Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.2.25

Chapter 10, Problem 10.2.25

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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Verified step by step guidance

Step 1: Identify the variables. The problem specifies two variables: the annual number of cars sold (in thousands) as the predictor variable (x) and the number of points scored in the Super Bowl as the response variable (y).

Step 2: Calculate the regression equation. Use the formula for the least-squares regression line: y = b₀ + b₁x, where b₁ (the slope) is calculated as b₁ = Σ((xᵢ - x̄)(yᵢ - ȳ)) / Σ((xᵢ - x̄)²), and b₀ (the y-intercept) is calculated as b₀ = ȳ - b₁x̄. Here, x̄ and ȳ are the means of the x and y values, respectively.

Step 3: Plug in the given data to compute b₁ and b₀. Use the provided data set of car sales and Super Bowl points to calculate the necessary summations (Σ(xᵢ), Σ(yᵢ), Σ(xᵢyᵢ), Σ(xᵢ²)) and substitute them into the formulas for b₁ and b₀.

Step 4: Use the regression equation to predict the number of Super Bowl points for a year with 8423 thousand cars sold. Substitute x = 8423 into the regression equation y = b₀ + b₁x to calculate the predicted value of y.

Step 5: Compare the predicted value to the actual result of 37 points. Calculate the difference between the predicted value and the actual value to assess how close the prediction is to the observed result.

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Key Concepts

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

Regression Analysis

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In this context, it helps to determine how the number of cars sold (independent variable) can predict the points scored in the Super Bowl (dependent variable). The result is a regression equation that can be used for making predictions based on new data.

Predictor and Response Variables

In regression analysis, the predictor variable (also known as the independent variable) is the one used to predict the value of another variable, called the response variable (dependent variable). In this case, car sales serve as the predictor variable, while Super Bowl points are the response variable. Understanding the roles of these variables is crucial for interpreting the regression results accurately.

Prediction Procedure

The prediction procedure involves using the regression equation to estimate the value of the response variable for a given value of the predictor variable. This typically includes substituting the predictor value into the regression equation to calculate the predicted response. In this scenario, the procedure will help determine the expected number of Super Bowl points based on car sales of 8423 thousand.

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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

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.

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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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Textbook Question

Appendix B Data Sets

In Exercises 29–32, use the data from Appendix B to construct a scatterplot, find the value of the linear correlation coefficient r, and find either the P-value or the critical values of r from Table A-6 using a significance level of α = 0.05. Determine whether there is sufficient evidence to support the claim of a linear correlation between the two variables.

Taxis Repeat Exercise 15 using all of the time/tip data from the 703 taxi rides listed in Data Set 32 “Taxis” from Appendix B. Compare the results to those found in Exercise 15.

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