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 10.5.13

Chapter 10, Problem 10.5.13

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.
Stock Market Listed below in order by row are the annual high values of the Dow Jones Industrial Average for each year beginning with 2000. Find the best model and then predict the value for the last year listed. Is the predicted value close to the actual value of 26,828.4?

Verified step by step guidance

Step 1: Organize the data by assigning each year a corresponding x-value starting from 1 for the year 2000, 2 for 2001, and so on, and the annual high values as y-values. This will help in plotting the scatterplot.

Step 2: Construct a scatterplot by plotting the years (x-values) on the horizontal axis and the Dow Jones annual high values (y-values) on the vertical axis to visually inspect the pattern or trend in the data.

Step 3: Consider the five types of models: linear, quadratic, logarithmic, exponential, and power. For each model, fit the data using appropriate regression techniques or software to find the best-fitting curve.

Step 4: Compare the goodness of fit for each model using criteria such as the coefficient of determination (R-squared) or residual plots to identify which model best captures the trend in the data within the given range.

Step 5: Use the best-fitting model to predict the Dow Jones value for the last year listed (corresponding to the x-value for that year). Then, compare this predicted value to the actual value of 26,828.4 to assess the accuracy of the model.

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Key Concepts

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

Scatterplot Construction

A scatterplot is a graphical representation of data points plotted 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.

Model Selection and Types

Choosing the best model involves comparing different types such as linear, quadratic, logarithmic, exponential, and power models. Each model describes data behavior differently, and the best fit minimizes errors and accurately represents the data within the given scope.

Prediction and Model Validation

After fitting a model, it is used to predict values within the data range. Comparing predicted values to actual data, like the Dow Jones value of 26,828.4, validates the model's accuracy and reliability for forecasting.

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Prediction Intervals

Related Practice

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.

Dirt Cheap The Cherry Hill Construction company in Branford, CT sells screened topsoil by the “yard,” which is actually a cubic yard. Let the variable x be the length (yd) of each side of a cube of screened topsoil. The table below lists the values of x along with the corresponding cost (dollars).

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Textbook Question

Making Predictions

In Exercises 5–8, let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 10-5. Use a 0.05 significance level.

Bear Measurements Head widths (in.) and weights (lb) were measured for 20 randomly selected bears (from Data Set 18 “Bear Measurements” in Appendix B). The 20 pairs of measurements yield xbar = 6.9 in., ybar = 214.3 lb, r = 0.879 P-value = 0.000 and y^ = -212 + 61.9x. Find the best predicted weight of a bear given that the bear has a head width of 6.5 in.

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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 The table below includes data from New York City taxi rides (from Data Set 32 “Taxis” in Appendix B). The distances are in miles, the times are in minutes, the fares are in dollars, and the tips are in dollars. Is there sufficient evidence to support the claim that there is a linear correlation between the time of the ride and the tip amount? Does it appear that riders base their tips on the time of the ride?

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Textbook Question

Super Bowl and R^2 Let x represent years coded as 1,1,3,... for years starting in 1980, and let y represent the numbers of points scored in each annual Super Bowl beginning in 1980. Using the data from 1980 to the last Super Bowl at the time of this writing, we obtain the following values of R^2 for the different models: linear: 0.008; quadratic: 0.023; logarithmic: 0.0004; exponential: 0.027; power: 0.007. Based on these results, which model is best? Is the best model a good model? What do the results suggest about predicting the number of points scored in a future Super Bowl game?

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Textbook Question

Testing for a Linear Correlation

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 tip amount? Does it appear that riders base their tips on the distance of the ride?

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Textbook Question

Variation and Prediction Intervals

In Exercises 17–20, find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions.

Weighing Seals with a Camera The table below lists overhead widths (cm) of seals measured from photographs and the weights (kg) of the seals (based on “Mass Estimation of Weddell Seals Using Techniques of Photogrammetry,” by R. Garrott of Montana State University). For the prediction interval, use a 99% confidence level with an overhead width of 9.0 cm.

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