Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.5.7

Chapter 10, Problem 10.5.7

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.
CD Yields The table lists the value y (in dollars) of \$1000 deposited in a certificate of deposit at Bank of New York (based on rates currently in effect).
$Table showing years and corresponding CD values in dollars for \$1000 deposits at Bank of New York.$

Verified step by step guidance

Step 1: Construct a scatterplot by plotting the given data points with 'Year' on the x-axis and 'Value' 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 form 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 characteristics of each model type: - Linear: $y = a + bx$ - Quadratic: $y = a + bx + cx^2$ - Exponential: $y = a \cdot b^x$ - Logarithmic: $y = a + b \ln(x)$ - Power: $y = a \cdot x^b$

Step 4: Use the data to test which model fits best by either fitting the models using regression techniques or by comparing the residuals (differences between observed and predicted values) for each model.

Step 5: Select the model with the best fit based on the scatterplot pattern and the smallest residuals, keeping in mind the model should only be used within the scope of the given data.

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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 on a coordinate plane, showing the relationship between two variables. Plotting the given years on the x-axis and corresponding CD values on the y-axis helps visualize trends and patterns, which is essential for selecting an appropriate mathematical model.

Types of Mathematical Models

Different models such as linear, quadratic, logarithmic, exponential, and power describe various types of relationships between variables. Understanding the characteristics of each model helps in fitting the data accurately, for example, linear models show constant rate changes, while exponential models show growth proportional to the current value.

Model Selection and Scope

Choosing the best model involves comparing how well each fits the data within the given range, considering residuals and overall trend. The model should be used only within the data's scope to avoid inaccurate predictions outside this range, ensuring reliable interpretation and application of the results.

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Related Practice

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

Moore’s Law In 1965, Intel cofounder Gordon Moore initiated what has since become known as Moore’s law: The number of transistors per square inch on integrated circuits will double approximately every 18 months. In the table below, the first row lists different years and the second row lists the number of transistors (in thousands) for different years.

Ignoring the listed data and assuming that Moore’s law is correct and transistors per square inch double every 18 months, which mathematical model best describes this law: linear, quadratic, logarithmic, exponential, power? What specific function describes Moore’s law?

Which mathematical model best fits the listed sample data?

Compare the results from parts (a) and (b). Does Moore’s law appear to be working reasonably well?

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Textbook Question

Large Data Sets

Exercises 29–32 use the same Appendix B data sets as Exercises 29–32 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted values following the prediction procedure summarized in Figure 10-5.

Taxis Repeat Exercise 16 using all of the distance/tip data from the 703 taxi rides listed in Data Set 32 “Taxis” from Appendix B.

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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.

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?

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