for each model using statistical measures such as the coefficient of determination (\[R^2\]) or residual analysis to determine which model best describes the data within the given range.

Step 5: Using the best-fitting model, predict the magnitude for the earthquake with a depth of 3.78 km by substituting this value into the model's equation. Then, compare the predicted magnitude to the actual magnitude of 7.1 to assess the model's accuracy.

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Key Concepts

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

Scatterplot

A scatterplot is a graphical representation that displays the relationship between two quantitative variables. Each point represents a pair of values, helping to visualize patterns, trends, or correlations. It is essential for identifying the type of relationship (linear, quadratic, etc.) between earthquake depth and magnitude.

Mathematical Models (Linear, Quadratic, Logarithmic, Exponential, Power)

Mathematical models describe relationships between variables using specific equations. Linear models show a straight-line relationship, quadratic models form parabolas, logarithmic models involve log functions, exponential models show rapid growth or decay, and power models use variables raised to a power. Choosing the best model depends on how well it fits the data pattern.

Prediction and Model Evaluation

Prediction uses the chosen model to estimate unknown values, such as the magnitude for a given earthquake depth. Model evaluation involves comparing predicted values to actual data to assess accuracy. This step ensures the model is reliable within the data scope and helps determine if the prediction for depth 3.78 km is close to the actual magnitude 7.1.

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Related Practice

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

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

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?

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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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