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 10.5.11

Chapter 10, Problem 10.5.11

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.
Richter Scale The table lists different amounts (metric tons) of the explosive TNT and the corresponding value measured on the Richter scale resulting from explosions of the TNT.

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Step 1: Construct a scatterplot by plotting the given data points with TNT amounts on the x-axis and Richter scale values on the y-axis. This visual representation will help identify the pattern or trend in the data.

Step 2: Observe the scatterplot to determine the general shape of the data points. Check if the points appear to follow a straight line (linear), a curve that opens up or down (quadratic), a curve that increases or decreases rapidly (exponential), a curve that increases slowly (logarithmic), or a curve that follows a power relationship.

Step 3: Consider the nature of the Richter scale and TNT amounts. Since the Richter scale is logarithmic by definition, suspect that a logarithmic or power model might fit well, but verify this by comparing the scatterplot shape to the typical shapes of each model.

Step 4: Fit each candidate model (linear, quadratic, logarithmic, exponential, power) to the data using regression techniques or software, and calculate the goodness of fit measures such as R-squared values to quantify how well each model fits the data.

Step 5: Compare the goodness of fit values and the scatterplot fit for each model, then select the model that best captures the trend in the data within the given range, keeping in mind the context and the expected behavior of the Richter scale.

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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 on the plot corresponds to a pair of values from the data set, helping to visualize patterns, trends, or correlations. It is essential for identifying the type of relationship before choosing a mathematical model.

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

Mathematical models describe relationships between variables using specific functions. Linear models show constant rate changes, quadratic models show parabolic trends, logarithmic models grow quickly then level off, exponential models grow or decay rapidly, and power models follow a polynomial relationship. Selecting the best model depends on how well it fits the data pattern.

Model Fitting and Scope of Data

Model fitting involves choosing a mathematical model that best represents the data within the observed range. It is important to use the model only within the scope of the given data to avoid inaccurate predictions. Evaluating residuals and goodness-of-fit measures helps determine the most appropriate model.

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Guided course

10:17

Goodness of Fit Test

Related Practice

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.

Powerball Jackpots and Tickets Sold Listed below are the same data from Table 10-1 in the Chapter Problem, but an additional pair of values has been added from actual Powerball results. (Jackpot amounts are in millions of dollars, ticket sales are in millions.) Find the best predicted number of tickets sold when the jackpot was actually 345 million dollars. How does the result compare to the value of 55 million tickets that were actually sold?

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.

Subway and the CPI Use the subway/CPI data from the preceding exercise. What is the best predicted value of the CPI when the subway fare is \$3.00?

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Textbook Question

Finding the Equation of the Regression Line

In Exercises 9 and 10, use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.

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Textbook Question

Randomization

For Exercises 33–36, repeat the indicated exercise using the resampling method of randomization.

Powerball Jackpots and Tickets Sold Exercise 14

Textbook Question

Coefficient of Determination Using the heights and weights described in Exercise 1, the linear correlation coefficient r is 0.394. Find the value of the coefficient of determination. What practical information does the coefficient of determination provide?

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Textbook Question

Standard Error of Estimate A random sample of 118 different female statistics students is obtained and their weights are measured in kilograms and in pounds. Using the 118 paired weights (weight in kg, weight in lb), what is the value of se? For a female statistics student who weighs 100 lb, the predicted weight in kilograms is 45.4 kg. What is the 95% prediction interval?