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 10.5.10

Chapter 10, Problem 10.5.10

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.
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.

Verified step by step guidance

Step 1: Organize the data into two variables: 'Year' and 'Deaths'. Convert the years into a numerical format relative to the starting year (e.g., 1975 becomes 0, 1980 becomes 5, etc.) to simplify calculations.

Step 2: Construct a scatterplot using the 'Year' (independent variable) and 'Deaths' (dependent variable). Plot the points to visually assess the trend and relationship between the variables.

Step 3: Test different mathematical models (linear, quadratic, logarithmic, exponential, and power) by fitting each model to the data. Use statistical software or graphing tools to calculate the goodness-of-fit metrics (e.g., R-squared values) for each model.

Step 4: Identify the model with the best fit based on the highest R-squared value or other relevant criteria. Ensure the chosen model is appropriate for the scope of the data and does not overfit.

Step 5: Use the best-fit model to project the number of deaths for the year 2025. Substitute the year 2025 (converted to the numerical format relative to 1975) into the equation of the chosen model to calculate the projected value.

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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 of two variables, where each point represents an observation in the dataset. It helps visualize the relationship between the variables, allowing for the identification of patterns, trends, or correlations. In this context, plotting the years against the number of deaths from motor vehicle crashes will help determine the nature of the relationship and guide the selection of an appropriate mathematical model.

Mathematical Models

Mathematical models are equations or functions that describe the relationship between variables in a dataset. Common types include linear, quadratic, logarithmic, exponential, and power models. Each model has distinct characteristics and is suitable for different types of data trends. Choosing the best model involves analyzing the scatterplot and determining which equation best fits the observed data points.

Extrapolation

Extrapolation is the process of estimating unknown values by extending a known sequence of values or facts. In this case, once the best-fitting model is identified, it can be used to project future values, such as the number of deaths from motor vehicle crashes in 2025. However, caution is needed, as extrapolation assumes that the identified trend will continue, which may not always be the case.

Related Practice

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

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?

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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

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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Textbook Question

Interpreting a Computer Display

In Exercises 9–12, refer to the display obtained by using the paired data consisting of weights (pounds) and highway fuel consumption amounts (mi/gal) of the large cars included in Data Set 35 “Car Data” in Appendix B. Along with the paired weights and fuel consumption amounts, StatCrunch was also given the value of 4000 pounds to be used for predicting highway fuel consumption.

Finding a Prediction Interval For a car weighing 4000 pounds (x = 4000) identify the 95% prediction interval estimate of the highway fuel consumption. Write a statement interpreting that interval.

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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.

Taxis Use the distance/fare data from Exercise 15 and find the best predicted fare amount for a distance of 3.10 miles. How does the result compare to the actual fare of \$15.30?

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