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Ch. 10 - Correlation and Regression
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 10 - Correlation and RegressionProblema 10.5.16
Capitolo 10, Problema 10.5.16

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.
Global Warming Listed below are mean annual temperatures (°C) of the earth for each decade, beginning with the decade of the 1880s. Find the best model and then predict the value for 2090–2099. Comment on the result.
Table showing mean annual Earth temperatures (°C) by decade from 1880s to 1970s for model fitting.

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Step 1: Organize the data by pairing each decade with its corresponding mean annual temperature. For example, assign x-values as 1880, 1890, ..., 1970 and y-values as the temperatures given: 13.819, 13.692, ..., 14.636.
Step 2: Construct a scatterplot by plotting the decades on the x-axis and the mean annual temperatures on the y-axis. This visual will help identify the trend and the type of model that fits best.
Step 3: Consider the five types of models: linear, quadratic, logarithmic, exponential, and power. Fit each model to the data using appropriate regression techniques or software to find the best fit.
Step 4: Compare the goodness of fit for each model, such as by looking at the coefficient of determination (\[R^2\]) or residual plots, to determine which model best represents the data within the given range.
Step 5: Use the best-fitting model to predict the mean annual temperature for the decade 2090–2099 by substituting the corresponding x-value (e.g., 2090) into the model's equation. Then, interpret the prediction in the context of global warming.

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Scatterplot Construction

A scatterplot is a graphical representation of data points on a coordinate plane, showing the relationship between two variables. It helps visualize patterns, trends, or correlations, which is essential for selecting an appropriate mathematical model.
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Scatterplots & Intro to Correlation

Model Types and Selection

Different models like linear, quadratic, logarithmic, exponential, and power describe various types of relationships between variables. Choosing the best model involves comparing how well each fits the data, often using visual fit and statistical measures like residuals or R-squared.
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Types of Data

Prediction and Extrapolation

Once a model is selected, it can be used to predict values within the data range or slightly beyond (extrapolation). Predictions should be interpreted cautiously, especially when extending far beyond the observed data, as model accuracy may decrease.
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Prediction Intervals
Pratica correlata
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Notation What is the difference between the regression equation y^ = b0 + b1x and the regression equation y = β0 + β1x.

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Interpreting the Coefficient of Determination

In Exercises 5–8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Times of Taxi Rides and Fares r = 0.953 (x = time in minutes, y = fare in dollars)

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Dummy Variable Refer to Data Set 18 “Bear Measurements” in Appendix B and use the sex, age, and weight of the bears. For sex, let 0 represent female and let 1 represent male. Letting the response variable represent weight, use the variable of age and the dummy variable of sex to find the multiple regression equation. Use the equation to find the predicted weight of a bear with the characteristics given below. Does sex appear to have much of an effect on the weight of a bear?


Female bear that is 20 years of age

Male bear that is 20 years of age

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Interpreting r

In Exercises 5–8, use a significance level of α = 0.05 and refer to the accompanying displays.

Bear Length and Weight The lengths (inches) and weights (pounds) of 54 bears are obtained from Data Set 18 “Bear Measurements” in Appendix B, and results are shown in the accompanying XLSTAT display. Is there sufficient evidence to support the claim that there is a linear correlation between length and weight?

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